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GraphData
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- The specified entity in GraphData can be an Entity, a list of entities or an entity canonical name such as such as "PetersenGraph" or "FosterCage".
- GraphData[entity] gives a Graph object.
- The specified property can be an EntityProperty, a property canonical name or a list of properties.
- GraphData[patt] gives a list of all graph standard names that match the string pattern patt.
- GraphData[] gives a list of all standard named graphs.
- GraphData[All] gives all available graphs.
- GraphData[;;n] gives a list of available standard named graphs with ≤n vertices.
- GraphData[m;;n] gives a list of available standard named graphs with m through n vertices.
- GraphData[class,nspec] gives a list of available graphs with nspec vertices in the specified class.
- GraphData["Classes"] gives a list of all supported classes.
- GraphData["Properties"] gives a list of properties available for graphs.
- GraphData[{n,i},…] gives data for the i
simple graph with n vertices. - GraphData[{"type",id},…] gives data for the graph of the specified type with identifier id. The type is typically a string and the identifier is typically an integer or lists of integers.
- Basic graph properties include:
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"AdjacencyMatrix" adjacency matrix "EdgeCount" total number of edges "Edges" edges "FaceCount" total number of faces (for a planar graph) "Faces" faces "IncidenceMatrix" incidence matrix "VertexCount" total number of vertices "Vertices" vertices - Properties related to graph connectivity include:
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"AdjacencyLists" adjacency lists "ArticulationVertexCount" number of articulation vertices "ArticulationVertices" list of vertices whose removal would disconnect the graph "BlockCount" number of blocks "Blocks" minimal 2-connected components "BridgeCount" number of bridges "Bridges" list of edges whose removal would disconnect the graph "Connected" connected "ConnectedComponentCount" number of connected components "ConnectedInducedSubgraphCount" connected induced subgraph count "ConnectedComponents" connected components "CyclicEdgeConnectivity" cyclic edge connectivity "Disconnected" disconnected "EdgeConnectivity" minimum edge deletions to disconnect the graph "EdgeCutCount" number of edge cuts "EdgeCuts" edge cuts "IncidenceLines" indices of collinear vertices in a configuration graph "LambdaComponents" lambda components "LuccioSamiComponents" Luccio–Sami components "MinimalEdgeCutCount" number of minimal edge cuts "MinimalEdgeCuts" minimal edge cuts "MinimalVertexCutCount" number of minimal vertex cuts "MinimalVertexCuts" minimal vertex cuts "MinimumCyclicEdgeCutCount" minimum cyclic edge cut count "MinimumCyclicEdgeCuts" minimum cyclic edge cuts "MinimumEdgeCutCount" number of minimum edge cuts "MinimumEdgeCut" minimum edge cuts "MinimumVertexCutCount" minimum vertex cut count "MinimumVertexCuts" minimum vertex cuts "MinorCount" graph minor count "SpanningTrees" spanning trees "Strength" graph strength "Toughness" graph toughness "Triangulated" triangulated (maximally planar) "VertexConnectivity" minimum vertex deletions to disconnect the graph - Properties related to graph minors include:
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"HadwigerNumber" Hadwiger number "MinorCount" graph minor count - Properties related to graph display include:
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"EmbeddingClasses" list of embedding class tags, one to an embedding "Embeddings" vertex coordinates for all available layouts "Graph" graph object "Graphics" graphic "Graphics3D" 3D graphics "Image" image "MeshRegion" mesh region "Polyhedron" polyhedron "VertexCoordinates" vertex coordinates for the default layout - Annotatable properties related to list-type output include:
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"AdjacencyMatrix","outputtype" one or all adjacency matrices or count of all possible adjacency matrices "ConnectedComponents","outputtype" connected components as a graph, list, count, list of edge counts, or list of vertex counts "Cycles","outputtype" undirected, directed, or count of cycles "Edges","outputtype" edges as a list, edge list, index pair list, rule list, graphic, or count of edges "EulerianCycles","outputtype" undirected, directed, or count of Eulerian cycles "Faces","outputtype" faces as a list, index list, graphic, or count of faces "HamiltonianCycles","outputtype" undirected, directed, or count of Hamiltonian cycles "HamiltonianPaths","outputtype" undirected, directed, or count of Hamiltonian paths "HamiltonianWalks","outputtype" undirected, directed, or count of Hamiltonian walks "VertexCoordinates","outputtype" one or all vertex coordinates or embedding count "Vertices","outputtype" vertices as a list, indexed coordinate rule list, graphic, or count of vertices - Properties giving pure functions representing graph polynomials include:
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"CharacteristicPolynomial" characteristic polynomial of the adjacency matrix "ChordlessCyclePolynomial" polynomial encoding the counts of chordless cycles by length "ChromaticPolynomial" chromatic polynomial "CliquePolynomial" clique polynomial "CoboundaryPolynomial" coboundary polynomial "ComplementChordlessCyclePolynomial" polynomial encoding the counts of chordless cycles in the graph complement by length "ComplementOddChordlessCyclePolynomial" polynomial encoding the counts of odd chordless cycles in the graph complement by length "ConnectedDominationPolynomial" connected domination polynomial "ConnectedInducedSubgraphPolynomial" polynomial encoding the counts of connected induced subgraphs by size "CyclePolynomial" cycle polynomial "DetourPolynomial" characteristic polynomial of the detour matrix "DistancePolynomial" distance polynomial "DominationPolynomial" domination polynomial "EdgeCoverPolynomial" edge cover polynomial "EdgeCutPolynomial" edge cut polynomial "FlowPolynomial" flow polynomial "IdiosyncraticPolynomial" Tutte's idiosyncratic polynomial "IndependencePolynomial" independence polynomial "IrredundancePolynomial" irredundance polynomial "LaplacianPolynomial" Laplacian polynomial "MatchingGeneratingPolynomial" matching generating polynomial "MatchingPolynomial" matching polynomial "MaximalCliquePolynomial" polynomial encoding the counts of maximal cliques by size "MaximalIndependencePolynomial" polynomial encoding the counts of maximal independent vertex sets by size "MaximalIrredundancePolynomial" polynomial encoding the counts of maximal irredundant sets by size "MaximalMatchingGeneratingPolynomial" polynomial encoding the counts of maximal independent edge sets by size "MinimalConnectedDominationPolynomial" polynomial encoding the counts of minimal connected dominating sets by size "MinimalDominationPolynomial" polynomial encoding the counts of minimal dominating sets by size "MinimalEdgeCoverPolynomial" polynomial encoding the counts of minimal edge covers by size "MinimalEdgeCutPolynomial" polynomial encoding the counts of minimal edge cuts by size "MinimalTotalDominationPolynomial" polynomial encoding the counts of minimal total dominating sets by size "MinimalVertexCoverPolynomial" polynomial encoding the counts of minimal vertex covers by size "MinimalVertexCutPolynomial" polynomial encoding the counts of minimal vertex cuts by size "OddChordlessCyclePolynomial" polynomial encoding the counts of odd chordless cycles by length "PathPolynomial" path polynomial "QChromaticPolynomial" Q-chromatic polynomial "RankPolynomial" rank polynomial "ReliabilityPolynomial" reliability polynomial "SigmaPolynomial" chromatic polynomial in falling factorial basis "TotalDominationPolynomial" polynomial encoding the counts of total dominating sets by size "TuttePolynomial" Tutte polynomial "VertexCoverPolynomial" polynomial encoding the counts of vertex covers by size "VertexCutPolynomial" polynomial encoding the counts of vertex cuts by size - Coloring‐related graph properties include:
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"ChromaticInvariant" chromatic invariant "ChromaticNumber" chromatic number "ChromaticPolynomial" chromatic polynomial "ChromaticRoots" chromatic roots "CyclicChromaticNumber" cyclic chromatic number "EdgeChromaticNumber" edge chromatic number "FractionalChromaticNumber" fractional chromatic number "FractionalEdgeChromaticNumber" fractional edge chromatic number "MinimumVertexColoringCount" number of minimum vertex colorings "MinimumVertexColorings" minimum vertex colorings "MinimumEdgeColoring" minimum edge coloring "MinimumWeightFractionalColoring" minimum weight fractional coloring "QChromaticPolynomial" Q‐chromatic polynomial "WeisfeilerLemanDimension" Weisfeiler–Leman dimension - Graph index properties include:
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"ABCIndex" atom-bond connectivity index "ArithmeticGeometricIndex" arithmetic-geometric index "BalabanIndex" Balaban index "CircuitRank" minimum number of edges to remove to turn acyclic "DetourIndex" detour index "HararyIndex" Harary index "HosoyaIndex" Hosoya index "KirchhoffIndex" Kirchhoff index "KirchhoffSumIndex" Kirchhoff sum index "MolecularTopologicalIndex" molecular topological (second Schultz) index "RandicIndex" Randić index "SomborIndex" Sombor index "StabilityIndex" stability index "TopologicalIndex" topological (first Schultz) index "WienerIndex" Wiener index "WienerSumIndex" Wiener sum index "ZagrebIndex1" first Zagreb index "ZagrebIndex2" second Zagreb index - Matrix graph properties include:
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"ABCMatrix" atom-bond connectivity matrix "AdjacencyMatrix" adjacency matrix "ArithmeticGeometricMatrix" arithmetic-geometric matrix "DetourMatrix" matrix of longest path distances "DistanceMatrix" distance matrix "IncidenceMatrix" incidence matrix "LaplacianMatrix" Laplacian matrix "MaximumFlowMatrix" matrix flow matrix "MinimumCostFlowMatrix" minimum cost flow matrix "NormalizedLaplacianMatrix" normalized Laplacian matrix "RandicMatrix" Randić matrix "ResistanceMatrix" resistances between pairs of vertices for unit‐resistance edges "SomborMatrix" Sombor matrix - Local graph properties include:
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"ChordCount" number of chords "Chords" chords "Curvatures" curvatures "IsolatedPointCount" number of isolated points "IsolatedPoints" vertices of degree 0 "LeafCount" number of leaves "Leaves" vertices of degree 1 - Global graph properties include:
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"Anarboricity" maximum number of edge‐disjoint nonacyclic subgraphs whose union is the original graph "ApexCount" apex count "Apices" list of vertices whose removal renders the graph planar "Arboricity" minimum number of edge‐disjoint acyclic subgraphs whose union is the original graph "Center" indices of vertices with graph eccentricity equal to radius "Circumference" circumference of the graph "Coarseness" maximum number of line‐disjoint nonplanar subgraphs "Corank" edge count minus vertex count plus connected component count "Degeneracy" graph degeneracy "DegreeSequence" vertex degrees in monotonic nonincreasing order "DeterminedByResistance" no other graph shares the same multiset of resistances "DeterminedBySpectrum" no other graph shares the spectrum "Diameter" diameter of the graph "Eccentricities" eccentricities of each vertex "FractionalArboricity" fractional arboricity "IntersectionArray" intersection array "LinearArboricity" linear arboricity "MaximumLeafNumber" largest number of tree leaves in any spanning trees "MaximumVertexDegree" largest vertex degree "MeanCurvature" mean curvature "MeanDistance" mean distance between vertices "MinimumLeafNumber" smallest number of tree leaves in any spanning trees "MinimumVertexDegree" smallest vertex degree "Periphery" indices of vertices with graph eccentricity equal to diameter "Pseudoarboricity" pseudoarboricity "QuadraticEmbeddingConstant" quadratic embedding constant "Rank" vertex count minus connected component count "RegularParameters" parameters describing common neighbor counts of vertices "Skewness" minimum number of edges whose removal would result in a planar graph "SpanningTreeCount" number of spanning trees "StarArboricity" star arboricity "Thickness" minimum number of planar subgraphs whose union is the original graph "TreeNumber" minimal number of trees covering the edges of a graph "Triameter" graph triameter (generalization of graph diameter) "VertexDegrees" vertex degeres - Spectral graph properties include:
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"ABCEnergy" atom-bond connectivity energy "ABCSpectralRadius" atom-bond connectivity spectral radius "AlgebraicConnectivity" second smallest eigenvalue of the Laplacian matrix "ArithmeticGeometricEnergy" arithmetic-geometric energy "ArithmeticGeometricSpectralRadius" arithmetic-geometric spectral radius "Energy" graph energy "LaplacianSpectralRadius" Laplacian spectral radius "LaplacianSpectralRatio" Laplacian spectral ratio "LaplacianSpectrum" eigenvalues of the Laplacian matrix "RandicEnergy" Randić energy "SomborEnergy" Sombor energy "SomborSpectralRadius" Sombor spectral radius "SpectralRadius" spectral radius "Spectrum" eigenvalues of the adjacency matrix "SpectrumSignature" tallies of adjacency matrix eigenvalues - Labeled graph properties include:
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"AverageDisorderNumber" average disorder number "DisorderNumber" disorder number "ErdosSequence" Erdős sequence "GracefulLabelingCount" number of fundamentally distinct graceful labelings "GracefulLabelings" fundamentally distinct graceful labelings "IrregularityStrength" irregularity strength "PinnacleSetCount" number of pinnacle sets "PinnacleSets" pinnacle sets "RadioLabelingCount" number of fundamentally distinct optimal radio labelings "RadioLabelings" fundamentally distinct optimal radio labelings "RadioNumber" radio number - Graph construction properties include:
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"AssemblyNumber" assembly number "ConstructionNumber" construction number - Topological graph properties include:
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"CrossingNumber" minimum crossings in an embedding of the graph "Dimension" graph dimension "Genus" minimum number of handles yielding a non-edge-crossing embedding "KleinBottleCrossingNumber" minimum crossings in a Klein bottle embedding "LocalCrossingNumber" local crossing number "MapNumber" map number "MetricDimension" metric dimension "ProjectivePlaneCrossingNumber" minimum crossings in a projective plane embedding "RectilinearCrossingNumber" minimum crossings in a straight‐line embedding "RectilinearLocalCrossingNumber" rectilinear local crossing number "ToroidalCrossingNumber" minimum crossings in a toroidal embedding - Clique‐related properties include:
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"CliquePolynomial clique polynomial "Cliques cliques "BicliqueNumber" biclique number "BipartiteDimension" bipartite dimension "CliqueCount" number of cliques "CliqueCoveringNumber" minimum number of maximum cliques needed to cover the vertex set "CliqueNumber" number of vertices in a maximum clique "DelsarteCliqueCount" number of Delsarte cliques "DelsarteCliques cliques in a distance-regular graph for which the Delsarte bound is achieved "FractionalCliqueNumber" fractional clique number "LowerCliqueNumber" size of smallest maximal clique "MaximalCliqueCount" number of distinct maximal cliques "MaximalCliquePolynomial" polynomial encoding tallies of maximal clique sizes "MaximalCliques" maximal cliques "MaximumCliqueCount" number of maximum cliques "MaximumCliques" maximum cliques "MinimumCoveringsByMaximalCliques" smallest coverings by maximal cliques "MinimumCoveringsByMaximalCliquesCount" number of smallest coverings by maximal cliques "SimplexGraph" simplex graph - Cover‐related properties include:
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"CliqueCoveringNumber" minimum number of maximum cliques needed to cover the vertex set "EdgeCoverCount" number of edge covers "EdgeCoverNumber" size of the minimum edge cover "EdgeCoverPolynomial" edge cover polynomial "EdgeCovers" edge covers "MinimalEdgeCoverCount" number of minimal edge covers "MinimalEdgeCoverPolynomial" polynomial encoding tallies of minimal edge cover sizes "MinimalEdgeCovers" minimal edge covers "MinimalVertexCoverCount" number of minimal vertex covers "MinimalVertexCoverPolynomial" polynomial encoding tallies of minimal vertex cover sizes "MinimalVertexCovers" minimal vertex covers "MinimumCliqueCoveringCount" number of minimum clique coverings "MinimumCliqueCoverings" minimum clique coverings "MinimumEdgeCoverCount" number of minimum edge covers (matchings) "MinimumEdgeCovers" minimum edge covers (matchings) "MinimumPathCoveringCount" number of minimum path coverings "MinimumPathCoverings" minimum path coverings "MinimumVertexCoverCount" number of minimum vertex covers "MinimumVertexCovers" minimum vertex covers "PathCoveringNumber" path covering number "VertexCoverCount" number of vertex covers "VertexCoverNumber" size of a minimum vertex cover "VertexCoverPolynomial" vertex cover polynomial - Independent set‐related properties include:
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"BipartiteDimension" bipartite dimension "EdgeIndependenceNumber" tallies of independent edge set sizes "FractionalIndependenceNumber" fractional independence number "IndependenceNumber" size of the largest independent vertex set "IndependencePolynomial" independence polynomial "IndependenceRatio" independence ratio "IndependentEdgeSetCount" number of independent edge sets "IndependentEdgeSets" independent edge sets "IndependentVertexSetCount" number of independent vertex sets "IndependentVertexSets" independent vertex sets "IntersectionNumber" intersection number "LowerIndependenceNumber" size of the smallest maximal independent vertex set "LowerMatchingNumber" size of the smallest maximal independent edge set "MatchingGeneratingPolynomial" matching‐generating polynomial "MatchingNumber" degree of the matching‐generating polynomial "MatchingPolynomial" matching‐generating polynomial "MaximalIndependencePolynomial" polynomial encoding the numbers of maximal independent vertex sets by size "MaximalIndependentEdgeSetCount" number of maximal independent edge sets (matchings) "MaximalIndependentEdgeSets" maximal independent edge sets (matchings) "MaximalIndependentVertexSetCount" number of maximal independent vertex sets "MaximalIndependentVertexSets" maximal independent vertex sets "MaximalMatchingGeneratingPolynomial" polynomial encoding the numbers of maximal independent edge sets by size "MaximumIndependentEdgeSetCount" number of maximum independent edge sets (matchings) "MaximumIndependentEdgeSets" maximum independent edge sets (matchings) "MaximumIndependentVertexSetCount" number of maximum independent vertex sets "MaximumIndependentVertexSets" maximum independent vertex sets - Irredundant set‐related properties include:
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"IrredundanceNumber" (lower) irredundance number "IrredundancePolynomial" irredundance polynomial "IrredundantSetCount" number of irredundant sets "IrredundantSets" irredundant sets "MaximalIrredundancePolynomial" polynomial encoding tallies of maximal irredundant set sizes "MaximalIrredundantSetCount" number of maximal irredundant sets "MaximalIrredundantSets" maximal irredundant sets "MaximumIrredundantSetCount" number of maximum irredundant sets "MaximumIrredundantSets" maximum irredundant sets "UpperIrredundanceNumber" upper irredundance number - Dominating set‐related properties include:
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"ConnectedDominatingSetCount" connected dominating set count "ConnectedDominatingSets" connected dominating sets "ConnectedDominationNumber" size of smallest possible connected dominating set "ConnectedDominationPolynomial" polynomial encoding tallies of connected dominating set sizes "DomaticNumber" maximum number of disjoint dominating sets in a domatic partition of a graph "DominatingSetCount" number of dominating sets "DominatingSets" dominating sets "DominationNumber" size of smallest possible dominating set "DominationPolynomial" polynomial encoding tallies of dominating set sizes "MinimalConnectedDominatingSetCount" number of minimal connected dominating sets "MinimalConnectedDominatingSets" minimal connected dominating sets "MinimalConnectedDominationPolynomial" polynomial encoding tallies of minimal connected dominating set sizes "MinimalDominatingSetCount" number of minimal dominating sets "MinimalDominatingSets" minimal dominating sets "MinimalDominationPolynomial" polynomial encoding tallies of minimal dominating set sizes "MinimalTotalDominatingSetCount" number of minimal total dominating sets "MinimalTotalDominatingSets" minimal total dominating sets "MinimalTotalDominationPolynomial" polynomial encoding tallies of minimal total dominating set sizes "MinimumConnectedDominatingSetCount" minimum connected dominating set count "MinimumConnectedDominatingSets" minimum connected dominating sets "MinimumDominatingSetCount" number of minimum dominating sets "MinimumDominatingSets" minimum dominating sets "MinimumTotalDominatingSetCount" number of minimum total dominating sets "MinimumTotalDominatingSets" minimum total dominating sets "TotalDominatingSetCount" number of total dominating sets "TotalDominatingSets" total dominating sets "TotalDominationNumber" size of smallest possible total dominating set "TotalDominationPolynomial" polynomial encoding the tallies of total dominating set sizes "UpperDominationNumber" size of the largest maximal dominating set - Symmetry‐related properties include:
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"ArcTransitivity" maximal order s of an s‐arc-transitive graph "AutomorphismCount" order of the vertex automorphism group "AutomorphismGroup" graph automorphism permutation group "CanonicalPolyhedronDistinctEdgeLengthCount" number of distinct edge lengths in the corresponding canonical polyhedron "CayleyGraphGeneratingGroupNames" names of groups that generate the graph as a Cayley graph "CayleyGraphGeneratingGroups" groups that generate the graph as a Cayley graph "DistinguishingNumber" distinguishing number "FixingNumber" fixing number "MinimumDistinguishingLabelingCount" number of minimum distinguishing labelings "MinimumDistinguishingLabelings" minimum distinguishing labelings "PlanarEmbeddingCount" number of planar embeddings "SymmetricallyDistinctFaceCount" number of symmetrically distinct faces "SymmetricallyDistinctFaces" list of symmetrically distinct face representatives "SymmetricallyDistinctVertexPairCount" number of symmetrically distinct vertex pairs "SymmetricallyDistinctVertexPairSignature" signature of symmetrically distinct vertex pairs "SymmetricallyDistinctVertices" list of symmetrically distinct vertex representatives "SymmetricallyDistinctVertexCount" number of symmetrically distinct vertices "SymmetricallyEquivalentFaces" lists of symmetrically equivalent faces "SymmetricallyEquivalentVertices" lists of symmetrically equivalent vertices - Information‐related properties include:
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"Bandwidth" graph bandwidth "BroadcastTime" graph broadcast time "BroadcastTimes" list of vertex broadcast times "BurningNumber" burning number "CheegerConstant" Cheeger constant "Conductance" graph conductance "CoolingNumber" cooling number "Gonality" gonality "Likelihood" probability for graph to be generated by random number and corresponding vertex 
-subset picking"LovaszNumber" Lovász number (estimate of Shannon capacity) "Pathwidth" graph pagewidth "PebblingNumber" pebbling number "ScrambleNumber" scramble number "ShannonCapacity" effective alphabet size in a graph‐represented communication model "TreeDepth" tree depth "Treewidth" graph treewidth - Path‐ and cycle‐related properties include:
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"ChordlessCycleCount" number of chordless cycles of length at least 4 "ChordlessCyclePolynomial" polynomial encoding tallies of chordless cycle by length "ChordlessCycles" cycles of length at least 4 with no chords "ComplementChordlessCycleCount" number of chordless cycles in the graph complement "ComplementChordlessCyclePolynomial" polynomial encoding tallies of chordless cycles of the graph complement by length "ComplementChordlessCycles" chordless cycles of length at least 4 in the graph "ComplementOddChordlessCycleCount" number of odd chordless cycles in the graph complement "ComplementOddChordlessCyclePolynomial" polynomial encoiding tallies of odd chordless cycles of the graph complement by length "ComplementOddChordlessCycles" odd chordless cycles of length at least 4 in the graph complement "CubeGraph" graph cube "CycleCount" number of distinct cycles "CyclePolynomial" cycle polynomial "Cycles" lists of cycles "EulerianCycleCount" number of distinct Eulerian cycles "EulerianCycles" lists of Eulerian cycles "FaceSignature" tallies of face lengths "Girth" length of the shortest cycle "HamiltonDecompositionCount" number of Hamilton decompositions "HamiltonDecompositions" partitions of edge set into Hamiltonian cycles "HamiltonianCycleCount" number of distinct Hamiltonian cycles "HamiltonianCycles" lists of Hamiltonian cycles "HamiltonianNumber" length of a shortest Hamiltonian walk "HamiltonianPathCount" number of distinct Hamiltonian paths "HamiltonianPaths" lists of Hamiltonian paths "HamiltonianWalkCount" number of distinct Hamiltonian walks "HamiltonianWalks" lists of Hamiltonian walks "HexagonCount" number of 6-cycles "KCyclicIndices" indices 
that label the graph as the 
th 
-cyclic graph"LCFSignature" tally of the orders of all possible LCF signatures "LongestCycleCount" number of longest cycles "LongestCycles" longest cycles "LongestPathCount" number of longest paths "LongestPathLength" length of longest path "LongestPaths" longest paths "MinimumPathCoveringCount" number of minimum path coverings "MinimumPathCoverings" minimum path coverings "OddChordlessCycleCount" number of odd chordless cycles of length greater than 3 "OddChordlessCyclePolynomial" polynomial encoding tallies of numbers of odd chordless cycles by length "OddChordlessCycles" chordless cycles of odd length greater than 3 "PathCount" number of distinct paths "PathCoveringNumber" path covering number "PathPolynomial" path polynomial "PathPolynomialMatrix" matrix function of path polynomials "Paths" lists of paths "PentagonCount" number of 5-cycles "Radius" radius of the graph "SquareCount" number of 4-cycles "SquareGraph" square graph "TriangleCount" number of 3-cycles - Graph centralities include:
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"BetweennessCentralities" betweenness centralities "ClosenessCentralities" closeness centralities "DegreeCentralities" vertex degrees "EccentricityCentralities" reciprocal of vertex eccentricities "EdgeBetweennessCentralities" edge betweenness centralities "EigenvectorCentralities" eigenvector centralities "HITSCentralities" hub centralities "KatzCentralities" Katz centralities "LinkRankCentralities" link rank centralities "PageRankCentralities" page rank centralities "RadialityCentralities" radiality centralities "StatusCentralities" status centralities - Graph clustering coefficients include:
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"GlobalClusteringCoefficient" global clustering coefficient "LocalClusteringCoefficients" local clustering coefficients "MeanClusteringCoefficient" mean clustering coefficient - Naming‐related properties include:
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"AlternateNames" alternate English names "AlternateStandardNames" alternate standard Wolfram Language names "Entity" graph entity "Name" English name "Names" English name and alternate names "StandardName" standard Wolfram Language name "StandardNames" standard and alternate Wolfram Language names - Notation‐related properties include:
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"HouseOfGraphID" House of Graphs identifier "LCFNotations" graph notations for embeddings based on Hamiltonian cycles "Notation" primary notation used for graph "NotationRules" rules for notations specifying the graph "WikidataID" Wikidata identifier - GraphData["class"] gives a list of named graphs in the specified class. GraphData[name,"class"] gives True or False, depending on whether the graph corresponding to name is in the specified class.
- GraphData[name,"Classes"] gives a list of the classes in which the graph corresponding to name appears.
- Basic classes of graphs include:
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"Bipartite" bipartite (two components connected by every edge) "Nonplanar" nonplanar (requires crossings) "Nonsimple" nonsimple graph "Planar" planar (no crossings) "Simple" simple graph (unlabeled, undirected) "Tree" tree (no cycles) - Classes based on crossings include:
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"Apex" apex graph "CriticalNonplanar" nonplanar and removal of any vertex gives a planar graph "Doublecross" crossing number 2 "DoubleToroidal" genus 2 "IntrinsicallyLinked" intrinically linked "LinklesslyEmbeddable" linklessly embeddable "Map" map graph "Nonplanar" crossing number ≥ 1 "Planar" crossing number 0 "Pretzel" genus 3 "Singlecross" crossing number 1 "Toroidal" minimally embeddable on a torus - Classes based on vertex degrees include:
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"Cubic" each vertex is of degree 3 "HighlyIrregular" neighbors of each vertex have distinct vertex degrees "Multigraphic" one or more other graphs share the degree sequence "Octic" each vertex is of degree 8 "Quartic" each vertex is of degree 4 "QuasiRegular" each vertex is of the same degree except a single one with degree one larger than the others "Quintic" each vertex is of degree 5 "Regular" each vertex is of the same degree "Septic" each vertex is of degree 7 "Sextic" each vertex is of degree 6 "Switchable" can be reduced to another graph with the same degree sequence by edge switching "TwoRegular" each vertex is of degree 2 "Unigraphic" no other graph shares the degree sequence "Unswitchable" not switchable - Classes based on traversals include:
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"Acyclic" free of cycles "AlmostHamiltonian" 
-node graph with Hamiltonian number 
"AlmostHypohamiltonian" almost Hamiltonian "Antipodal" each vertex has exactly one maximum-distance vertex "Bridged" contains at least one bridge "Bridgeless" free of bridges "Chordal" free of chordless cycles of length at least 4 "Chordless" free of chords "Cyclic" contains at least one cycle "Eulerian" has a closed cycle containing every edge once "Geodetic" has a unique shortest path between any two vertices "HamiltonConnected" every pair of vertices bounds a Hamiltonian path "HamiltonDecomposable" has a partition of its edge set into Hamiltonian cycles "Hamiltonian" has a closed cycle containing every vertex once "HamiltonLaceable" Hamilton‐connected with bipartitioned endpoints "HStarConnected" either Hamilton‐connected or Hamilton‐laceable "Hypohamiltonian" one‐vertex‐removed graphs are Hamiltonian "Hypotraceable" one‐vertex‐removed graphs are traceable "KempeCounterexample" counterexample to Kempe's 4‐coloring algorithm "MaximallyNonhamiltonian" maximally nonhamiltonian "Median" median graph "Meyniel" every odd cycle of length ≥5 has at least 2 chords "Noneulerian" not Eulerian "Nonhamiltonian" not Hamiltonian "Pancyclic" contains cycles of all lengths from 3 to vertex count "SquareFree" free of 4‐cycles "Traceable" contains a Hamiltonian path "TriangleFree" free of 3‐cycles "Unicyclic" possesses a unique cycle "UniquelyHamiltonian" possessing a unique Hamiltonian cycle "UniquelyPancyclic" uniquely pancyclic "Untraceable" not traceable - Classes based on chess boards include:
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"Antelope" moves of an antelope generalized chess piece "Bishop" moves of two (white and black) chess bishops "BlackBishop" moves of a chess black bishop "Fiveleaper" moves of a fiveleaper generalized chess piece "Camel" moves of a (1,3)-leaper graph "Giraffe" moves of a (1,4)-leaper graph "King" moves of a chess king "Knight" moves of a chess knight "Queen" moves of a chess queen "Rook" moves of a chess rook "RookComplement" graph complement of a rook graph "TriangularHoneycombAcuteKnight" moves of an acute knight on a triangular honeycomb chess board "TriangularHoneycombBishop" moves of a bishop on a triangular honeycomb chess board "TriangularHoneycombKing" moves of a king on a triangular honeycomb chess board "TriangularHoneycombObtuseKnight" moves of an obtuse knight on a triangular honeycomb chess board "TriangularHoneycombQueen" moves of a queen on a triangular honeycomb chess board "TriangularHoneycombRook" moves of a rook on a triangular honeycomb chess board "WhiteBishop" moves of a chess white bishop "Zebra" moves of a (2,3)-leaper graph - Classes based on symmetry and regularity include:
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"ArcTransitive" ordered pairs of adjacent vertices have identical environments "Asymmetric" not symmetric "Chang" strongly regular on 28 vertices "ConformallyRigid" conformally rigid "DistanceRegular" all vertices have identical distance sets "DistanceTransitive" all pairs of vertices have identical distance environments "EdgeTransitive" all edges have identical environments "Geometric" every edge of a distance-regular graph lies in a unique Delsarte clique "Identity" automorphism group is of order unity "LocallyPetersen" locally Petersen "Nongeometric" nongeometric "Paulus" strongly regular on 25 or 26 vertices "Semisymmetric" regular and edge transitive but not vertex transitive "StronglyRegular" strongly regular "Symmetric" both edge transitive and vertex transitive "Taylor" distance regular with intersection array of form 
"UniquelyEmbeddable" uniquely embeddable "VertexTransitive" all vertices have identical environments "WeaklyRegular" regular, but not strongly regular "ZeroSymmetric" vertex‐transitive cubic with edges partitioned into three orbits "ZeroTwo" every two vertices have either 0 or 2 common neighbors - Spectral classes include:
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"Integral" spectrum consists of integers "Line" line graph "Maverick" maverick graph - Classes based on forbidden graphs include:
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"Beineke" Beineke graph (line graph forbidden induced subgraph) "Kuratowski" Kuratowski graph (planar graph forbidden induced subgraph) "Metelsky" Metelsky graph (ine graph forbidden if 
induced subgraph)"Pathwidth1ForbiddenMinor" pathwidth 1 forbidden minor "Pathwidth2ForbiddenMinor" pathwidth 2 forbidden minor "PetersenFamily" linklessly embeddable forbidden minor "ProjectivePlanarForbiddenMinor" projective planar forbidden minor "ProjectivePlanarForbiddenTopologicalMinor" projective planar forbidden topological minor "ToroidalForbiddenMinor" toroidal graph forbidden minor "UnitDistanceForbiddenSubgraph" minimal unit-distance forbidden subgraph - Special classes include:
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"DistanceHereditary distance-hereditary graph "AlmostControllable" almost-controllable graph "Bicolorable" two or fewer vertex colors needed "Bicubic" bipartite and cubic "Biplanar" biplanar "Block" block graph "BracedPolygon" braced regular polygon graph "Cage" smallest graph of a given girth "Cayley" Cayley graph "ChromaticallyUnique" no other graph shares the chromatic polynomial "ClawFree" free of the claw graph "Conference" conference graph "Configuration" graph represents a point-line configuration "Controllable" controllable graph "Flexible" infinitesimally flexible graph "Fullerene" planar cubic with all bounded faces pentagons or hexagons "Fusene" planar 2‐connected with all bounded faces hexagons "Graceful" graceful graph "Imperfect" imperfect (i.e. not perfect) graph "Incidence" incidence graph of a configuration "Laman" minimal rigid (Laman) graph "LCF" describable in LCF notation (regular Hamiltonian) "Local" graph is locally a particular graph for all vertices "Matchstick" embeddable with a planar drawing having unit edge lengths "Moore" graphs with the Moore property "Nonempty" nonempty graph "NoPerfectMatching" has no perfect matching "Nuciferous" nuciferous graph "Nut" adjacency matrix has rank 1 and contains no 0 element "Ore" Ore graph "Outerplanar" outerplanar graph "Perfect" perfect graph "PerfectMatching" has a matching with 
vertices"Polyhex" polyhex graph "Polyiamond" polyiamond graph "Polyomino" polyomino graph "ProjectivePlanar" graph can be drawn on the real projective plane "Ptolemaic" Ptolemaic graph "QuadraticallyEmbeddable" quadratically embeddable graph "Rigid" infinitesimally rigid graph "SelfComplementary" isomorphic to its complement "SelfDual" isomorphic to its dual "Split" split graph "StronglyPerfect" every induced subgraph 
has an independent set meeting all maximal cliques of 
"UniquelyColorable" vertex-colorable in a single way modulo graph symmetry and permutation of colors "UnitDistance" embeddable with edges of unit length "WeaklyPerfect" clique number equals chromatic number "WellCovered" every minimal vertex cover has the same size - Classes associated with polyhedra include:
-
"Antiprism" skeleton of an antiprism "Archimedean" skeleton of one of the 13 Archimedean solids "ArchimedeanDual" skeleton of one of the 13 Archimedean duals "Dipyramid" skeleton of a dipyramid "JohnsonSkeleton" skeleton of one of the 92 Johnson solids "Platonic" skeleton of one of the five Platonic solids "Polyhedral" skeleton of a polyhedron "Prism" skeleton of a prism "RegularPolychoron" skeleton of one of the six regular four‐dimensional solids "Trapezohedron" skeleton of a trapezohedron "UniformSkeleton" skeleton of a uniform polyhedron "Wheel" skeleton of a pyramid - Snark-related classes include:
-
"Flower" flower graph (snark for n5, 7, …) "Goldberg" Goldberg graph (snark for n5, 7, …) "Snark" snark (cyclically 4-edge connected cubic graph with edge chromatic number 4 and girth at least 5) "WeakSnark" weak snark (cyclically 4-edge connected cubic graph with edge chromatic number 4 and girth at least 4) - Special classes of trees and their generalization include:
-
"Cactus" connected graph in which any two cycles have no edge in common "Caterpillar" vertices are on a central stalk or only one edge away from a stalk "Centipede" vertices and edges correspond to the configuration of a comb "Forest" a collection of trees (same as "Acyclic") "FullyReconstructibleC1" determined from its 1‐dimensional measurement variety "FullyReconstructibleC2" determined from its 2‐dimensional measurement variety "FullyReconstructibleC3" determined from its 3‐dimensional measurement variety "Halin" Halin graph "KTree" 
-tree"Lobster" removal of leaves gives a caterpillar "Pseudoforest" contains at most one cycle per connected component "Pseudotree" a connected pseudoforest "SeriesReduced" series-reduced tree (no vertices of degree 2) "Spider" one vertex of degree at most 3 and all others ≤2 "Tripod" a tree having exactly three tree leaves - Classes of graphs indexed by one or more integers include:
-
"Accordion" accordion graph "Alkane" n alkane graph "Apollonian" connectivity graph of a 2D Apollonian gasket "BipartiteKneser" vertices represent k subsets and n‐k subsets of 
"Book" graph Cartesian product of a star and a two‐path graph "Bouwer" regular graphs including members that are symmetric but not arc transitive "Bruhat" graph whose vertices are permutations on n symbols with edges for permutations differing by an adjacent transposition "Caveman" caveman graph "Circulant" n vertices each with identical relative adjacencies "Complete" all pairs of vertices are connected "CompleteBipartite" all pairs connected across two disjoint sets of vertices "CompleteKPartite" all pairs connected across 
disjoint sets of vertices"CompleteTripartite" all neighboring pairs connected across three disjoint sets of vertices "Cone" graph join of a cycle graph and empty graph "Crown" complete bipartite 
with horizontal edges removed"Cycle" a single cycle through n vertices "CycleComplement" complement graph of the cycle graph 
"Cyclotomic" graph with vertices adjacent if their difference is a cube in 
"DiagonalIntersection" graph with vertices formed from the vertices of a regular n-gon and the intersection of its diagonals "Dipyramid" skeleton graph of an n-dipyramid "Doob" Cartesian product of Shrikhande graphs and a Hamming graph "DorogovtsevGoltsevMendes" Dorogovtsev-Goltsev-Mendes graph "DoubleCone" double cone graph with n-gonal bases "Egawa" Egawa graph "Empty" n vertices with no edges "Fan" graph join of an empty graph with a path graph "FibonacciCube" a Fibonacci cube graph "Flower" flower graph (snark for n5, 7, …) "FoldedCube" folded n‐hypercube graph "Gear" a wheel with vertices added between the vertices of the outer cycle "GeneralizedPolygon" an incidence plane based on a symmetric binary relation "GoethalsSeidelBlockDesign" Goethals–Seidel block design graph "Goldberg" Goldberg graph (snark for n5, 7, …) "Grassmann" Grassmann graph "Grid" an array of points with grid connectivity "Haar" Haar (regular bipartite) graph of index n "Hadamard" graph corresponding to a 
matrix satisfying 
"HalvedCube" halved n‐hypercube graph "Hamming" direct product of m complete graphs of size n "Hanoi" a Hanoi graph "Harary" a Harary graph "Helm" a wheel with a pendant edge adjoined at each cycle vertex "HexagonalGrid" hexagonal grid graph "HoneycombToroidal" honeycomb toroidal graph "Hypercube" an n‐dimensional hypercube "IGraph" generalization of a generalized Petersen graph "Jahangir" Jahangir graph "Johnson" graph describing adjacencies in the m‐subsets of an n‐set "JohnsonSkeleton" skeleton graph of the n 
Johnson solid"KayakPaddle" kayak paddle graph "Keller" Keller graph "KleinBottleTriangulation" a regular triangulation of a Klein bottle "Kneser" vertices represent k‐subsets of {1,…,n} "Ladder" a 2n‐vertex ladder graph "LadderRung" graph union of n two‐paths "LindgrenSousselier" Lindgren-Sousselier graphs "LucasCube" Lucas cube graph "Mathon" Mathon graph "MengerSponge" connectivity graph of a Menger sponge "MiddleLayer" middle layer graph "MoebiusLadder" an n‐sided prism graph with a half‐twist "Mycielski" a triangle‐free graph with chromatic number n "Odd" an odd graph "Paley" graph with vertices adjacent if their difference is a square in 
"Pan" an n‐cycle connected to a singleton graph by a bridge "Pasechnik" Pasechnik graph "Path" an n‐vertex tree with no branches "PathComplement" complement graph of the path graph 
"Pell" Pell graph "PermutationStar" a "star graph" on permutations of {1,…,n} with edges at swaps "PlummerToft" Plummer-Toft graphs "SierpinskiCarpet" connectivity graph of the Sierpiński carpet "SierpinskiSieve" connectivity graph of the Sierpiński sieve "SierpinskiTetrahedron" connectivity graph of the Sierpiński tetrahedron (tetrix) "Spoke" spoke graph with 
arms and 
vertices per arm"StackedBook" graph Cartesian product of a star and a path graph "StackedPrism" a stacked prism graph "Star" a center vertex connected to n‐1 vertices "Sun" a complete graph with erected triangles on outer edges "Sunlet" a cycle with pendant edges "Tetrahedral" an 
‐Johnson graph"TorusGrid" grid graph on a torus "TorusTriangulation" a regular triangulation of a torus "Transposition" a transposition graph "Triangular" an 
‐Johnson graph"TriangularGrid" a triangular grid graph "TriangularSnake" triangular snake graph "Turan" a Turán graph on n vertices that is (k+1)‐clique free "Wheel" a cycle with all vertices connected to a center "WheelComplement" complement of a wheel graph 
"Windmill" m copies of the complete graph 
with a vertex in common"Wreath" n collections of k nodes arranged around a circle such that all nodes in adjacent collections are connected - GraphData[name,"property","type"] gives a set of specific graphs, images, or embeddings, where in 2D "type" may include "3D", "All", "Circulant", "Circular", "Degenerate", "Gear", "GeneralizedPetersen", "Grid", "Halin", "IGraph", "IntegerCoordinates", "Integral", "LCF", "Linear", "Matchstick", "MinimalCrossing", "MinimalIntegral", "MinimalLocalCrossing", "MinimalPlanarIntegral", "MinimalRectilinearCrossing", "MinimalRectilinearLocalCrossing", "Perspective", "Planar", "Polyiamond", "Polyomino", "Primary", "Torus", "TriangularGrid", "UnitDistance", and "XYZ"; and in 3D "type" may include "Grid", "IntegerCoordinates", "Integral", "Polyhedron", "Primary", "TetrahedralGrid", "UnitDistance", and "XYZ".
- Type properties related to graph display include:
-
"Embeddings" embeddings of a given type "EmbeddingClasses" list of embedding classifications "Graph" graph of a given type "Graphics" graphics "Graphics3D" 3D graphics "Image" images of a given type "MeshRegion" mesh region - GraphData[name,"property","outputtype"] gives graph properties in the format specified by "outputtype", which, depending on "property", may be "All", "Count", "Directed", "Edge", "Entity", "Graph", "Graphics", "Group", "Image", "Labeled", "List", "Name", "Pair", "Polyhedron", "Rule", or "Undirected".
- Output type properties related to graph output include:
-
"BipartiteDoubleGraph" bipartite double graph "CanonicalGraph" canonical version of graph that is isomorphic to the original "CayleyGraphGeneratingGroups" groups generating corresponding to graph as a Cayley graph as groups, entities, or entity names "CochromaticGraphs" cochromatic graphs as graphs, entities, entity names, 3D graphs, graphics, or images "CodegreeSequenceGraphs" codegree sequence graphs as graphs, entities, entity names, 3D graphs, graphics, or images "ComplementGraph" complement graph as a graph, entity, entity name, graphics, or image "ConnectedComponents" connected components as graphs, entities, entity names, 3D graphs, graphics, or images "CoresistanceGraphs" coresistance graphs as graphs, entities, entity names, 3D graphs, graphics, or images "CospectralGraphs" cospectral graphs as graphs, entities, entity names, 3D graphs, graphics, or images "CubeGraph" cube graph as a graph, entity, entity name, graphics, or image "DualGraph" dual graph as a graph, entity, entity name, graphics, or image "LeviGraph" Levi graph as a graph, entity, entity name, graphics, or image "LineGraph" line graph as a graph, entity, entity name, graphics, or image "LocalGraph" local graph as a graph, entity, entity name, graphics, or image "OrdinaryLineGraph" ordinary line graph "PolyhedralEmbeddings" polyhedra having given graph as a skeleton as polyhedra, entities, or entity names "RootGraph" root graph as a graph, entity, entity name, graphics, or image "SimplexGraph" simplex graph as a graph, entity, entity name, graphics, or image "SquareGraph" square graph as a graph, entity, entity name, graphics, or image - GraphData[name,"property","ann"] or GraphData["property","ann"] gives various annotations associated with a property. Typical annotations include:
-
"Description" short textual description of the property "Information" hyperlink to additional information "LongDescription" longer textual description of the property "Note" additional information about the property "Value" the value of the property - Using GraphData may require internet connectivity.
Examples
open allclose allBasic Examples (4)Summary of the most common use cases
Return a list of standard names for all simple graphs on 5 vertices:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-30qni8
Direct link to example
Out[1]=1
Give the corresponding Graph objects:
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In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-kmt7rh
Direct link to example
Out[2]=2
Do the same without explicitly specifying the default "Graph" property:
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In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-hvjyv0
Direct link to example
Out[3]=3
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-fmz5ge
Direct link to example
Out[1]=1
Show all available 2D embeddings of the graph:
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In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-za3wzb
Direct link to example
Out[2]=2
Show the spectrum of the icosahedral graph:
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In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-md5inq
Direct link to example
Out[1]=1
Generate a list of named snark graphs:
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In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-fswpwi
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-jyom9b
Direct link to example
Out[2]=2
Scope (741)Survey of the scope of standard use cases
Names and Classes (7)
Obtain a list of all standard implemented graphs:
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In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-yfb41f
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-p6u93n
Direct link to example
Out[2]=2
Obtain a list of all implemented graphs:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-jp7blu
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-eqig3n
Direct link to example
Out[2]=2
Find the English name of a graph:
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In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-dxyruk
Direct link to example
Out[1]=1
A list of alternate names can also be found:
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In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-typed
Direct link to example
Out[2]=2
Additional names acceptable as input can be found:
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In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-p1hvl4
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-crdphk
Direct link to example
Out[2]=2
Find the list of graph classes:
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In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-rf8cr
Direct link to example
Out[1]=1
Find the list of named graphs belonging to a class:
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In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-nfye2n
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-n286w
Direct link to example
Out[2]=2
Test whether a graph belongs to a class:
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In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-j4935
Direct link to example
Out[1]=1
Properties and Annotations (2)
Get a list of properties for a particular graph:
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In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-c9swca
Direct link to example
Get a short textual description of a property:
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In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-4v7haq
Direct link to example
Out[1]=1
Get a longer textual description:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-cwfej
Direct link to example
Out[2]=2
Property Values (4)
The values of properties may take a variety of forms as Wolfram Language expressions:
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In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-i086aw
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-1ju5lf
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-6fjaqu
Direct link to example
Out[3]=3
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0e7p5nb2-kc6f2g
Direct link to example
Out[4]=4
Copy to clipboard.
In[5]:=5
✖
https://wolfram.com/xid/0e7p5nb2-jknk8w
Direct link to example
Copy to clipboard.
In[6]:=6
✖
https://wolfram.com/xid/0e7p5nb2-g61b66
Direct link to example
Out[6]=6
Copy to clipboard.
In[7]:=7
✖
https://wolfram.com/xid/0e7p5nb2-g46w3f
Direct link to example
A property that is not available for a graph has the value Missing["NotAvailable"]:
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In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-c8hvnz
Direct link to example
Out[1]=1
Some graph properties may be Missing but still include partial information:
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In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-iygno7
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-nmht48
Direct link to example
Out[2]=2
A property whose value is too large to include has the value Missing["TooLarge"]:
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In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-kfzued
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-o9roqr
Direct link to example
Out[2]=2
Detailed Properties (728)
Basic Graph Properties (9)
Give the adjacency matrix, returned as a SparseArray object:
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In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-gegvxc
Direct link to example
Out[1]=1
Convert to an explicit matrix:
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In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-znkdm9
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-835lrf
Direct link to example
Plot the matrix using ArrayPlot:
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In[4]:=4
✖
https://wolfram.com/xid/0e7p5nb2-qhva5x
Direct link to example
Out[4]=4
Verify that the positions of 1s in the adjacency matrix correspond to graph edges:
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In[5]:=5
✖
https://wolfram.com/xid/0e7p5nb2-s6xsvl
Direct link to example
Out[5]=5
Copy to clipboard.
In[6]:=6
✖
https://wolfram.com/xid/0e7p5nb2-vjw84m
Direct link to example
Out[6]=6
Copy to clipboard.
In[7]:=7
✖
https://wolfram.com/xid/0e7p5nb2-g6behz
Direct link to example
Out[7]=7
List the number of distinct adjacency matrices possible for the cubical graph:
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In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-25hiiq
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-wu98c0
Direct link to example
Out[2]=2
Return the number of edges of the octahedral graph:
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In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-uac5bl
Direct link to example
Out[1]=1
Compare with taking the length of the edge list:
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In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-ktm3pc
Direct link to example
Out[2]=2
Return the edges of the octahedral graph:
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In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-it4xkn
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-3tefd5
Direct link to example
Out[2]=2
Give the face count of the octahedral graph:
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In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-q00ni5
Direct link to example
Out[1]=1
Give the faces of the octahedral graph:
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In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-c4p0fp
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-q5pxv9
Direct link to example
Out[2]=2
Give the incidence matrix of the octahedral graph:
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In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-srhu8z
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-0pywg4
Direct link to example
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-oz05v6
Direct link to example
Out[3]=3
Give the number of vertices of the octahedral graph:
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In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-g6q7q7
Direct link to example
Out[1]=1
Return the vertices of the octahedral graph:
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In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-7cuiov
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-2pvcwm
Direct link to example
Out[2]=2
Properties Related to Graph Connectivity (30)
Give the block count of the butterfly graph:
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In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-3i4xvs
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-8ibd3a
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-hooo3i
Direct link to example
Out[3]=3
Give the blocks of the butterfly graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-1ige5x
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-5xgcy8
Direct link to example
Out[2]=2
Show the bridges of the 3-barbell graph:
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In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-eqg7wu
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-uasytr
Direct link to example
Out[2]=2
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-4gj1e8
Direct link to example
List connected graphs on five vertices:
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In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-2su929
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-dwk2vn
Direct link to example
Out[3]=3
Check if the graph 
is connected:
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In[4]:=4
✖
https://wolfram.com/xid/0e7p5nb2-ivbpf1
Direct link to example
Out[4]=4
Copy to clipboard.
In[5]:=5
✖
https://wolfram.com/xid/0e7p5nb2-ngd6bs
Direct link to example
Out[5]=5
Check if the graph 
is connected:
Copy to clipboard.
In[6]:=6
✖
https://wolfram.com/xid/0e7p5nb2-lp9n20
Direct link to example
Out[6]=6
Copy to clipboard.
In[7]:=7
✖
https://wolfram.com/xid/0e7p5nb2-r99437
Direct link to example
Out[7]=7
Return the number of connected components in the 
graph:
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In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-mnrng4
Direct link to example
Out[1]=1
Do the same using the "Count" annotation:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-66ey8w
Direct link to example
Out[2]=2
Return the connected components of the 
graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-gy6g5n
Direct link to example
Out[1]=1
List the names of the connected components:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-tucjey
Direct link to example
Out[2]=2
List the indices of connected components:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-84gx7k
Direct link to example
Out[3]=3
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0e7p5nb2-6rboce
Direct link to example
Out[4]=4
Give the number of connected induced subgraphs of the diamond graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-2o0er9
Direct link to example
Out[1]=1
Compare with the value obtained from the connected induced subgraph polynomial:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-yno4x6
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-nbn01z
Direct link to example
Out[3]=3
Display the diamond graph and compare the above counts with those computed from scratch:
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0e7p5nb2-jknsg0
Direct link to example
Out[4]=4
Copy to clipboard.
In[5]:=5
✖
https://wolfram.com/xid/0e7p5nb2-mgco39
Direct link to example
Out[5]=5
Copy to clipboard.
In[6]:=6
✖
https://wolfram.com/xid/0e7p5nb2-dyfeuo
Direct link to example
Out[6]=6
Give the connected induced subgraph polynomial of the diamond graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-byg9ka
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-k9ywv0
Direct link to example
Out[2]=2
Compare to the actual subgraphs:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-bb56hz
Direct link to example
Out[3]=3
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0e7p5nb2-cty4ph
Direct link to example
Out[4]=4
Give the cyclic edge connectivity of the Petersen graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-0mi46w
Direct link to example
Out[1]=1
Show the two cyclic components of a edge cut set of size 5:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-lvrse
Direct link to example
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-1tc3lb
Direct link to example
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0e7p5nb2-zib81m
Direct link to example
Copy to clipboard.
In[5]:=5
✖
https://wolfram.com/xid/0e7p5nb2-e4tlmb
Direct link to example
Out[5]=5
List disconnected graphs on five vertices:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-yz1l5k
Direct link to example
Out[1]=1
Check if the graph 
is disconnected:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-dnov37
Direct link to example
Out[2]=2
Check if the graph 
is disconnected:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-drxfjo
Direct link to example
Out[3]=3
Find the edge connectivity of a complete binary tree of order 4:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-fdngul
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-bhwqi7
Direct link to example
Out[2]=2
Return the number of edge cuts in the triangle graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-cupmlw
Direct link to example
Out[1]=1
Do the same with the "Count" annotation:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-7xyoif
Direct link to example
Out[2]=2
Compare with the length of edge cuts:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-gcpm29
Direct link to example
Out[3]=3
Show the edge cuts in the triangle graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-yjnykb
Direct link to example
Out[1]=1
Visualize how the cuts disconnect the graph:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-6s06uh
Direct link to example
Out[2]=2
Give the incidence lines that generate the Pappus graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-7nfhy8
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-6s3mhm
Direct link to example
Out[2]=2
Show the Levi graph generated by these incidences:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-171m0d
Direct link to example
Out[3]=3
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0e7p5nb2-0twvkc
Direct link to example
Out[4]=4
Give a realization of the Pappus configuration showing the incidences:
Copy to clipboard.
In[5]:=5
✖
https://wolfram.com/xid/0e7p5nb2-6z7pct
Direct link to example
Out[5]=5
Give the lambda components of the domino graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-5tuxpl
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-o5w5bs
Direct link to example
Out[2]=2
Give the Luccio–Sami components of the domino graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-esluwf
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-lbro72
Direct link to example
Out[2]=2
Return the number of minimal edge cuts in the triangle graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-jlcm80
Direct link to example
Out[1]=1
Do the same with the "Count" annotation:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-9cfkqw
Direct link to example
Out[2]=2
Compare with the length of edge cuts:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-wknsdb
Direct link to example
Out[3]=3
Show the minimal edge cuts in the triangle graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-zob2kx
Direct link to example
Out[1]=1
Visualize how the cuts disconnect the graph:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-lxo6x0
Direct link to example
Out[2]=2
Return the number of minimal vertex cuts in the butterfly graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-vhorjg
Direct link to example
Out[1]=1
Do the same with the "Count" annotation:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-bqk7uu
Direct link to example
Out[2]=2
Compare with the length of edge cuts:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-edscll
Direct link to example
Out[3]=3
Show the minimal edge cuts in the butterfly graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-irqkqk
Direct link to example
Out[1]=1
Visualize how the cuts disconnect the graph:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-no6n5y
Direct link to example
Out[2]=2
Return the number of minimum vertex cuts of the square graph with triangles erected on two opposite sides:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-1i17rc
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-gxs4tm
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-w4xefd
Direct link to example
Out[3]=3
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0e7p5nb2-gbsghh
Direct link to example
Out[4]=4
Give the number of minimum cyclic edge cuts of the Petersen graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-hihyn4
Direct link to example
Out[1]=1
Do the same as an annotation to "MinimumCyclicEdgeCuts":
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-z3d524
Direct link to example
Out[2]=2
Compare with the number of minimum cyclic edge cuts:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-qmehi5
Direct link to example
Out[3]=3
Return the minimum cyclic edge cuts of the Petersen graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-hm5gml
Direct link to example
Out[1]=1
Compare with the cyclic edge connectivity:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-qsska
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-qn6z5
Direct link to example
Out[3]=3
Visualize the cuts to reveal the resulting two cyclic components:
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0e7p5nb2-o88bu0
Direct link to example
Copy to clipboard.
In[5]:=5
✖
https://wolfram.com/xid/0e7p5nb2-4w0pbr
Direct link to example
Out[5]=5
Give the minimum vertex cuts of the square graph with triangles erected on two opposite sides:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-x3menr
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-m4m5wj
Direct link to example
Out[2]=2
Show the disconnected graphs resulting from these cuts:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-42isyq
Direct link to example
Out[3]=3
Give spanning trees for the tetrahedral graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-byw63y
Direct link to example
Out[1]=1
Give the strength of the butterfly graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-pvele2
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-k7t8tn
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-f2al33
Direct link to example
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0e7p5nb2-1bxzwp
Direct link to example
Out[4]=4
Copy to clipboard.
In[5]:=5
✖
https://wolfram.com/xid/0e7p5nb2-sspvwh
Direct link to example
Out[5]=5
Give the toughness of the butterfly graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-xwz20p
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-tvpzvj
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-h7lsu7
Direct link to example
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0e7p5nb2-v3i8nj
Direct link to example
Out[4]=4
Copy to clipboard.
In[5]:=5
✖
https://wolfram.com/xid/0e7p5nb2-kpa8oa
Direct link to example
Out[5]=5
Give the vertex connectivity of Tietze's graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-q7ak8x
Direct link to example
Out[1]=1
Return the number of vertex cuts in the butterfly graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-4evspw
Direct link to example
Out[1]=1
Do the same with the "Count" annotation:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-q9lm5m
Direct link to example
Out[2]=2
Compare with the length of edge cuts:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-h5fbo2
Direct link to example
Out[3]=3
Show the vertex cuts of the butterfly graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-vxwjpm
Direct link to example
Out[1]=1
Visualize how the cuts disconnect the graph:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-qt8gc6
Direct link to example
Out[2]=2
Properties Related to Graph Minors (2)
Show the number of graph minors the octahedral graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-gi6v4i
Direct link to example
Out[1]=1
Give the Hadwiger number of a wheel complement graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-7en80z
Direct link to example
Out[1]=1
Properties Related to Graph Display (12)
Show the default embedding of the octahedral graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-6kav16
Direct link to example
Out[1]=1
Show classes of tabulated 2D embeddings of the cubical graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-lunnq3
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-bcj8qc
Direct link to example
Out[2]=2
Show classes of known 3D embeddings of the cubical graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-pkb1rd
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-zyyoaa
Direct link to example
Out[2]=2
List the known 2D embeddings of the octahedral graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-vticzz
Direct link to example
Out[1]=1
List the known 3D embeddings of the octahedral graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-2rpweo
Direct link to example
Out[1]=1
Show the default embedding of the octahedral graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-09icc3
Direct link to example
Out[1]=1
Show the default 3D embedding of the octahedral graph:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-2ehrjj
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-cz2dw3
Direct link to example
Out[3]=3
Give all cataloged 3D embeddings of the cubical graph:
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0e7p5nb2-j390oy
Direct link to example
Out[4]=4
Show a graphic of the octahedral graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-yn2s6w
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-jw4pty
Direct link to example
Out[2]=2
Give all cataloged embeddings of the octahedral graph as graphics:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-xmmmyf
Direct link to example
Out[3]=3
Show a 3D graphic of the octahedral graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-cu8cg3
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-8rpn0a
Direct link to example
Out[2]=2
Give all cataloged embeddings of the cubical graph as 3D graphics:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-1e0p1t
Direct link to example
Out[3]=3
Show an image of the default embedding of the octahedral graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-fve7u2
Direct link to example
Out[1]=1
Show the three-dimensional embedding of the cubical graph as an image:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-8izhp
Direct link to example
Out[2]=2
Show the Nauru graph as a mesh region:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-s4ol80
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-p7z8xy
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-ztxb16
Direct link to example
Out[3]=3
Show all available embeddings as mesh regions:
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0e7p5nb2-povjs9
Direct link to example
Out[4]=4
Show a mesh region embedded in 3D:
Copy to clipboard.
In[5]:=5
✖
https://wolfram.com/xid/0e7p5nb2-eqi98n
Direct link to example
Out[5]=5
Copy to clipboard.
In[6]:=6
✖
https://wolfram.com/xid/0e7p5nb2-v15xrg
Direct link to example
Out[6]=6
Show the icosahedral graph as a polyhedron:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-tkpud6
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-7g4389
Direct link to example
Out[2]=2
Return the vertex coordinates for the default embedding of the octahedral graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-qiv8hq
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-qt0tk7
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-si884q
Direct link to example
Out[3]=3
Show vertex coordinates for all cataloged embeddings:
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0e7p5nb2-dmi65n
Direct link to example
Out[4]=4
Annotatable Properties Related to Graph Output (20)
Return the bipartite double graph of the Clebsch graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-r53s9j
Direct link to example
Out[1]=1
Give the standard name of the bipartite double graph:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-7qysm4
Direct link to example
Out[2]=2
Return the cubical graph in canonical form:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-49mrx4
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-bquaw4
Direct link to example
Out[2]=2
The edge lists are not necessarily the same:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-tznjus
Direct link to example
Out[3]=3
Verify the graphs are isomorphic:
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0e7p5nb2-u3apnr
Direct link to example
Out[4]=4
Return groups whose Cayley graphs generate the cubical graph using the default output type:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-p341ze
Direct link to example
Out[1]=1
Return them explicitly as groups:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-qdhdvc
Direct link to example
Out[2]=2
Return FiniteGroupData standard names whose Cayley graphs generate the cubical graph:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-n4jgq
Direct link to example
Out[3]=3
Return FiniteGroupData entities whose Cayley graphs generate the cubical graph:
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0e7p5nb2-iuqaae
Direct link to example
Out[4]=4
Give cochromatic graphs of the claw graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-wc7s4y
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-fuo1n4
Direct link to example
Out[2]=2
Return the names of the cochromatic graphs:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-ufndpa
Direct link to example
Out[3]=3
Verify these share the same chromatic polynomial:
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0e7p5nb2-bjflls
Direct link to example
Out[4]=4
Copy to clipboard.
In[5]:=5
✖
https://wolfram.com/xid/0e7p5nb2-hr8bj8
Direct link to example
Out[5]=5
Show graph names for graphs that are cochromatic with the 5-star graph:
Copy to clipboard.
In[6]:=6
✖
https://wolfram.com/xid/0e7p5nb2-b1h0ki
Direct link to example
Out[6]=6
Copy to clipboard.
In[7]:=7
✖
https://wolfram.com/xid/0e7p5nb2-f4r1xm
Direct link to example
Out[7]=7
Copy to clipboard.
In[8]:=8
✖
https://wolfram.com/xid/0e7p5nb2-sww8ow
Direct link to example
Out[8]=8
Give the names of graphs cochromatic with the bull graph:
Copy to clipboard.
In[9]:=9
✖
https://wolfram.com/xid/0e7p5nb2-r2ccoj
Direct link to example
Out[9]=9
Copy to clipboard.
In[10]:=10
✖
https://wolfram.com/xid/0e7p5nb2-if5pdt
Direct link to example
Out[10]=10
Copy to clipboard.
In[11]:=11
✖
https://wolfram.com/xid/0e7p5nb2-tbvdpb
Direct link to example
Out[11]=11
Copy to clipboard.
In[12]:=12
✖
https://wolfram.com/xid/0e7p5nb2-isz4tm
Direct link to example
Out[12]=12
Return the co-degree sequence graphs of the 5-path graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-bf9wgp
Direct link to example
Out[1]=1
Since this graph shares a degree sequence, it is not unigraphic:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-nb956x
Direct link to example
Out[2]=2
Return the graph complement of the icosahedral graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-kjliky
Direct link to example
Out[1]=1
Do the same using the annotation "Graph":
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-lx1ieo
Direct link to example
Out[2]=2
Return as an Image:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-43mxam
Direct link to example
Out[3]=3
Return as a Graphics object:
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0e7p5nb2-8hvygi
Direct link to example
Out[4]=4
Give the name of the complement graph:
Copy to clipboard.
In[5]:=5
✖
https://wolfram.com/xid/0e7p5nb2-g6uach
Direct link to example
Out[5]=5
Return the entity corresponding to the complement graph:
Copy to clipboard.
In[6]:=6
✖
https://wolfram.com/xid/0e7p5nb2-o4jzfv
Direct link to example
Out[6]=6
Show the graph names for the complement of graphs on four or fewer vertices:
Copy to clipboard.
In[7]:=7
✖
https://wolfram.com/xid/0e7p5nb2-n2gmr2
Direct link to example
Out[7]=7
The complement graph name of a self-complementary graph is identical to the StandardName:
Copy to clipboard.
In[8]:=8
✖
https://wolfram.com/xid/0e7p5nb2-rt8dgf
Direct link to example
Show graph and complement names (should be the same) for "interesting" small self-complementary graphs:
Copy to clipboard.
In[9]:=9
✖
https://wolfram.com/xid/0e7p5nb2-mhjcet
Direct link to example
Out[9]=9
Return the connected components graph of the 
graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-ja3xnk
Direct link to example
Out[1]=1
Do the same using the "Graph" annotation:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-nxpwgd
Direct link to example
Out[2]=2
Return the indices of the connected components:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-g7b2gk
Direct link to example
Out[3]=3
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0e7p5nb2-099onh
Direct link to example
Out[4]=4
Return the names of the connected components:
Copy to clipboard.
In[5]:=5
✖
https://wolfram.com/xid/0e7p5nb2-57t5hn
Direct link to example
Out[5]=5
Return the number of connected components:
Copy to clipboard.
In[6]:=6
✖
https://wolfram.com/xid/0e7p5nb2-xwuci0
Direct link to example
Out[6]=6
Do the same using the "Count" annotation:
Copy to clipboard.
In[7]:=7
✖
https://wolfram.com/xid/0e7p5nb2-rw4ep6
Direct link to example
Out[7]=7
Give the vertex counts of the connected components:
Copy to clipboard.
In[8]:=8
✖
https://wolfram.com/xid/0e7p5nb2-cu622c
Direct link to example
Out[8]=8
Do the same using the "List" annotation:
Copy to clipboard.
In[9]:=9
✖
https://wolfram.com/xid/0e7p5nb2-po6dnk
Direct link to example
Out[9]=9
Do the same using the "VertexCount" annotation:
Copy to clipboard.
In[10]:=10
✖
https://wolfram.com/xid/0e7p5nb2-ckd1y4
Direct link to example
Out[10]=10
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-yjhavr
Direct link to example
Out[1]=1
Do the same using the "Graph" annotation:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-6uwev
Direct link to example
Out[2]=2
Get the names of the coresistance graphs:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-hpk1eh
Direct link to example
Out[3]=3
Show the graph and its coresistance graph together:
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0e7p5nb2-ehju7u
Direct link to example
Out[4]=4
Verify that the graphs in question have the same resistance sets:
Copy to clipboard.
In[5]:=5
✖
https://wolfram.com/xid/0e7p5nb2-6hejf1
Direct link to example
Out[5]=5
Show graphs that share a resistance multiset with at least one other distinct graph:
Copy to clipboard.
In[6]:=6
✖
https://wolfram.com/xid/0e7p5nb2-fkm90r
Direct link to example
List the names of graphs sharing the same multiset of resistances with a given graph:
Copy to clipboard.
In[7]:=7
✖
https://wolfram.com/xid/0e7p5nb2-u0ug0s
Direct link to example
Out[7]=7
Copy to clipboard.
In[8]:=8
✖
https://wolfram.com/xid/0e7p5nb2-19uyjs
Direct link to example
Out[8]=8
Show graph names for graphs that are equivalent with a particular 20-vertex graph:
Copy to clipboard.
In[9]:=9
✖
https://wolfram.com/xid/0e7p5nb2-divg51
Direct link to example
Out[9]=9
Copy to clipboard.
In[10]:=10
✖
https://wolfram.com/xid/0e7p5nb2-fds6q0
Direct link to example
Out[10]=10
Copy to clipboard.
In[11]:=11
✖
https://wolfram.com/xid/0e7p5nb2-n9i0pw
Direct link to example
Out[11]=11
Return the cospectral graphs of the tesseract graph using the default output type:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-2cntp2
Direct link to example
Out[1]=1
Return the cospectral graphs of the tesseract graph explicitly as a graph:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-xqt5pi
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-d2geqp
Direct link to example
Out[3]=3
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0e7p5nb2-cneplv
Direct link to example
Out[4]=4
Copy to clipboard.
In[5]:=5
✖
https://wolfram.com/xid/0e7p5nb2-lkr2io
Direct link to example
Out[5]=5
Copy to clipboard.
In[6]:=6
✖
https://wolfram.com/xid/0e7p5nb2-b8itjm
Direct link to example
Out[6]=6
Give the names of graphs cospectral with the Shrikhande graph:
Copy to clipboard.
In[7]:=7
✖
https://wolfram.com/xid/0e7p5nb2-eheqz3
Direct link to example
Out[7]=7
Show graph names for graphs that are cospectral with the tesseract graph:
Copy to clipboard.
In[8]:=8
✖
https://wolfram.com/xid/0e7p5nb2-kpq5ha
Direct link to example
Out[8]=8
Copy to clipboard.
In[9]:=9
✖
https://wolfram.com/xid/0e7p5nb2-lxlo6c
Direct link to example
Out[9]=9
Copy to clipboard.
In[10]:=10
✖
https://wolfram.com/xid/0e7p5nb2-l1l34d
Direct link to example
Out[10]=10
Return a graph object for the cube graph of the 7-antiprism graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-eyi5uy
Direct link to example
Out[1]=1
Compare with an explicit computation:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-86zka5
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-csvzwy
Direct link to example
Out[3]=3
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0e7p5nb2-mnpa58
Direct link to example
Out[4]=4
Return a graph object for the dual graph of the icosahedral graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-3ku1er
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-vno82z
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-5wv6hv
Direct link to example
Out[3]=3
Not all graphs have unique duals:
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0e7p5nb2-5gqt97
Direct link to example
Out[4]=4
Show the graph name for the graph that is dual to the icosahedral graph:
Copy to clipboard.
In[5]:=5
✖
https://wolfram.com/xid/0e7p5nb2-xizb71
Direct link to example
Out[5]=5
It is in turn dual to the icosahedral graph:
Copy to clipboard.
In[6]:=6
✖
https://wolfram.com/xid/0e7p5nb2-edt99e
Direct link to example
Out[6]=6
List graphs with a tabulated graph dual:
Copy to clipboard.
In[7]:=7
✖
https://wolfram.com/xid/0e7p5nb2-2c0p56
Direct link to example
Display the Johnson solid J8 skeleton:
Copy to clipboard.
In[8]:=8
✖
https://wolfram.com/xid/0e7p5nb2-vc0ky8
Direct link to example
Out[8]=8
Verify that it is dual to itself:
Copy to clipboard.
In[9]:=9
✖
https://wolfram.com/xid/0e7p5nb2-n91em4
Direct link to example
Out[9]=9
Tabulate cataloged self-dual graphs:
Copy to clipboard.
In[10]:=10
✖
https://wolfram.com/xid/0e7p5nb2-h17eni
Direct link to example
Out[10]=10
Return the Levi graph of the Desargues configuration graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-bmx77b
Direct link to example
Out[1]=1
Return the name of the Levi graph of the Desargues configuration graph:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-o5126
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-bmrbib
Direct link to example
Out[3]=3
Visualize the Desargues configuration:
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0e7p5nb2-b30iqs
Direct link to example
Out[4]=4
Copy to clipboard.
In[5]:=5
✖
https://wolfram.com/xid/0e7p5nb2-i78kft
Direct link to example
Out[5]=5
Compute the Levi graph explicitly:
Copy to clipboard.
In[6]:=6
✖
https://wolfram.com/xid/0e7p5nb2-fvwkh3
Direct link to example
Out[6]=6
Return the line graph of the icosahedral graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-hkncqc
Direct link to example
Out[1]=1
Return the name of the line graph:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-d1w9sm
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-pb9k1p
Direct link to example
Out[3]=3
Give the name of the line graph for the Petersen graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-d9v5mw
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-rhviuf
Direct link to example
Out[2]=2
Give the names of the line graphs of the Platonic graphs:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-1i4jue
Direct link to example
Out[3]=3
Show the Platonic graphs and their line graphs:
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0e7p5nb2-f2v0ix
Direct link to example
Out[4]=4
Show the graph names for the line graphs of non-empty graphs on four or fewer vertices:
Copy to clipboard.
In[5]:=5
✖
https://wolfram.com/xid/0e7p5nb2-lkz3gc
Direct link to example
Out[5]=5
Taking the line graph twice does not in general give back the original graph:
Copy to clipboard.
In[6]:=6
✖
https://wolfram.com/xid/0e7p5nb2-vtqk64
Direct link to example
Out[6]=6
Copy to clipboard.
In[7]:=7
✖
https://wolfram.com/xid/0e7p5nb2-p7mutk
Direct link to example
Out[7]=7
The line graph of a graph is isomorphic to itself only for cycle graphs or unions of identical cycle graphs:
Copy to clipboard.
In[8]:=8
✖
https://wolfram.com/xid/0e7p5nb2-9seo7e
Direct link to example
Out[8]=8
Return the local graph of the Conway–Smith graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-yvcg6j
Direct link to example
Out[1]=1
Return the name of the local graph:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-4gswn4
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-wxq36z
Direct link to example
Out[3]=3
Return the name of the ordinary line graph of the pentagram configuration:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-y0jq3x
Direct link to example
Out[1]=1
Return the ordinary line graph of the pentagram configuration:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-ciodjr
Direct link to example
Out[2]=2
Compute by finding the extraordinary lines of the pentagram configuration:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-dpb60m
Direct link to example
Out[3]=3
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0e7p5nb2-9e80h
Direct link to example
Out[4]=4
Complement with all possible pairs of points to get the ordinary lines:
Copy to clipboard.
In[5]:=5
✖
https://wolfram.com/xid/0e7p5nb2-wegc8c
Direct link to example
Out[5]=5
Copy to clipboard.
In[6]:=6
✖
https://wolfram.com/xid/0e7p5nb2-qefows
Direct link to example
Build the ordinary line graph:
Copy to clipboard.
In[7]:=7
✖
https://wolfram.com/xid/0e7p5nb2-gawdwv
Direct link to example
Out[7]=7
Identify using ToEntity:
Copy to clipboard.
In[8]:=8
✖
https://wolfram.com/xid/0e7p5nb2-t5xvge
Direct link to example
Out[8]=8
Return polyhedra whose skeletons are isomorphic to the cubical graph using the default output type:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-5v4txt
Direct link to example
Out[1]=1
Return explicitly as PolyhedronData standard names:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-vgetoz
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-670gr3
Direct link to example
Out[3]=3
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0e7p5nb2-j7zw1b
Direct link to example
Out[4]=4
Return the root graph of the icosahedral line graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-9fboe9
Direct link to example
Out[1]=1
Return the name of the line graph:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-m9zl73
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-v7oqyy
Direct link to example
Out[3]=3
Return a graph object for the simplex graph of the complete graph K4:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-86gjdr
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-4zg58g
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-ovalxg
Direct link to example
Out[3]=3
Return a graph object for the square graph of the 7-antiprism graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-kof05w
Direct link to example
Out[1]=1
Compare with an explicit computation:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-48oog8
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-n8qng7
Direct link to example
Out[3]=3
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0e7p5nb2-cm3n4l
Direct link to example
Out[4]=4
Annotatable Properties Related to List-Type Output (10)
Give the adjacency matrix of the default embedding for the square graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-r0sn1l
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-jgm57q
Direct link to example
Give the number of possible adjacency matrices:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-ri2nie
Direct link to example
Out[3]=3
Return the same using the "Count" annotation:
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0e7p5nb2-1tl4jo
Direct link to example
Out[4]=4
Give all possible adjacency matrices:
Copy to clipboard.
In[5]:=5
✖
https://wolfram.com/xid/0e7p5nb2-n7evq6
Direct link to example
Out[5]=5
Copy to clipboard.
In[6]:=6
✖
https://wolfram.com/xid/0e7p5nb2-i89x2k
Direct link to example
Out[6]=6
Give (undirected) cycles of the house graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-frtttr
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-qzckbw
Direct link to example
Out[2]=2
Do the same using the explicit "Undirected" annotation:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-6dxb24
Direct link to example
Out[3]=3
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0e7p5nb2-i1qjnk
Direct link to example
Out[4]=4
Give count of undirected cycles:
Copy to clipboard.
In[5]:=5
✖
https://wolfram.com/xid/0e7p5nb2-jksx1r
Direct link to example
Out[5]=5
Compare with the direct property:
Copy to clipboard.
In[6]:=6
✖
https://wolfram.com/xid/0e7p5nb2-1pbmlz
Direct link to example
Out[6]=6
Return the edges of the octahedral graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-qcau7g
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-irtr69
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-wlwuuj
Direct link to example
Out[3]=3
Verify the above agrees with the "EdgeCount" property:
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0e7p5nb2-uka5oa
Direct link to example
Out[4]=4
Return the edges as a set of rules, suitable for plotting in GraphPlot:
Copy to clipboard.
In[5]:=5
✖
https://wolfram.com/xid/0e7p5nb2-ligjii
Direct link to example
Out[5]=5
Copy to clipboard.
In[6]:=6
✖
https://wolfram.com/xid/0e7p5nb2-ya5a8e
Direct link to example
Out[6]=6
Return a graphic of the edges:
Copy to clipboard.
In[7]:=7
✖
https://wolfram.com/xid/0e7p5nb2-lhztwk
Direct link to example
Out[7]=7
Give (undirected) Eulerian cycles of the butterfly graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-u1gsbs
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-bjo4yt
Direct link to example
Out[2]=2
Compare with the output of FindEulerianCycle:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-d5oen7
Direct link to example
Out[3]=3
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0e7p5nb2-3aftxv
Direct link to example
Out[4]=4
Do the same using the explicit "Undirected" annotation:
Copy to clipboard.
In[5]:=5
✖
https://wolfram.com/xid/0e7p5nb2-7m2pgo
Direct link to example
Out[5]=5
Copy to clipboard.
In[6]:=6
✖
https://wolfram.com/xid/0e7p5nb2-861m67
Direct link to example
Out[6]=6
Give count of undirected cycles:
Copy to clipboard.
In[7]:=7
✖
https://wolfram.com/xid/0e7p5nb2-p4dqhx
Direct link to example
Out[7]=7
Compare with the direct property:
Copy to clipboard.
In[8]:=8
✖
https://wolfram.com/xid/0e7p5nb2-1uiwq
Direct link to example
Out[8]=8
Give the faces of the octahedral graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-5qsbvv
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-i0f2lo
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-myam2
Direct link to example
Out[3]=3
Verify the above agrees with the "FaceCount" property:
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0e7p5nb2-t9pgva
Direct link to example
Out[4]=4
Show the faces of the octahedral graph:
Copy to clipboard.
In[5]:=5
✖
https://wolfram.com/xid/0e7p5nb2-89tt0w
Direct link to example
Out[5]=5
Give (undirected) Hamiltonian cycles of the house X graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-s3ed0d
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-fjcy0j
Direct link to example
Out[2]=2
Do the same using the explicit "Undirected" annotation:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-s4awip
Direct link to example
Out[3]=3
Give directed Hamiltonian cycles:
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0e7p5nb2-vpyap8
Direct link to example
Out[4]=4
Give count of undirected Hamiltonian cycles:
Copy to clipboard.
In[5]:=5
✖
https://wolfram.com/xid/0e7p5nb2-pzfqn8
Direct link to example
Out[5]=5
Compare with the direct property:
Copy to clipboard.
In[6]:=6
✖
https://wolfram.com/xid/0e7p5nb2-gt4dbi
Direct link to example
Out[6]=6
Give (undirected) Hamiltonian paths of the butterfly graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-wkc2a3
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-6r4u2m
Direct link to example
Out[2]=2
Do the same using the explicit "Undirected" annotation:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-3ixh5z
Direct link to example
Out[3]=3
Give directed Hamiltonian paths:
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0e7p5nb2-4z6i7s
Direct link to example
Out[4]=4
Give count of undirected Hamiltonian paths:
Copy to clipboard.
In[5]:=5
✖
https://wolfram.com/xid/0e7p5nb2-0wvbh7
Direct link to example
Out[5]=5
Compare with the direct property:
Copy to clipboard.
In[6]:=6
✖
https://wolfram.com/xid/0e7p5nb2-kd8n95
Direct link to example
Out[6]=6
Give (undirected) Hamiltonian walks of the butterfly graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-y4iewt
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-futzs4
Direct link to example
Out[2]=2
Do the same using the explicit "Undirected" annotation:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-vitlds
Direct link to example
Out[3]=3
Give directed Hamiltonian paths:
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0e7p5nb2-zgkw96
Direct link to example
Out[4]=4
Give count of undirected Hamiltonian paths:
Copy to clipboard.
In[5]:=5
✖
https://wolfram.com/xid/0e7p5nb2-8tz0i7
Direct link to example
Out[5]=5
Compare with the direct property:
Copy to clipboard.
In[6]:=6
✖
https://wolfram.com/xid/0e7p5nb2-jbb4j
Direct link to example
Out[6]=6
Give the vertex coordinates of the default embedding of the octahedral graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-o99774
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-pj39uc
Direct link to example
Out[2]=2
This is equivalent to the "Embeddings" property:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-4yn0et
Direct link to example
Out[3]=3
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0e7p5nb2-gzmemj
Direct link to example
Out[4]=4
Return the vertices of the octahedral graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-qfpael
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-pufhn2
Direct link to example
Out[2]=2
Verify the above agrees with the "VertexCount" property:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-kj49yn
Direct link to example
Out[3]=3
Annotatable Properties Related to Graph Display (7)
Give all tabulated embeddings of the cubical graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-ncjdbu
Direct link to example
Out[1]=1
Show classes for each embedding:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-6oh31m
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-xoulqj
Direct link to example
Out[3]=3
Return the primary embedding of a graph:
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0e7p5nb2-xpex7o
Direct link to example
Out[4]=4
Copy to clipboard.
In[5]:=5
✖
https://wolfram.com/xid/0e7p5nb2-1bu0jy
Direct link to example
Out[5]=5
Return all tabulated planar embeddings of the graph:
Copy to clipboard.
In[6]:=6
✖
https://wolfram.com/xid/0e7p5nb2-omzesc
Direct link to example
Out[6]=6
Copy to clipboard.
In[7]:=7
✖
https://wolfram.com/xid/0e7p5nb2-yjzwey
Direct link to example
Out[7]=7
Return all tabulated LCF embeddings of the graph:
Copy to clipboard.
In[8]:=8
✖
https://wolfram.com/xid/0e7p5nb2-kojx5p
Direct link to example
Out[8]=8
Copy to clipboard.
In[9]:=9
✖
https://wolfram.com/xid/0e7p5nb2-d19hlq
Direct link to example
Out[9]=9
Give all tabulated 3D embeddings of the cubical graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-digie3
Direct link to example
Out[1]=1
Show classes for each 3D embedding:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-uwao46
Direct link to example
Out[2]=2
Return all tabulated Polyhedron embeddings of the graph:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-exsos0
Direct link to example
Out[3]=3
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0e7p5nb2-mq4grk
Direct link to example
Out[4]=4
Return the cubical graph as a graph object:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-rv0spo
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-9dcw7x
Direct link to example
Out[2]=2
Return tabulated planar embeddings:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-hc9cwn
Direct link to example
Out[3]=3
Return the cubical graph as a 3D graph object:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-3uxvaf
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-xybi89
Direct link to example
Out[2]=2
Return tabulated polyhedron embeddings:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-bpgizo
Direct link to example
Out[3]=3
Return the cubical graph as a graphics object:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-b8pwrp
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-wpc6ss
Direct link to example
Out[2]=2
Return tabulated planar embeddings:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-mac8h5
Direct link to example
Out[3]=3
Return the cubical graph as a 3D graph object:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-sek860
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-w2aj6e
Direct link to example
Out[2]=2
Return tabulated polyhedron embeddings as graphics objects:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-zkxz4b
Direct link to example
Out[3]=3
Return an image of the primary embedding of a graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-hbymli
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-0avyel
Direct link to example
Out[2]=2
Give all cataloged embeddings of the octahedral graph:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-cv1fav
Direct link to example
Out[3]=3
Show the three-dimensional embedding of the cubical graph as an image:
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0e7p5nb2-nlol71
Direct link to example
Out[4]=4
Copy to clipboard.
In[5]:=5
✖
https://wolfram.com/xid/0e7p5nb2-glu44u
Direct link to example
Out[5]=5
Give all cataloged 3D embeddings of the cubical graph as images:
Copy to clipboard.
In[6]:=6
✖
https://wolfram.com/xid/0e7p5nb2-o9t89i
Direct link to example
Out[6]=6
Properties Representing Graph Polynomials (43)
Display the characteristic polynomial of the Coxeter graph as a pure function:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-jxmzys
Direct link to example
Out[1]=1
As a function of a variable x:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-rm2lat
Direct link to example
Compare with the directly computed value:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-ehbqzq
Direct link to example
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0e7p5nb2-ujf6s2
Direct link to example
Out[4]=4
Give the chordless cycle polynomial of the cubical graph as a pure function:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-earlge
Direct link to example
Out[1]=1
Extract tallies by cycle length:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-zt33tj
Direct link to example
Out[2]=2
Extract tallies by cycle length a different way:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-c051an
Direct link to example
Out[3]=3
Compare with the tallies of chordless cycle lengths:
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0e7p5nb2-lz803w
Direct link to example
Out[4]=4
Give the chromatic polynomial of the cubical graph as a pure function:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-kozy30
Direct link to example
Out[1]=1
As a function of a variable x:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-mu1ni5
Direct link to example
The chromatic polynomial is a special case of the rank polynomial:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-7ic48i
Direct link to example
Out[3]=3
Give the chromatic polynomial of the icosahedral graph in terms of a variable x:
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0e7p5nb2-y2jus9
Direct link to example
Give the clique polynomial of the utility graph as a pure function:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-6dax0u
Direct link to example
Out[1]=1
Compare to the explicit cliques:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-dwb361
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-t194ek
Direct link to example
Out[3]=3
Give the complement chordless cycle polynomial of the 
circulant graph as a pure function:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-zbbuw
Direct link to example
Out[1]=1
Extract tallies by cycle length:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-f7c6wr
Direct link to example
Out[2]=2
Extract tallies by cycle length a different way:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-29utrj
Direct link to example
Out[3]=3
Compare with the tallies of complement chordless cycle lengths:
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0e7p5nb2-qkw7j1
Direct link to example
Out[4]=4
Give the complement odd chordless cycle polynomial of the 
circulant graph as a pure function:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-ksxiu8
Direct link to example
Out[1]=1
Extract tallies by cycle length:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-bburdp
Direct link to example
Out[2]=2
Extract tallies by cycle length a different way:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-cgrqer
Direct link to example
Out[3]=3
Compare with the tallies of complement odd chordless cycle lengths:
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0e7p5nb2-kgw0rc
Direct link to example
Out[4]=4
Give the coboundary polynomial of the complete graph K5 as a pure function:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-cv4wa2
Direct link to example
Out[1]=1
As a function of a variables q and t:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-f3xbkg
Direct link to example
The coboundary polynomial is a special case of the Tutte polynomial:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-ik622d
Direct link to example
Give the connected domination polynomial of the fish graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-ployvu
Direct link to example
Out[1]=1
Compare with the connected dominating set count:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-7mgh8q
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-d0lwds
Direct link to example
Out[3]=3
Compare with connected dominating set tallies:
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0e7p5nb2-jww2rf
Direct link to example
Out[4]=4
Copy to clipboard.
In[5]:=5
✖
https://wolfram.com/xid/0e7p5nb2-pae8wt
Direct link to example
Out[5]=5
Give the connected induced subgraph polynomial of the Eiffel Tower graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-up5r24
Direct link to example
Out[1]=1
Compare with the count of connected induced subgraphs:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-jkrwmu
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-pshboi
Direct link to example
Out[3]=3
Give the cycle polynomial of the cubical graph as a pure function:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-o3vlfg
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-2bzt53
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-okqzdh
Direct link to example
Out[3]=3
Give the detour polynomial of the cubical graph as a pure function:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-ir2ibe
Direct link to example
Out[1]=1
As a function of a variable x:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-tux3ab
Direct link to example
Give the distance polynomial of the cubical graph as a pure function:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-l5yqsh
Direct link to example
Out[1]=1
As a function of a variable x:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-slm1r3
Direct link to example
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-rqmjm5
Direct link to example
Out[3]=3
Give the domination polynomial of the Coxeter graph as a pure function:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-j22uf7
Direct link to example
Out[1]=1
As a function of a variable x:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-srrmh9
Direct link to example
Give the edge cover polynomial of the utility graph as a pure function:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-e4w8qm
Direct link to example
Out[1]=1
As a function of a variable x:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-o8302v
Direct link to example
Out[2]=2
Compare to the explicit covers:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-bcb6m4
Direct link to example
Out[3]=3
Give the edge cut polynomial of the utility graph as a pure function:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-kggh31
Direct link to example
Out[1]=1
As a function of a variable x:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-74w9y2
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-kjk2he
Direct link to example
Out[3]=3
Give the flow polynomial of the cubical graph as a function of a variable u:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-enoc84
Direct link to example
The flow polynomial is a special case of the rank polynomial:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-hu1vdl
Direct link to example
Give the idiosyncratic polynomial of the cubical graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-4vj9hf
Direct link to example
Compare with a direct computation:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-mlq9yt
Direct link to example
Give the independence polynomial of the cubical graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-0cwj4c
Direct link to example
Give the irredundance polynomial of the cubical graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-8gs3on
Direct link to example
Compute from the irredundant sets:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-cuwogc
Direct link to example
Give the Laplacian polynomial of the cubical graph as a pure function:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-qnsizq
Direct link to example
Out[1]=1
As a function of a variable x:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-j0jupi
Direct link to example
Compute from Laplacian polynomial:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-frictb
Direct link to example
Out[3]=3
Give the matching generating polynomial of the cubical graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-q4iq25
Direct link to example
Give the matching polynomial of the cubical graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-6j4zmu
Direct link to example
Give the maximal clique polynomial of the house graph as a pure function:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-575ixy
Direct link to example
Out[1]=1
Extract tallies by maximal clique size:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-qc12vr
Direct link to example
Out[2]=2
Extract tallies by maximal clique size a different way:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-s6tpuz
Direct link to example
Out[3]=3
Compare with the tallies of maximal clique sizes:
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0e7p5nb2-h77u4o
Direct link to example
Out[4]=4
Give the maximal independence polynomial of the cricket graph as a pure function:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-bhxg3d
Direct link to example
Out[1]=1
Extract tallies by maximal independent vertex set size:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-wntcfi
Direct link to example
Out[2]=2
Extract tallies by maximal independent vertex set size a different way:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-so7mtw
Direct link to example
Out[3]=3
Compare with the tallies of maximal independent edge sets:
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0e7p5nb2-cqnoqd
Direct link to example
Out[4]=4
Give the maximal irredundance polynomial of the cricket graph as a pure function:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-zgi5hk
Direct link to example
Out[1]=1
Extract tallies by maximal irredundant set size:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-bj7ndy
Direct link to example
Out[2]=2
Extract tallies by maximal irredundant set size a different way:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-58aeyj
Direct link to example
Out[3]=3
Compare with the tallies of maximal irredundant sets:
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0e7p5nb2-484uuq
Direct link to example
Out[4]=4
Give the maximal matching generating polynomial of the cricket graph as a pure function:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-m33kzd
Direct link to example
Out[1]=1
Extract tallies by maximal independent edge set size:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-0v1ixh
Direct link to example
Out[2]=2
Extract tallies by maximal independent edge set size a different way:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-cg59w0
Direct link to example
Out[3]=3
Compare with the tallies of maximal independent edge sets:
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0e7p5nb2-b4cv49
Direct link to example
Out[4]=4
Give the minimal connected domination polynomial of the 5-wheel graph as a pure function:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-ij7odo
Direct link to example
Out[1]=1
Extract tallies by minimal connected dominating set size:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-odbmp4
Direct link to example
Out[2]=2
Extract tallies by minimal connected dominating set size a different way:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-l6b1kp
Direct link to example
Out[3]=3
Compare with the tallies of minimal connected dominating sets:
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0e7p5nb2-w09b9f
Direct link to example
Out[4]=4
Give the minimal domination polynomial of the cricket graph as a pure function:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-31chqo
Direct link to example
Out[1]=1
Extract tallies by minimal dominating set size:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-tyzl5y
Direct link to example
Out[2]=2
Extract tallies by minimal dominating set size a different way:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-4flbuv
Direct link to example
Out[3]=3
Compare with the tallies of minimal dominating sets:
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0e7p5nb2-2u62ms
Direct link to example
Out[4]=4
Give the minimal edge cover polynomial of the cricket graph as a pure function:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-ch2ii8
Direct link to example
Out[1]=1
Extract tallies by minimal edge cover size:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-mtq840
Direct link to example
Out[2]=2
Extract tallies by minimal edge cover size a different way:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-23zay8
Direct link to example
Out[3]=3
Compare with the tallies of minimal edge covers:
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0e7p5nb2-v1gkmc
Direct link to example
Out[4]=4
Give the minimal edge cut polynomial of the utility graph as a pure function:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-pzcb4e
Direct link to example
Out[1]=1
As a function of a variable x:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-c58zmz
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-6ykslw
Direct link to example
Out[3]=3
Give the minimal total domination polynomial of the butterfly graph as a pure function:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-7raqdv
Direct link to example
Out[1]=1
Extract tallies by total dominating set size:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-jk80qd
Direct link to example
Out[2]=2
Extract tallies by total dominating set size a different way:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-vaa53i
Direct link to example
Out[3]=3
Compare with the tallies of total dominating sets:
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0e7p5nb2-qo94qi
Direct link to example
Out[4]=4
Give the minimal vertex cover polynomial of the cricket graph as a pure function:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-7vh4kb
Direct link to example
Out[1]=1
Extract tallies by minimal vertex cover size:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-cw28ng
Direct link to example
Out[2]=2
Extract tallies by minimal vertex cover size a different way:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-6rss3d
Direct link to example
Out[3]=3
Compare with the tallies of minimal vertex covers:
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0e7p5nb2-pnizfq
Direct link to example
Out[4]=4
Give the minimal vertex cut polynomial of the utility graph as a pure function:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-9sry2x
Direct link to example
Out[1]=1
As a function of a variable x:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-n7k520
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-6498ou
Direct link to example
Out[3]=3
Give the odd chordless cycle polynomial of the Golomb graph as a pure function:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-qf3dii
Direct link to example
Out[1]=1
Extract tallies by odd chordless cycle lengths:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-6g8e87
Direct link to example
Out[2]=2
Extract tallies by odd chordless cycle lengths in a different way:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-c4si3b
Direct link to example
Out[3]=3
Compare with the tallies of odd chordless cycle lengths:
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0e7p5nb2-uvwmme
Direct link to example
Out[4]=4
Give the path polynomial of the cubical graph as a pure function:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-jvq1t4
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-ff579q
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-ogkydb
Direct link to example
Out[3]=3
Give the Q-chromatic polynomial of the Chvátal graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-rlz0k1
Direct link to example
Out[1]=1
Give the rank polynomial of the cubical graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-62c8x8
Direct link to example
Give the reliability polynomial of the cubical graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-wq45k6
Direct link to example
The reliability polynomial is a special case of the Tutte polynomial:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-zudrs8
Direct link to example
Give the sigma polynomial of the cubical graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-1sa94v
Direct link to example
Give the total domination polynomial of the 5-wheel graph as a pure function:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-76k8sf
Direct link to example
Out[1]=1
Extract tallies by total dominating set size:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-pin01
Direct link to example
Out[2]=2
Extract tallies by total dominating set size a different way:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-dmw259
Direct link to example
Out[3]=3
Compare with the tallies of total dominating sets:
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0e7p5nb2-i67c1e
Direct link to example
Out[4]=4
Give the Tutte polynomial of the cubical graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-isbl76
Direct link to example
The Tutte polynomial is a special case of the rank polynomial:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-ewscza
Direct link to example
Give the vertex cover polynomial of the utility graph as a pure function:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-pn1ou2
Direct link to example
Out[1]=1
Compare to the explicit covers:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-ernh1h
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-h03t4w
Direct link to example
Out[3]=3
Give the vertex cut polynomial of the utility graph as a pure function:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-nj7i7h
Direct link to example
Out[1]=1
As a function of a variable x:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-udya5e
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-e6q265
Direct link to example
Out[3]=3
Coloring-Related Graph Properties (13)
Give the chromatic invariant cubical graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-23tfzt
Direct link to example
Out[1]=1
Give the chromatic number of the cubical graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-cx8fan
Direct link to example
Out[1]=1
Compare with the built-in function:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-6lzjn
Direct link to example
Out[2]=2
Visualize the chromatic number:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-3r6ux9
Direct link to example
Out[3]=3
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0e7p5nb2-sgvei4
Direct link to example
Out[4]=4
Give the edge chromatic number of the cubical graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-8jaeqg
Direct link to example
Out[1]=1
Compare with the built-in function:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-44zawc
Direct link to example
Out[2]=2
Visualize the edge chromatic number:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-qsmqm3
Direct link to example
Out[3]=3
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0e7p5nb2-3tfmv9
Direct link to example
Out[4]=4
Give the fractional chromatic numbers of the Mycielski graphs:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-knxsm7
Direct link to example
Out[1]=1
Compare with the closed form known for this fractional chromatic number:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-3z3r70
Direct link to example
Out[2]=2
Return the chromatic roots of the first Harvey–Royle graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-jhzwlt
Direct link to example
Out[1]=1
Verify equality with the roots of the chromatic polynomial:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-t6odgc
Direct link to example
Out[2]=2
Return the cyclic chromatic number of of 2-Plummer–Toft graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-g7d7d4
Direct link to example
Out[1]=1
Give the fractional edge chromatic numbers of the flower snark 
:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-9r93k4
Direct link to example
Out[1]=1
Give a minimum edge coloring of the octahedral graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-gjw16l
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-bhcret
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-odrbau
Direct link to example
Out[3]=3
Give the number of minimum vertex colorings of the octahedral graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-jotl5m
Direct link to example
Out[1]=1
Do the same using an annotation:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-6vmdoj
Direct link to example
Out[2]=2
Compare with the number of actual colorings:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-8fqwkx
Direct link to example
Out[3]=3
Give the minimum vertex colorings of the octahedral graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-mspech
Direct link to example
Out[1]=1
Verify the colorings are consistent with the chromatic number:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-vm3wh4
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-89nteg
Direct link to example
Out[3]=3
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0e7p5nb2-c7lwah
Direct link to example
Out[4]=4
Show a minimum-weight fractional coloring of the Petersen graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-7mvdeu
Direct link to example
Out[1]=1
Compute the fractional chromatic number:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-rmy4um
Direct link to example
Out[2]=2
Compare with the direct property:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-jj3ibu
Direct link to example
Out[3]=3
Give the Q-chromatic polynomial of the Chvátal graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-heuc2s
Direct link to example
Out[1]=1
The Q-chromatic polynomial has smaller degree and coefficients than the usual chromatic polynomial for graphs with chromatic number at least 3:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-haxg9r
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-38a29f
Direct link to example
Out[3]=3
The Q-chromatic polynomial is not defined for graphs with chromatic number less than 3:
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0e7p5nb2-c2h6e5
Direct link to example
Out[4]=4
Copy to clipboard.
In[5]:=5
✖
https://wolfram.com/xid/0e7p5nb2-ek9k7o
Direct link to example
Out[5]=5
Give the Weisfeiler–Leman dimension of the tetrahedral graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-u4hc6a
Direct link to example
Out[1]=1
Graph Index Properties (18)
Give the ABC index of the butane graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-3hy5ws
Direct link to example
Out[1]=1
Compare with the sum of ABC matrix elements:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-7yjrhz
Direct link to example
Out[2]=2
Give the arithmetic-geometric index of the propane graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-czyfiv
Direct link to example
Out[1]=1
Compare with the sum of arithmetic-geometric matrix elements:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-4evd5r
Direct link to example
Out[2]=2
Display the Balaban index of the Coxeter graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-0zk564
Direct link to example
Out[1]=1
Give the Balaban index of the isobutane graph:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-ji1im0
Direct link to example
Out[2]=2
Give the circuit rank of the Coxeter graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-wnl1zx
Direct link to example
Out[1]=1
Display the circuit rank of the icosahedral graph:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-e3upow
Direct link to example
Out[2]=2
Compare with the value obtained from other properties:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-lsofe7
Direct link to example
Out[3]=3
Give the detour index of the cubical graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-5n9itd
Direct link to example
Out[1]=1
Give the Harary index of the cubical graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-fpkd9q
Direct link to example
Out[1]=1
Give the Hosoya index of the cubical graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-5a91ap
Direct link to example
Out[1]=1
The Hosoya index is identical to the independent edge set count:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-copbc4
Direct link to example
Out[2]=2
Give the Kirchhoff index of the cubical graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-lf9sfs
Direct link to example
Out[1]=1
Give the Kirchhoff index of the isobutane graph:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-ckhb2n
Direct link to example
Out[2]=2
Give the Kirchhoff sum index of the cubical graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-mcgbgq
Direct link to example
Out[1]=1
Give the Kirchhoff sum index of the isobutane graph:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-2cvpb2
Direct link to example
Out[2]=2
Give the molecular topological index of the cubical graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-joqrhs
Direct link to example
Out[1]=1
Give the stability index of the cubical graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-c8cg7k
Direct link to example
Out[1]=1
Give the Randić index of the octahedral graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-x2yrkp
Direct link to example
Out[1]=1
Compare with the sum of Randić matrix elements:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-wg2l6z
Direct link to example
Out[2]=2
Give the Sombor index of the octahedral graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-rhwhv8
Direct link to example
Out[1]=1
Compare with the sum of Sombor matrix elements:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-t1cw8n
Direct link to example
Out[2]=2
Give the topological index of the cubical graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-kutyqx
Direct link to example
Out[1]=1
Give the Wiener index of the cubical graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-setrl4
Direct link to example
Out[1]=1
Give the Wiener index of the isobutane graph:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-uocnq4
Direct link to example
Out[2]=2
Give the Wiener sum index of the cubical graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-f7vqvz
Direct link to example
Out[1]=1
Give the Wiener sum index of the isobutane graph:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-ibpsdu
Direct link to example
Out[2]=2
Give the first Zagreb index of the cubic graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-8gmbyg
Direct link to example
Out[1]=1
The first Zagreb index is defined as the sum of squared vertex degrees:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-1mpw2c
Direct link to example
Out[2]=2
Give the second Zagreb index of the cubic graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-9pvrct
Direct link to example
Out[1]=1
The second Zagreb index is defined as the sum of squared vertex degrees:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-7itn0p
Direct link to example
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-9ji174
Direct link to example
Out[3]=3
Matrix Graph Properties (12)
Give the ABC matrix of the propane graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-omxnxj
Direct link to example
Give the arithmetic-geometric matrix of the propane graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-06a30f
Direct link to example
Give the adjacency matrix, returned as a SparseArray object:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-5cvaaa
Direct link to example
Out[1]=1
Convert to an explicit matrix:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-5s79cr
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-gqhbnr
Direct link to example
Plot the matrix using ArrayPlot:
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0e7p5nb2-ofhaou
Direct link to example
Out[4]=4
Verify that the positions of 1s in the adjacency matrix correspond to graph edges:
Copy to clipboard.
In[5]:=5
✖
https://wolfram.com/xid/0e7p5nb2-d9nqbk
Direct link to example
Out[5]=5
Copy to clipboard.
In[6]:=6
✖
https://wolfram.com/xid/0e7p5nb2-g1op4a
Direct link to example
Out[6]=6
Copy to clipboard.
In[7]:=7
✖
https://wolfram.com/xid/0e7p5nb2-d8uid4
Direct link to example
Out[7]=7
Give the detour matrix of the cubical graph:
Copy to clipboard.
In[8]:=8
✖
https://wolfram.com/xid/0e7p5nb2-m17h8a
Direct link to example
Return the distance matrix of the octahedral graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-6lri5i
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-8h0fnx
Direct link to example
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-wbzuy8
Direct link to example
Out[3]=3
Give the incidence matrix of the octahedral graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-r60me3
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-ralb2s
Direct link to example
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-bbgtbq
Direct link to example
Out[3]=3
Give the Laplacian matrix of the octahedral graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-vd5q3
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-1h820q
Direct link to example
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-1gxfly
Direct link to example
Out[3]=3
Show the maximum flow matrix for the E graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-tm0urf
Direct link to example
Show the minimum cost flow matrix for the E graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-d7rgtn
Direct link to example
Give the normalized Laplacian matrix of the octahedral graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-8c6owr
Direct link to example
Out[1]=1
Give the matrix in expanded form:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-fks4un
Direct link to example
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-cpqg98
Direct link to example
Out[3]=3
Give the Randić matrix of the propane graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-q86ta4
Direct link to example
Give the resistance matrix of the cubical graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-jszi3y
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-nftkcu
Direct link to example
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-dyryyz
Direct link to example
Out[3]=3
Give the Sombor matrix of the propane graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-ix74n8
Direct link to example
Local Graph Properties (6)
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-evtx3c
Direct link to example
List the bridges of the Walther graph:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-gqqyh0
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-h294fu
Direct link to example
Out[3]=3
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0e7p5nb2-qe4khl
Direct link to example
Out[4]=4
Do the same using the "Count" annotation:
Copy to clipboard.
In[5]:=5
✖
https://wolfram.com/xid/0e7p5nb2-sikmwl
Direct link to example
Out[5]=5
Give the number of chords in the house graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-s8eioh
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-svmy9e
Direct link to example
Out[2]=2
List the chords of the house graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-xdwei8
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-5hls5f
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-oq087t
Direct link to example
Out[3]=3
Do the same using the "Count" annotation:
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0e7p5nb2-1vuzah
Direct link to example
Out[4]=4
Display the curvatures of the A graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-lae6ts
Direct link to example
Out[1]=1
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-w4t2r6
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-t0622n
Direct link to example
Out[2]=2
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-c43wnc
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-181nnn
Direct link to example
Out[2]=2
Show number of leaves directly:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-0bah5g
Direct link to example
Out[3]=3
Do the same using the "Count" annotation:
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0e7p5nb2-07z4bg
Direct link to example
Out[4]=4
Copy to clipboard.
In[5]:=5
✖
https://wolfram.com/xid/0e7p5nb2-li09kt
Direct link to example
Out[5]=5
Global Graph Properties (29)
Show the anarboricity of the cycle graph 
:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-7s8516
Direct link to example
Out[1]=1
Give the number of apices of the Wagner graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-y69a9e
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-p8qco
Direct link to example
Out[2]=2
Compare with the length of the list of apices:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-z7lzye
Direct link to example
Out[3]=3
List named connected (nonplanar) apex graphs:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-oxm9up
Direct link to example
Out[1]=1
Find all apices of the Wagner graph:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-h5l64s
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-6dujd3
Direct link to example
Out[3]=3
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0e7p5nb2-ntw4we
Direct link to example
Out[4]=4
Delete apices and verify planarity of the resulting graphs:
Copy to clipboard.
In[5]:=5
✖
https://wolfram.com/xid/0e7p5nb2-bwvq2l
Direct link to example
Out[5]=5
Copy to clipboard.
In[6]:=6
✖
https://wolfram.com/xid/0e7p5nb2-jsm53u
Direct link to example
Out[6]=6
Show the arboricity of the complete graphs:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-v9r97b
Direct link to example
Out[1]=1
Compare with the value form theory:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-srd713
Direct link to example
Out[2]=2
Give the number of articulation vertices in the A-graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-9mow9m
Direct link to example
Out[1]=1
Compare with the list of articulation vertices:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-lfj85l
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-0levas
Direct link to example
Out[3]=3
Find graphs having articulation vertices:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-60xmkd
Direct link to example
Out[1]=1
Give the indices of the articulation vertices of the isobutane graph:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-1p7iyy
Direct link to example
Out[2]=2
Visualize the articulation vertices:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-qojtoo
Direct link to example
Out[3]=3
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-ivav62
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-zyxgjw
Direct link to example
Out[2]=2
Display the circumference of the icosahedral graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-mzk35b
Direct link to example
Out[1]=1
Verify that this corresponds to the length of the longest cycle:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-rj9x0h
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-zo3fvz
Direct link to example
Out[3]=3
Display the corank of the icosahedral graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-lq81kj
Direct link to example
Out[1]=1
Compute the corank from other graph properties:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-9f9sgw
Direct link to example
Out[2]=2
Give the graphs on four or fewer vertices that are determined by resistance:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-kva5l6
Direct link to example
Out[1]=1
Check if the cubical graph is determined by spectrum:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-jvt0mj
Direct link to example
Out[1]=1
Check if the tesseract graph is determined by spectrum:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-2icxyd
Direct link to example
Out[2]=2
Give the names of the graphs with the same spectrum as the tesseract graph:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-f6z61g
Direct link to example
Out[3]=3
Give the diameter of the Pappus graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-73rwyy
Direct link to example
Out[1]=1
Give the eccentricities of the Pappus graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-jijkyn
Direct link to example
Out[1]=1
Give the intersection array of the cubical graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-8gkn2k
Direct link to example
Out[1]=1
Display the maximum leaf number of the A graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-cm2jy2
Direct link to example
Out[1]=1
The maximum leaf number is the maximum possible leaf count of a spanning tree:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-hc2x86
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-qq4a39
Direct link to example
Out[3]=3
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0e7p5nb2-zwi15l
Direct link to example
Out[4]=4
Give the maximum vertex degree of the E graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-fpveeq
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-clc4tb
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-71tpz
Direct link to example
Out[3]=3
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0e7p5nb2-f23zgg
Direct link to example
Out[4]=4
Copy to clipboard.
In[5]:=5
✖
https://wolfram.com/xid/0e7p5nb2-ldu3hf
Direct link to example
Out[5]=5
Copy to clipboard.
In[6]:=6
✖
https://wolfram.com/xid/0e7p5nb2-smwl39
Direct link to example
Out[6]=6
Display the mean curvature of the A graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-xonkhc
Direct link to example
Out[1]=1
The mean curvature is the mean of the curvatures associated with individual vertices:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-bry1rd
Direct link to example
Out[2]=2
Display the mean distance of the Petersen graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-lluo1c
Direct link to example
Out[1]=1
Display the minimum leaf number of the A graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-lgk1s3
Direct link to example
Out[1]=1
The maximum leaf number is the minimum possible leaf count of a spanning tree:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-0ngn7
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-ok3qym
Direct link to example
Out[3]=3
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0e7p5nb2-vemzxz
Direct link to example
Out[4]=4
Give the minimum vertex degree of the E graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-t5ccoo
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-irx2iv
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-p8n731
Direct link to example
Out[3]=3
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0e7p5nb2-xxh9u7
Direct link to example
Out[4]=4
Copy to clipboard.
In[5]:=5
✖
https://wolfram.com/xid/0e7p5nb2-kwx2jw
Direct link to example
Out[5]=5
Copy to clipboard.
In[6]:=6
✖
https://wolfram.com/xid/0e7p5nb2-j9lbai
Direct link to example
Out[6]=6
Display the periphery of a tree:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-6ehx8i
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-tolk1q
Direct link to example
Out[2]=2
Display the quadratic embedding constant of the icosahedral graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-5nahgn
Direct link to example
Out[1]=1
For graphs whose distance matrices have constant row sums, the quadratic embedding constant equals the second largest eigenvalue of the distance matrix:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-meg5yl
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-nfufq8
Direct link to example
Out[3]=3
Display the rank of the icosahedral graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-3eh2zy
Direct link to example
Out[1]=1
Compute the rank from other graph properties:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-7w6eas
Direct link to example
Out[2]=2
Display the regular parameters of the cubical graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-mdkqrb
Direct link to example
Out[1]=1
Display the skewness of the tesseract graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-okiqrm
Direct link to example
Out[1]=1
Compare with the result from theory:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-virmf5
Direct link to example
Out[2]=2
Display the number of spanning trees in the 120-cell graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-opa520
Direct link to example
Out[1]=1
Show the triameter of the Coxeter graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-72pe5
Direct link to example
Out[1]=1
Compare with the directly computed value:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-krsq6p
Direct link to example
Out[2]=2
Show the vertex degrees of the claw graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-nm3izs
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-xm3nu2
Direct link to example
Out[2]=2
Give the toroidal crossing numbers for complete graphs:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-ni27g0
Direct link to example
Out[1]=1
Spectral Graph Properties (13)
Give the ABC energy of isobutane:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-ndnl03
Direct link to example
Out[1]=1
Compare with the sum of absolute values of the eigenvalue of the ABC matrix:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-twgs1y
Direct link to example
Out[2]=2
Give the ABC spectral radius of isobutane:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-tn3ghm
Direct link to example
Out[1]=1
Compare with the largest eigenvalue of the ABC matrix:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-y4m1at
Direct link to example
Out[2]=2
Give the algebraic connectivity of isobutane:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-mtgbhb
Direct link to example
Out[1]=1
Algebraic connectivity is defined as the second smallest member of the Laplacian spectrum:
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In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-5adw8e
Direct link to example
Out[2]=2
Compare with the second smallest eigenvalue of the Laplacian matrix:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-eay39l
Direct link to example
Out[3]=3
Give the arithmetic-geometric energy of methane:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-bu0gdn
Direct link to example
Out[1]=1
Compare with the sum of absolute values of the eigenvalue of the arithmetic-geometric matrix:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-0989g9
Direct link to example
Out[2]=2
Give the arithmetic-geometric spectral radius of isobutane:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-kn1of
Direct link to example
Out[1]=1
Compare with the largest eigenvalue of the arithmetic-geometric matrix:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-b2pxtk
Direct link to example
Out[2]=2
Display the Laplacian spectral radius of the tesseract graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-hfcwpv
Direct link to example
Out[1]=1
The Laplacian spectral radius is the largest eigenvalue of the Kirchhoff matrix:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-joy06l
Direct link to example
Out[2]=2
Display the Laplacian spectral ratio of the tesseract graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-pq5ifg
Direct link to example
Out[1]=1
The Laplacian spectral ratio is defined as the ratio of the Laplacian spectral radius to the algebraic connectivity:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-pqj4s8
Direct link to example
Out[2]=2
Display the Laplacian spectrum of the 600-cell graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-uae0yo
Direct link to example
Out[1]=1
Display a nicely formatted version:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-muhysq
Direct link to example
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-q9f68y
Direct link to example
Give the Randić energy of isobutane:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-hn2r7v
Direct link to example
Out[1]=1
Compare with the sum of absolute values of the eigenvalue of the Randić matrix:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-ygbgtp
Direct link to example
Out[2]=2
Give the Sombor energy of isobutane:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-f5uuo2
Direct link to example
Out[1]=1
Compare with the sum of absolute values of the eigenvalue of the Sombor matrix:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-ft4ffk
Direct link to example
Out[2]=2
Give the Sombor spectral radius of isobutane:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-bysqb9
Direct link to example
Out[1]=1
Compare with the largest eigenvalue of the Sombor matrix:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-vy6xpx
Direct link to example
Out[2]=2
Give the spectral radius of the 600-cell graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-5xt0n4
Direct link to example
Out[1]=1
Compare with the graph spectrum:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-245xpw
Direct link to example
Out[2]=2
Display the spectrum of the 600-cell graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-2vws8w
Direct link to example
Out[1]=1
Display a nicely formatted version:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-dik3to
Direct link to example
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-mk8tmi
Direct link to example
Labeled Graph Properties (11)
Return the average disorder number of the cubical graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-lgdwy4
Direct link to example
Out[1]=1
Compare with the Wiener index:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-f80z64
Direct link to example
Out[2]=2
Return the disorder number of the cubical graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-3defdv
Direct link to example
Out[1]=1
Return the Erdős sequence of the 2-triangular grid graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-sms4ak
Direct link to example
Out[1]=1
Construct the corresponding Erdős graph:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-r594l6
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-gudj98
Direct link to example
Out[3]=3
Return the number of fundamentally distinct graceful labelings for the tetrahedral graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-0a6x7v
Direct link to example
Out[1]=1
Do the same using the "Count" annotation:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-u788d5
Direct link to example
Out[2]=2
Compare with the number of distinct labelings:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-3ps1jo
Direct link to example
Out[3]=3
Show the fundamentally distinct graceful labelings of the tetrahedral graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-glwl6v
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-ko1za1
Direct link to example
Out[2]=2
Verify the labeling is graceful:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-rxd2qc
Direct link to example
Out[3]=3
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0e7p5nb2-oev81e
Direct link to example
Out[4]=4
Give the irregularity strength of the 5-star graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-kd9fje
Direct link to example
Out[1]=1
Return the connected graphs for which the irregularity strength equals one less than the vertex count:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-nucg2b
Direct link to example
Out[2]=2
Give the number of pinnacle sets in the banner graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-u19bv4
Direct link to example
Out[1]=1
Do the same with the "Count" annotation:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-f60dz0
Direct link to example
Out[2]=2
Compare with the length of actual sets:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-g4gu9q
Direct link to example
Out[3]=3
Return the pinnacle sets of the banner graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-xb9xoj
Direct link to example
Out[1]=1
Show labelings of the graph corresponding to these pinnacle sets:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-gsu7qo
Direct link to example
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-z9jfrr
Direct link to example
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0e7p5nb2-6soebt
Direct link to example
Out[4]=4
Show the fundamentally distinct optimal radio labelings of the 5-path graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-z5yyr0
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-m7b9tx
Direct link to example
Out[2]=2
Return the number of fundamentally distinct optimal radio labelings for the 5-path graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-q4acuu
Direct link to example
Out[1]=1
Do the same using the "Count" annotation:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-zp6r4d
Direct link to example
Out[2]=2
Compare with the number of distinct optimal labelings:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-la530j
Direct link to example
Out[3]=3
Give the radio number of the 5-path graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-e97b1b
Direct link to example
Out[1]=1
Compare again the maximum label of all distinct optimal radio labelings:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-re4zza
Direct link to example
Out[2]=2
Graph Construction Properties (2)
Display the assembly number of the pentatope graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-9sityi
Direct link to example
Out[1]=1
Display the construction number of the pentatope graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-wrbzln
Direct link to example
Out[1]=1
Topological Graph Properties (10)
Display the crossing number of the icosahedral graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-oo0qhz
Direct link to example
Out[1]=1
The crossing number is 0 since the graph is planar:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-o78av6
Direct link to example
Out[2]=2
Return the dimension of the cubical graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-y0inur
Direct link to example
Out[1]=1
Show unit-distance embeddings in the plane corresponding to dimension 2:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-2a2t0j
Direct link to example
Out[2]=2
Give the genus of the cubical graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-pr8x2h
Direct link to example
Out[1]=1
Give the genus of the unique graph that triangulates the torus and Klein bottle:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-poisa1
Direct link to example
Out[2]=2
Give the Klein bottle crossing number of the unique graph that triangulates the torus and Klein bottle:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-8gdgl3
Direct link to example
Out[1]=1
Display the local crossing number of the complete graph K6:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-ppw5s9
Direct link to example
Out[1]=1
Display embeddings with this number of crossings:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-iicu4v
Direct link to example
Out[2]=2
The crossed dodecahedral graph has a unique 2-planar embedding, but does not admit a straight-line 2-planar drawing:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-pel3ib
Direct link to example
Out[3]=3
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0e7p5nb2-v9mkkx
Direct link to example
Out[4]=4
Return the metric dimension of the cubical graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-zxahvi
Direct link to example
Out[1]=1
Give the projective plane crossing numbers of the complete bipartite graphs 
:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-fqapa6
Direct link to example
Out[1]=1
Compare with the known closed-form values:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-9sdt7a
Direct link to example
Out[2]=2
Give the rectilinear crossing numbers for complete graphs:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-b3f8fu
Direct link to example
Out[1]=1
Display the rectilinear local crossing number of the complete graph K6:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-slzabs
Direct link to example
Out[1]=1
Display embeddings with this number of rectilinear crossings:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-y3jnf4
Direct link to example
Out[2]=2
Give the toroidal crossing number of the unique graph that triangulates the torus and Klein bottle:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-9hktx0
Direct link to example
Out[1]=1
Clique‐Related Graph Properties (15)
Return the clique count of the octahedral graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-z3re74
Direct link to example
Out[1]=1
Compare with the explicit listing of cliques:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-ofxck
Direct link to example
Out[2]=2
Return the clique number of the icosahedral graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-y0qtz7
Direct link to example
Out[1]=1
Compare with the lengths of the maximum cliques:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-pf4fh8
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-n6w51r
Direct link to example
Out[3]=3
Give the clique polynomial of the utility graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-zmqsvr
Direct link to example
Out[1]=1
Compare to the explicit cliques:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-sryxht
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-wa5uhl
Direct link to example
Out[3]=3
Return the cliques of the octahedral graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-rvh26z
Direct link to example
Out[1]=1
Return the Delsarte cliques of the octahedral graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-106tw9
Direct link to example
Out[1]=1
Compare with number of cliques directly:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-6u3g2s
Direct link to example
Out[2]=2
Return the Delsarte cliques of the octahedral graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-fivzbe
Direct link to example
Out[1]=1
Verify the graph is distance-regular:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-3x9tui
Direct link to example
Out[2]=2
Verify these cliques achieve the Delsarte bound:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-e3a1px
Direct link to example
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0e7p5nb2-fluu2u
Direct link to example
Out[4]=4
Return the fractional clique number of the Heawood graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-x13jlm
Direct link to example
Out[1]=1
Return the lower clique number of the Krackhardt kite graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-h5spn2
Direct link to example
Out[1]=1
Compare with the size of the smallest maximal clique:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-lhc6tj
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-m1zirm
Direct link to example
Out[3]=3
Return the number of maximal cliques of the cubical graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-imusgd
Direct link to example
Out[1]=1
Compare with the number of maximal cliques:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-lz45g4
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-za927d
Direct link to example
Out[3]=3
Return the maximal clique polynomial of the cubical graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-y1qkvj
Direct link to example
Out[1]=1
Return the maximal cliques of the bull graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-s7756q
Direct link to example
Out[1]=1
Return the maximum cliques of the bull graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-gb8yqt
Direct link to example
Out[1]=1
Give the smallest covers of the cubical graph by maximal cliques:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-8sksjr
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-2bdsml
Direct link to example
Out[2]=2
Give the number of smallest covers of the cubical graph by maximal cliques:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-rd7lj5
Direct link to example
Out[1]=1
Give the name of the simplex graph of the cycle graph C5, i.e. the graph with vertices given by the cliques of the base graph and edges between pairs of cliques that differ by insertion/deletion of exactly one vertex:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-822lvi
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-rzt2qn
Direct link to example
Out[2]=2
Cover‐Related Graph Properties (20)
Give the biclique cover of the cubical graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-bmdqlo
Direct link to example
Out[1]=1
Return the clique covering number of the icosahedral graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-tlj0wu
Direct link to example
Out[1]=1
Verify it is the chromatic number of the graph complement:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-nh51go
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-k73l6b
Direct link to example
Out[3]=3
Give the number of edge covers of the cubical graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-bolf89
Direct link to example
Out[1]=1
Compare to the length of the list of edge covers:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-7wyx6a
Direct link to example
Out[2]=2
Return the edge cover number of the icosahedral graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-ugadmm
Direct link to example
Out[1]=1
Compare with the length of a minimum edge cover:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-861k0n
Direct link to example
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-dk8dnr
Direct link to example
Out[3]=3
Give the edge cover polynomial of the utility graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-z6ep64
Direct link to example
Out[1]=1
Compare to the explicit covers:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-2b5cgf
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-s1qvuy
Direct link to example
Out[3]=3
Give the edge covers of the utility graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-rpd5g
Direct link to example
Return the number of minimal edge covers for the cubical graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-vbz4o3
Direct link to example
Out[1]=1
Compare to the length of the list:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-5dbcww
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-5yrucm
Direct link to example
Out[3]=3
Return the minimal edge cover polynomial of the tetrahedral graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-o7vawk
Direct link to example
Out[1]=1
Compute from the minimal edge covers:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-3rx269
Direct link to example
Out[2]=2
Return the minimal edge covers of the tetrahedral graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-hfmvsx
Direct link to example
Out[1]=1
Return the number of minimal vertex covers for the cubical graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-7hn00d
Direct link to example
Out[1]=1
Compare to the length of the list:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-38zvgn
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-mlslt
Direct link to example
Out[3]=3
Return the minimal vertex cover polynomial of the tetrahedral graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-d2ajj
Direct link to example
Out[1]=1
Compute from the minimal vertex covers:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-vg4vnl
Direct link to example
Out[2]=2
Return the minimal vertex covers of the tetrahedral graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-slyqjh
Direct link to example
Out[1]=1
Return the number of minimum clique coverings for the cubical graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-kerpnj
Direct link to example
Out[1]=1
Compare to the length of the list:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-ole4s1
Direct link to example
Out[2]=2
Give the number of minimum path coverings of the 4-star graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-1lo3v8
Direct link to example
Out[1]=1
Return using a "Count" qualifier:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-2hdg1n
Direct link to example
Out[2]=2
Compare with minimum path coverings:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-2wrcyx
Direct link to example
Out[3]=3
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0e7p5nb2-ja7ds8
Direct link to example
Out[4]=4
Show the minimum path coverings of the 4-star graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-r0zumf
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-60tr7w
Direct link to example
Out[2]=2
Give the path covering number of the cross graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-8xxw5g
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-hk9nsb
Direct link to example
Out[2]=2
Append "paths" of length 0 to the paths on the graph:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-5livlk
Direct link to example
Out[3]=3
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0e7p5nb2-1r3gzk
Direct link to example
Out[4]=4
The path covering number is the smallest number of vertex-disjoint paths that cover the given graph:
Copy to clipboard.
In[5]:=5
✖
https://wolfram.com/xid/0e7p5nb2-2xqy81
Direct link to example
Out[5]=5
Compare with the minimum path coverings:
Copy to clipboard.
In[6]:=6
✖
https://wolfram.com/xid/0e7p5nb2-w6z0tl
Direct link to example
Out[6]=6
Give the number of vertex covers of the cubical graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-sow338
Direct link to example
Out[1]=1
Compare to the length of the list of edge covers:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-v0zp3c
Direct link to example
Out[2]=2
Return the vertex cover number:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-jbjjyu
Direct link to example
Out[1]=1
Compare with the size of a minimum vertex cover:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-oqq0d7
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-s7yzng
Direct link to example
Out[3]=3
Give the vertex cover polynomial of the utility graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-y58vrs
Direct link to example
Out[1]=1
Compare to the explicit covers:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-cmdes9
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-mkug58
Direct link to example
Out[3]=3
Give the vertex covers of the cubical graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-rhw8ye
Direct link to example
Out[1]=1
Independent Set‐Related Graph Properties (27)
Give the biclique cover of the cubical graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-5q6rbt
Direct link to example
Out[1]=1
Give the bipartite dimension of the cubical graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-u3dgm9
Direct link to example
Out[1]=1
Return the fractional independence number of the Petersen graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-3rhpso
Direct link to example
Out[1]=1
Compare with the usual independence number:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-5yug29
Direct link to example
Out[2]=2
Return the independence number of the Heawood graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-dbxbbk
Direct link to example
Out[1]=1
Compare with the size of a maximum independent vertex set:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-jm9un0
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-r74urp
Direct link to example
Out[3]=3
Return the independence polynomial of the utility graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-w9nkd6
Direct link to example
Out[1]=1
Compare to the explicit independent sets:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-j8ymnc
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-u6qelq
Direct link to example
Out[3]=3
Return the independence ratio of the Petersen graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-m9u78m
Direct link to example
Out[1]=1
Compare to the direct ratio of properties:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-45idot
Direct link to example
Out[2]=2
Return the number of independent edge sets of the Heawood graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-nhngjk
Direct link to example
Out[1]=1
Compare to the explicit independent sets:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-ee6ksa
Direct link to example
Out[2]=2
Return the independent edge sets of the utility graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-9oxht2
Direct link to example
Out[1]=1
Return the number of independent vertex sets of the cubical graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-6fl7k5
Direct link to example
Out[1]=1
Compare to the explicit independent sets:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-06fnbn
Direct link to example
Out[2]=2
Return the independent vertex sets of the Heawood graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-x7jk4x
Direct link to example
Out[1]=1
Return the intersection number of the Heawood graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-1k5ps9
Direct link to example
Out[1]=1
Give the lower independence number of the Petersen graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-y79ck4
Direct link to example
Out[1]=1
Compare with the smallest exponent of the maximal independence polynomial:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-ebre87
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-ceekk8
Direct link to example
Out[3]=3
Give the lower matching number of the Coxeter graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-iptg4c
Direct link to example
Out[1]=1
Compare with the smallest exponent of the maximal matching-generating polynomial:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-desagz
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-icsnuq
Direct link to example
Out[3]=3
Return the matching-generating polynomial of the utility graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-2r7nec
Direct link to example
Out[1]=1
Compare to the explicit independent edge sets:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-eakoig
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-fr362r
Direct link to example
Out[3]=3
Return the matching number of the cubical graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-t0v2j2
Direct link to example
Out[1]=1
Compare with the matching-generating polynomial:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-o46evj
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-jvh9hq
Direct link to example
Out[3]=3
Return the maximal independence polynomial for the Heawood graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-vatz1c
Direct link to example
Out[1]=1
Compare with the numbers of maximal independent vertex sets:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-4mm9xi
Direct link to example
Out[2]=2
Return the number of maximal independent edge sets for the cubical graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-g0j7xx
Direct link to example
Out[1]=1
Compare with the number of maximal independent edge sets:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-mknlfs
Direct link to example
Out[2]=2
Return the maximal independent edge sets for the cubical graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-tmtajw
Direct link to example
Out[1]=1
Return the maximal matching-generating polynomial for the Heawood graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-d607ij
Direct link to example
Out[1]=1
Compare with the explicit edge sets:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-cekji6
Direct link to example
Out[2]=2
Return the number of maximal independent vertex sets for the Heawood graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-mav14e
Direct link to example
Out[1]=1
Compare with the number of maximal independent vertex sets:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-rhtllx
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-4bancl
Direct link to example
Out[3]=3
Return the maximal independent vertex sets for the cubical graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-z7tfqv
Direct link to example
Out[1]=1
Return the maximal independence polynomial for the cubical graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-7uscrn
Direct link to example
Out[1]=1
Compare with the explicit edge sets:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-oohtxn
Direct link to example
Out[2]=2
Return the maximal matching generating polynomial for the cubical graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-tc8vrx
Direct link to example
Out[1]=1
Compare with the numbers of maximal independent edge sets:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-jdblie
Direct link to example
Out[2]=2
Return the number of maximum independent edge sets for the Heawood graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-qgduic
Direct link to example
Out[1]=1
Do the same with the "Count" annotation:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-g2aa8o
Direct link to example
Out[2]=2
Return the maximum independent edge sets for the Heawood graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-cgb7a7
Direct link to example
Return the number of maximum independent vertex sets for the Heawood graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-83y22q
Direct link to example
Out[1]=1
Do the same with the "Count" annotation:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-dlcob2
Direct link to example
Out[2]=2
Return the maximum independent vertex sets for the Heawood graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-t6tar6
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-kx3j9b
Direct link to example
Out[2]=2
Irredundant Set‐Related Graph Properties (9)
Give the irredundance number of the Petersen graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-4ow1l8
Direct link to example
Out[1]=1
Compare with the smallest exponent of the maximal irredundance polynomial:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-s2ppz9
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-wcog4a
Direct link to example
Out[3]=3
Give the number of irredundant sets in the cubical graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-svbav6
Direct link to example
Out[1]=1
Compare with the length of the list of sets:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-crxob4
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-hk6r4f
Direct link to example
Out[3]=3
Give the irredundant sets in the cubical graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-ydq6xb
Direct link to example
Out[1]=1
Give the maximal irredundant polynomial of the cubical graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-fkkuaz
Direct link to example
Out[1]=1
Compute from the maximal irredundant sets:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-kmi8r0
Direct link to example
Out[2]=2
Give the number of maximal irredundant sets in the cubical graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-wuyk42
Direct link to example
Out[1]=1
Compare with the length of the list of sets:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-616vzs
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-7yru9i
Direct link to example
Out[3]=3
Give the maximal irredundant sets in the cubical graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-cbxre0
Direct link to example
Out[1]=1
Give the number of maximum irredundant sets in the cubical graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-gqxz6l
Direct link to example
Out[1]=1
Compare with the length of the list of sets:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-od4uc7
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-if1et3
Direct link to example
Out[3]=3
Give the maximum irredundant sets in the cubical graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-h0wn4w
Direct link to example
Out[1]=1
Give the upper irredundance number of the Petersen graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-lq77rj
Direct link to example
Out[1]=1
Compare with the largest exponent of the maximal irredundance polynomial:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-gv9kp3
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-jfaas1
Direct link to example
Out[3]=3
Dominating Set‐Related Graph Properties (29)
Return the connected dominating set count of the cubical graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-pj0tzi
Direct link to example
Out[1]=1
Compare with the connected domination polynomial:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-de4m1l
Direct link to example
Out[2]=2
Compare with the connected dominating sets:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-ybgq63
Direct link to example
Out[3]=3
Give the connected dominating sets of the square graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-ykztc5
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-kirlqx
Direct link to example
Out[2]=2
Return the connected domination number of the cubical graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-nfmo6y
Direct link to example
Out[1]=1
Compare with the connected domination polynomial:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-fa419m
Direct link to example
Out[2]=2
Compare with the connected dominating sets:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-7kh2a8
Direct link to example
Out[3]=3
Return the connected domination polynomial of the cubical graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-yjkn5e
Direct link to example
Out[1]=1
Compare with the sizes of connected dominating sets:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-88p2wg
Direct link to example
Out[2]=2
Give the domatic numbers of the first several hypercube graphs:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-5c2nei
Direct link to example
Out[1]=1
Return the number of dominating sets of the cubical graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-ua0iwa
Direct link to example
Out[1]=1
Compare with the domination polynomial:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-sanf52
Direct link to example
Out[2]=2
Compare with the dominating sets:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-82hiby
Direct link to example
Out[3]=3
Return the dominating sets of the cubical graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-xluehx
Direct link to example
Return the domination number of the cubical graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-jqd4cf
Direct link to example
Out[1]=1
Return the domination polynomial of the cubical graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-ffuegq
Direct link to example
Out[1]=1
Compare with the sizes of dominating sets:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-ofpnnx
Direct link to example
Out[2]=2
Give the number of minimal connected dominating sets in the Nauru graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-wqe21c
Direct link to example
Out[1]=1
Compute from the minimal connected dominating sets:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-ng343n
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-vvyzrj
Direct link to example
Out[3]=3
Return the minimal connected dominating sets in the Nauru graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-bvtcd4
Direct link to example
Out[1]=1
Give the minimal connected domination polynomial of the Nauru graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-b3pfil
Direct link to example
Out[1]=1
Compute from the minimal connected dominating sets:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-ba3c04
Direct link to example
Out[2]=2
Give the number of minimal dominating sets in the cubical graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-ngbugd
Direct link to example
Out[1]=1
Compute from the minimal dominating sets:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-hfusdf
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-lhnv92
Direct link to example
Out[3]=3
Return the minimal dominating sets in the cubical graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-2sp5hb
Direct link to example
Out[1]=1
Give the minimal domination polynomial of the cubical graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-b791vr
Direct link to example
Out[1]=1
Compute from the minimal dominating sets:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-60hmq8
Direct link to example
Out[2]=2
Give the minimal total dominating set count of the cubical graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-ldibsj
Direct link to example
Out[1]=1
Compute from the minimal total dominating sets:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-wcjwx7
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-pppwdl
Direct link to example
Out[3]=3
Give the minimal total dominating sets in the cubical graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-datwjz
Direct link to example
Out[1]=1
Give the minimal total domination polynomial of the cubical graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-nsz9vb
Direct link to example
Out[1]=1
Compute from the minimal total dominating sets:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-ybcz4q
Direct link to example
Out[2]=2
Return the minimum connected dominating set count of the cubical graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-we3vt6
Direct link to example
Out[1]=1
Compare with the connected domination polynomial:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-8j6nwt
Direct link to example
Out[2]=2
Compare with the minimum connected dominating sets:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-02k89v
Direct link to example
Out[3]=3
Give the minimum connected dominating sets of the square graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-k8tzb0
Direct link to example
Out[1]=1
Return the number of minimum dominating sets for the cubical graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-dcj297
Direct link to example
Out[1]=1
Do the same using the "Count" annotation:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-oj89ro
Direct link to example
Out[2]=2
Compare to the value computed from the domination polynomial:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-mmv3kc
Direct link to example
Out[3]=3
Return the minimum dominating sets for the cubical graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-oflhbz
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-2nujew
Direct link to example
Out[2]=2
Compare to the number from the direct property:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-bhqokg
Direct link to example
Out[3]=3
Return the number of minimum total dominating sets for the cubical graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-fkm193
Direct link to example
Out[1]=1
Do the same using the "Count" annotation:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-jio3ph
Direct link to example
Out[2]=2
Compare to the value computed from the total domination polynomial:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-g0kjtg
Direct link to example
Out[3]=3
Return the minimum total dominating sets for the cubical graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-f4rb08
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-8vnjdf
Direct link to example
Out[2]=2
Compare to the number from the direct property:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-xpet5l
Direct link to example
Out[3]=3
Return the number of total dominating sets of the cubical graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-59izsc
Direct link to example
Out[1]=1
Compare with the total domination polynomial:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-5ww6t2
Direct link to example
Out[2]=2
Compare with the total dominating sets:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-w2bn4j
Direct link to example
Out[3]=3
Return the total dominating sets of the cubical graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-uludfb
Direct link to example
Return the total domination number of the cubical graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-7tregq
Direct link to example
Out[1]=1
Return the total domination polynomial of the cubical graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-9mtwql
Direct link to example
Out[1]=1
Compare with the sizes of total dominating sets:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-yl7iho
Direct link to example
Out[2]=2
Return the upper domination number of the cubical graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-rmaacj
Direct link to example
Out[1]=1
Compare with the largest exponent of the minimal domination polynomial:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-pz8dfo
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-gq30ho
Direct link to example
Out[3]=3
Symmetry‐Related Graph Properties (18)
Display the arc transitivity of the Coxeter graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-6v4jfg
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-lj8txm
Direct link to example
Produce a table of the arc transitivities of some small arc-transitive graphs:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-ywzj7k
Direct link to example
Out[3]=3
Give the order of the automorphism group of the octahedral graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-ouaa3r
Direct link to example
Out[1]=1
Directly compute from the group:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-9ppoqe
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-qlsgyi
Direct link to example
Out[3]=3
Give the number of distinct edge lengths in the 4-trapezohedron:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-mj3fia
Direct link to example
Out[1]=1
Show the corresponding polyhedron:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-9stv2m
Direct link to example
Out[2]=2
List the names of groups generating the cubical graph as an (undirected) Cayley graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-or28q4
Direct link to example
Out[1]=1
Visualize the (directed) Cayley graphs:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-xhtya5
Direct link to example
Out[2]=2
Verify the undirected graphs are isomorphic with the cubical graph:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-l3o92h
Direct link to example
Out[3]=3
List group representations (not necessarily corresponding to distinct groups) that generate the cubical graph as an (undirected) Cayley graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-ehuf60
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-41wkku
Direct link to example
Out[2]=2
Verify these all generate the cubical graph:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-7v2sua
Direct link to example
Out[3]=3
Give the distinguishing number of the Petersen graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-4fj6jg
Direct link to example
Out[1]=1
Give the fixing number of the Petersen graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-ijg2no
Direct link to example
Out[1]=1
Give the number of minimum distinguishing labelings of the Petersen graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-nm53dz
Direct link to example
Out[1]=1
Do the same using the "Count" annotation:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-qh7pvn
Direct link to example
Out[2]=2
Give minimum distinguishing labelings of the Petersen graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-wdnw5i
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-du6md3
Direct link to example
Out[2]=2
Compare with the value from the direct property:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-7n72dy
Direct link to example
Out[3]=3
Give the number of planar embeddings in the Eiffel Tower graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-4csh5r
Direct link to example
Out[1]=1
Give the number of planar embeddings of a nonplanar graph:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-uu9dte
Direct link to example
Out[2]=2
Give the number of planar embeddings of a uniquely embeddable graph:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-2ns54u
Direct link to example
Out[3]=3
Give the number of symmetrically distinct faces in the truncated cubical graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-r8k43t
Direct link to example
Out[1]=1
Give representatives of the symmetrically distinct faces in the truncated cubical graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-lkew76
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-32etdd
Direct link to example
Out[2]=2
Give the number of symmetrically distinct vertices in the butterfly graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-8lrexc
Direct link to example
Out[1]=1
Give representatives of symmetrically distinct vertices:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-bv1cgf
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-4qi2ol
Direct link to example
Out[3]=3
Give the number of symmetrically distinct vertex pairs in the Petersen graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-8t07rw
Direct link to example
Out[1]=1
Compare with the signature of symmetrically distinct vertex pairs:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-spibpz
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-w0rzwm
Direct link to example
Out[3]=3
Give the signature of symmetrically distinct vertex pairs in the Petersen graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-nxru9m
Direct link to example
Out[1]=1
This means that there are 30 vertex pairs symmetrically equivalent to 
and 15 equivalent to 
:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-o3m1ok
Direct link to example
Out[2]=2
Compare the number of symmetrically distinct vertex pairs with the count:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-9sdfa5
Direct link to example
Out[3]=3
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0e7p5nb2-4qsa1l
Direct link to example
Out[4]=4
Give representatives of symmetrically distinct vertices in the butterfly graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-5944w2
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-fjjptr
Direct link to example
Out[2]=2
List all symmetrically equivalent faces in the truncated tetrahedral graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-sthuya
Direct link to example
Out[1]=1
Visualize via the graph's polyhedral embedding:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-mhxn45
Direct link to example
Out[2]=2
List the symmetrically equivalent vertices in the butterfly graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-3clr3l
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-s6av2f
Direct link to example
Out[2]=2
Information‐Related Graph Properties (12)
Give the bandwidth of the 5-hypercube graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-62ceqe
Direct link to example
Out[1]=1
Compare to the known closed form:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-okdbme
Direct link to example
Out[2]=2
Give the burning number of the 5-star graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-i6kb2f
Direct link to example
Out[1]=1
Give the cooling number of the 5-cycle graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-gl18kv
Direct link to example
Out[1]=1
The cooling number of the 
-cycle is given by ![TemplateBox[{{{(, {n, +, 2}, )}, /, 3}}, Ceiling]](Files/GraphData.en/45.png "TemplateBox[{{{(, {n, +, 2}, )}, /, 3}}, Ceiling]"){kind=small}
:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-cho6qr
Direct link to example
Out[2]=2
Give the gonality of the 5-star graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-d9amc5
Direct link to example
Out[1]=1
Return the likelihood of the Petersen graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-bx3ozp
Direct link to example
Out[1]=1
Likelihoods are normalized for fixed node count:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-5thc5r
Direct link to example
Out[2]=2
Return the Lovász number of the 5-cycle graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-pbj49w
Direct link to example
Out[1]=1
Give the pathwidths of the hypercube graphs:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-7umulg
Direct link to example
Out[1]=1
Compare to the known closed form:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-ifzdf0
Direct link to example
Out[2]=2
Give the pebbling number of the cubical graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-kw5kkx
Direct link to example
Out[1]=1
Give the scramble number of the cubical graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-q1pr6f
Direct link to example
Out[1]=1
Display the Shannon capacity of the cubical graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-f45nz
Direct link to example
Out[1]=1
Give the tree depth of a complete graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-0r38b9
Direct link to example
Out[1]=1
Give the treewidth of a complete graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-gm7r1y
Direct link to example
Out[1]=1
Give the treewidth of a complete bipartite graph:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-ptbt2w
Direct link to example
Out[2]=2
Path‐ and Cycle‐Related Properties (45)
Return the number of chordless cycles in the Petersen graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-sbwrct
Direct link to example
Out[1]=1
Verify against the chordless cycle polynomial:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-7510k4
Direct link to example
Out[2]=2
Verify against the chordless cycles:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-6pyx0x
Direct link to example
Out[3]=3
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0e7p5nb2-287y5b
Direct link to example
Out[4]=4
Return the chordless cycle polynomial of the Petersen graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-r5uzfm
Direct link to example
Out[1]=1
Give the chordless cycles of the Petersen graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-iqeziz
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-sdxe17
Direct link to example
Out[2]=2
Visually verify the returned cycles are chordless:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-cqlu1i
Direct link to example
Out[3]=3
Return the number of chordless cycles in the complement of the Petersen graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-otnva3
Direct link to example
Out[1]=1
Verify against the chordless cycle polynomial:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-3a5803
Direct link to example
Out[2]=2
Verify against the chordless cycles:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-5o1ja5
Direct link to example
Out[3]=3
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0e7p5nb2-hb5u0r
Direct link to example
Out[4]=4
Return the chordless cycle polynomial in the complement of the Petersen graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-2lpoeo
Direct link to example
Out[1]=1
Compare with the property computed from the complement:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-0v87wt
Direct link to example
Out[2]=2
Return the chordless cycles of the complement of the Petersen graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-0dff9p
Direct link to example
Out[1]=1
Return the number of chordless cycles in the complement of the Petersen graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-c97mh
Direct link to example
Out[1]=1
Verify against the chordless cycle polynomial:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-kc3ww3
Direct link to example
Out[2]=2
Verify against the chordless cycles:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-zqa802
Direct link to example
Out[3]=3
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0e7p5nb2-x7ffhg
Direct link to example
Out[4]=4
Return the odd chordless cycle polynomial in the complement of the Petersen graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-87lk3x
Direct link to example
Out[1]=1
Compare with the property computed from the complement:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-7f6ciz
Direct link to example
Out[2]=2
Return the odd chordless cycles of the complement of the Petersen graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-h8fy5m
Direct link to example
Out[1]=1
Give the name of the cube graph of the 7-Möbius ladder:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-qnsvx4
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-zao4q7
Direct link to example
Out[2]=2
Compare with the third graph power:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-51359o
Direct link to example
Out[3]=3
Verify the graph property and computed graphs are isomorphic:
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0e7p5nb2-4y2ti6
Direct link to example
Out[4]=4
Return the number of cycles of the cubical graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-ukkws0
Direct link to example
Out[1]=1
Do the same using the "Count" annotation:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-jeobhw
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-geczto
Direct link to example
Out[3]=3
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0e7p5nb2-v3vxdp
Direct link to example
Out[4]=4
Give the cycle polynomial of the cubical graph as a pure function:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-kfyn1w
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-8ixf7j
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-qowlyg
Direct link to example
Out[3]=3
Return the cycles in the cubical graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-mm3ul6
Direct link to example
Out[1]=1
Return the number of Eulerian cycles of the butterfly graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-0b9w0g
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-qp1xm
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-44ey6a
Direct link to example
Out[3]=3
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0e7p5nb2-iy9ss6
Direct link to example
Out[4]=4
Visualize the Eulerian cycles:
Copy to clipboard.
In[5]:=5
✖
https://wolfram.com/xid/0e7p5nb2-lmrwda
Direct link to example
Out[5]=5
Give the tally of cubical graph face sizes:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-i63200
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-mmdwlz
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-0lzniz
Direct link to example
Out[3]=3
Display the girth of the Petersen graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-mnxfjb
Direct link to example
Out[1]=1
Return the number of Hamilton decompositions of the octahedral graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-gmemrt
Direct link to example
Out[1]=1
Do the same using the "Count" annotation:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-vj0ykq
Direct link to example
Out[2]=2
Return Hamilton decompositions of the octahedral graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-3a32ji
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-7ln3wy
Direct link to example
Out[2]=2
Return the number of Hamiltonian cycles of the cubical graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-r8f4hf
Direct link to example
Out[1]=1
Do the same using the "Count" annotation:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-f948tx
Direct link to example
Out[2]=2
List the Hamiltonian cycles of the cubical graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-c91vwk
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-bzgrvm
Direct link to example
Out[2]=2
Display the Hamiltonian cycles:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-mns03w
Direct link to example
Out[3]=3
Return the Hamiltonian number of the octahedral graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-pe0oaj
Direct link to example
Out[1]=1
Return the number of Hamiltonian paths of the tetrahedral graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-f6azsk
Direct link to example
Out[1]=1
Do the same using the "Count" annotation:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-mr89bg
Direct link to example
Out[2]=2
Return the Hamiltonian paths of the cubical graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-i8garp
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-erboov
Direct link to example
Out[2]=2
Display the Hamiltonian paths:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-o247z1
Direct link to example
Out[3]=3
Return the number of Hamiltonian walks of the cubical graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-ic1vox
Direct link to example
Out[1]=1
Do the same using the "Count" annotation:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-uu033
Direct link to example
Out[2]=2
List the Hamiltonian walks of the cubical graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-ohpwso
Direct link to example
Out[1]=1
Give the hexagon count of the truncated tetrahedral graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-4w4if
Direct link to example
Out[1]=1
Compare with the value from the cycle polynomial:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-rkpzox
Direct link to example
Out[2]=2
Return the k-cyclic indices of the butterfly graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-18pma7
Direct link to example
Out[1]=1
Give the longest cycle count of the (3,4)-cone graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-q9rdyx
Direct link to example
Out[1]=1
Do the same using the "Count" annotation:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-glog5h
Direct link to example
Out[2]=2
Give the longest cycles of the (3,4)-cone graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-3phb2
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-3ycs1o
Direct link to example
Out[2]=2
Compare to the value from the direct property:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-0dcl4t
Direct link to example
Out[3]=3
Compare to the number of hexagons:
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0e7p5nb2-3m5znn
Direct link to example
Out[4]=4
Give the longest path count of the (3,4)-cone graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-7dv9na
Direct link to example
Out[1]=1
Do the same using the "Count" annotation:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-muvomr
Direct link to example
Out[2]=2
Give the longest path length of the (3,4)-cone graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-uouqgo
Direct link to example
Out[1]=1
Compare to the value obtained from the paths themselves (adjusting for the fact that a path specified by 
vertices has length 
):
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-q0bv9b
Direct link to example
Out[2]=2
Give the longest paths of the (3,4)-cone graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-jowuuf
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-rjk7cv
Direct link to example
Out[2]=2
Compare to the value from the direct property:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-dii5zc
Direct link to example
Out[3]=3
Give the number of minimum path coverings of the 5-wheel graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-prtqoo
Direct link to example
Out[1]=1
Give the minimum path coverings of the 5-wheel graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-koju26
Direct link to example
Out[1]=1
View some of the edge coverings:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-ku7oxb
Direct link to example
Out[2]=2
Since this graph is traceable, the minimum path coverings correspond to Hamiltonian paths:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-w9bd2c
Direct link to example
Out[3]=3
Return the number of odd chordless cycles in the Petersen graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-bmwens
Direct link to example
Out[1]=1
Verify against the chordless cycle polynomial:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-wl6p5f
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-g1s0da
Direct link to example
Out[3]=3
Verify against the chordless cycles:
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0e7p5nb2-udpc4p
Direct link to example
Out[4]=4
Copy to clipboard.
In[5]:=5
✖
https://wolfram.com/xid/0e7p5nb2-j3svrl
Direct link to example
Out[5]=5
Give the odd chordless cycle polynomial of the 9-antiprism graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-17kunn
Direct link to example
Out[1]=1
Compare with the length tallies of all odd chordless cycles:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-2bw2yl
Direct link to example
Out[2]=2
Give the odd chordless cycles of the Petersen graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-th381u
Direct link to example
Out[1]=1
Compare with the set of cycles extracted from all chordless cycles:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-1zkpbl
Direct link to example
Out[2]=2
Return the number of paths of the butterfly graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-1vgr02
Direct link to example
Out[1]=1
Do the same using the "Count" annotation:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-mt8n1o
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-81xdnk
Direct link to example
Out[3]=3
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0e7p5nb2-4jhd13
Direct link to example
Out[4]=4
Give the path polynomial of the butterfly graph as a pure function:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-t7wy47
Direct link to example
Out[1]=1
Compare to path length tallies:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-o62iwn
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-jlq3gr
Direct link to example
Out[3]=3
Give the path covering number of the cross graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-p21lpb
Direct link to example
Out[1]=1
Exhibit a path covering consisting of two paths:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-bjw44g
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-4c48fs
Direct link to example
Out[3]=3
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0e7p5nb2-mccadm
Direct link to example
Out[4]=4
Copy to clipboard.
In[5]:=5
✖
https://wolfram.com/xid/0e7p5nb2-rrrnl3
Direct link to example
Out[5]=5
Copy to clipboard.
In[6]:=6
✖
https://wolfram.com/xid/0e7p5nb2-cyb68q
Direct link to example
Out[6]=6
Return the paths in the butterfly graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-yrffgi
Direct link to example
Out[1]=1
Give the pentagon count of the snub cubical graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-721mxj
Direct link to example
Out[1]=1
Compare with the value from the cycle polynomial:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-7p0sia
Direct link to example
Out[2]=2
Give the square count of the snub cubical graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-gudc96
Direct link to example
Out[1]=1
Compare with the value from the cycle polynomial:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-r6nbzr
Direct link to example
Out[2]=2
Graphs with nonzero square count should not be square-free:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-3y3aji
Direct link to example
Out[3]=3
Give the name of the square graph of the 7-Möbius ladder:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-nmbrdf
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-bx65yw
Direct link to example
Out[2]=2
Compare with the second graph power:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-6j46gr
Direct link to example
Out[3]=3
Verify the graph property and computed graphs are isomorphic:
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0e7p5nb2-6fcjf
Direct link to example
Out[4]=4
Give the square count of the snub cubical graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-osqbyv
Direct link to example
Out[1]=1
Compare with the value from the cycle polynomial:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-dw4o9r
Direct link to example
Out[2]=2
Graphs with nonzero triangle count should not be triangle-free:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-xv96zi
Direct link to example
Out[3]=3
Graph Centralities (11)
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-ouqya
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-52in0o
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-dg49lr
Direct link to example
Out[3]=3
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-3rojer
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-k1ubki
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-g2u26u
Direct link to example
Out[3]=3
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0e7p5nb2-qu13dl
Direct link to example
Out[4]=4
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-so7k59
Direct link to example
Out[1]=1
Identical to reciprocal of vertex eccentricities for a connected graph:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-q5bvw8
Direct link to example
Out[2]=2
Edge betweenness centralities:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-bntr0u
Direct link to example
Out[1]=1
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-t1thka
Direct link to example
Out[1]=1
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-ita86v
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-jjtq8s
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-xdg96i
Direct link to example
Out[3]=3
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-isaehk
Direct link to example
Out[1]=1
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-f4rv7l
Direct link to example
Out[1]=1
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-pdyfwz
Direct link to example
Out[1]=1
Return using explicit variables:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-63rov4
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-euvcmg
Direct link to example
Out[3]=3
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0e7p5nb2-2f0nsj
Direct link to example
Out[4]=4
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-8ghgv3
Direct link to example
Out[1]=1
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-7s7c5w
Direct link to example
Out[1]=1
Graph Clustering Coefficients (3)
Give the global clustering coefficient of the house graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-phzkht
Direct link to example
Out[1]=1
Compare with the value computed from the graph by the built-in function:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-qbk1h2
Direct link to example
Out[2]=2
Give the local clustering coefficients of the house graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-od8y6y
Direct link to example
Out[1]=1
Compare with the value computed from the graph by the built-in function:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-5zw58a
Direct link to example
Out[2]=2
Give the mean clustering coefficient of the house graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-809e8r
Direct link to example
Out[1]=1
Compare with the value computed from the graph by the built-in function:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-6vw6fk
Direct link to example
Out[2]=2
Naming‐Related Properties (7)
List the alternate English names of the tesseract graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-fmb2n2
Direct link to example
Out[1]=1
Show the alternate standard names for the tesseract graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-l6pa98
Direct link to example
Out[1]=1
Give the entity for the Petersen graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-61yrpj
Direct link to example
Out[1]=1
Give the textual name of the octahedral graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-g75tv
Direct link to example
Out[1]=1
Give the name of the complete graph 
:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-ghg3ls
Direct link to example
Out[2]=2
Verify the standard name for this graph:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-gnmaou
Direct link to example
Out[3]=3
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0e7p5nb2-ha7r5e
Direct link to example
Out[4]=4
Give all textual names of the octahedral graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-69q6m2
Direct link to example
Out[1]=1
Query the standard name of the 4-hypercube graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-mc3ilh
Direct link to example
Out[1]=1
Show other alternate standard names corresponding to this standard name:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-bc9coy
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-9kb9zz
Direct link to example
Out[3]=3
Give the standard name of the complete graph 
:
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0e7p5nb2-ubvshh
Direct link to example
Out[4]=4
Copy to clipboard.
In[5]:=5
✖
https://wolfram.com/xid/0e7p5nb2-17helo
Direct link to example
Out[5]=5
Give all standard names of the octahedral graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-kcj451
Direct link to example
Out[1]=1
Notation‐Related Properties (3)
Give LCF notations for the octahedral graph (sorted by exponents):
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-qn8bim
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-ju4dhp
Direct link to example
Out[2]=2
Tally the LCF notation exponents:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-pn2mnf
Direct link to example
Out[3]=3
Display the nontrivial LCF embeddings:
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0e7p5nb2-baubqz
Direct link to example
Out[5]=5
Give the primary notation of the cubical graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-3snl8k
Direct link to example
Out[1]=1
Display the notation with traditional typesetting:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-2s3t5s
Direct link to example
Give a list of rules for notations associated with the complete graph 
:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-ztexke
Direct link to example
Out[1]=1
Give rules for various notations for the octahedral graph:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-bps542
Direct link to example
Out[2]=2
Basic Classes (4)
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-uvkt6k
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-w44vzu
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-zcw2d8
Direct link to example
Out[3]=3
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-x1i2wd
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-ppn93z
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-im2sut
Direct link to example
Out[3]=3
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-k2bqqm
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-ykq2y9
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-qfrw2h
Direct link to example
Out[3]=3
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0e7p5nb2-jrecoa
Direct link to example
Out[4]=4
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-6zfq7j
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-bv5qx7
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-g9gxjj
Direct link to example
Out[3]=3
Classes Based on Crossings (11)
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-6uvkdw
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-t35vtk
Direct link to example
List the doublecross graphs on 7 or fewer graphs:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-c4c5af
Direct link to example
Out[1]=1
Compare with the crossing numbers:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-rf982v
Direct link to example
Out[2]=2
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-wnc7fn
Direct link to example
List intrinsically linked graphs:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-qp3s6u
Direct link to example
List linklessly embeddable graphs:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-85vmxc
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-zbqqyq
Direct link to example
List the planar graphs on 5 or fewer graphs:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-l1qo15
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-uomem8
Direct link to example
Out[2]=2
Compare with the crossing numbers:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-74qzw4
Direct link to example
Out[3]=3
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-wad4cm
Direct link to example
List the singlecross graphs on 6 or fewer graphs:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-4m56se
Direct link to example
Out[1]=1
Compare with the crossing numbers:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-fcxsiv
Direct link to example
Out[2]=2
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-yls3ds
Direct link to example
Classes Based on Vertex Degrees (14)
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-m6erp9
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-c5i6uf
Direct link to example
Out[2]=2
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-e3659z
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-84tsld
Direct link to example
Out[2]=2
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-ba7yry
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-zmbwj0
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-narci9
Direct link to example
Out[2]=2
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-469j0s
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-qa6k3c
Direct link to example
Out[2]=2
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-vbqpdv
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-rma12u
Direct link to example
Out[2]=2
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-wi4tho
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-w32xpr
Direct link to example
Out[2]=2
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-63e67u
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-67hvnx
Direct link to example
Out[2]=2
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-du5sb5
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-11w89q
Direct link to example
Out[2]=2
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-383ikz
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-zstvf0
Direct link to example
Out[2]=2
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-5i1uv
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-sm55wn
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-7tyoe3
Direct link to example
Out[2]=2
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-0ebqgf
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-pfhx0d
Direct link to example
Classes Based on Traversals (32)
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-yvj6b
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-7rd2no
Direct link to example
Almost hypohamiltonian graphs:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-zulhmz
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-knl7a0
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-lajxm7
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-cfb67o
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-gsb0r8
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-2l54gd
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-coa2kn
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-gwji83
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-bz894n
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-2gxtte
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-2fupl
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-qjlgta
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-owqkx2
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-myz9sd
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-h6r8kd
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-14vl2d
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-lf592m
Direct link to example
Out[2]=2
Graphs that provide counterexamples to Kempe's purported proof of the four-color theorem:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-ux9y3z
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-5zzgr6
Direct link to example
Out[2]=2
Maximally nonhamiltonian graphs:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-0it69v
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-co7v8v
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-j8tlx6
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-fnc1xz
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-ksor43
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-xbso2
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-so8x2c
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-35ky44
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-i8nd67
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-54ydmw
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-cl7e35
Direct link to example
Pseudotrees that are not trees are equivalent to unicyclic graphs:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-rbmc7o
Direct link to example
Out[2]=2
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-jtpcs0
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-zndu88
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-f3dn6t
Direct link to example
Out[2]=2
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-emmr1i
Direct link to example
Classes Based on Chess Boards (19)
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-ik0myy
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-k31q35
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-pqi4an
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-xd67yn
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-eism8f
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-wh6go
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-v6dzj0
Direct link to example
Out[1]=1
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-6cgj45
Direct link to example
Out[1]=1
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-xsks2x
Direct link to example
Out[1]=1
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-hdxqj5
Direct link to example
Out[1]=1
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-b9j2bo
Direct link to example
Out[1]=1
Triangular honeycomb acute knight graphs:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-3yhmls
Direct link to example
Triangular honeycomb bishop graphs:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-yfftm
Direct link to example
Triangular honeycomb king graphs:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-1g5pti
Direct link to example
Triangular honeycomb obtuse knight graphs:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-6jrywc
Direct link to example
Triangular honeycomb queen graphs:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-skqhs8
Direct link to example
Triangular honeycomb rook graphs:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-u08oi5
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-6r8lxx
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-h3u40k
Direct link to example
Classes Based on Symmetry and Regularity (21)
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-uw00dt
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-z1n2i8
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-jbjmua
Direct link to example
Out[1]=1
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-sz6j6d
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-mxn3oh
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-t36i74
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-bp85gd
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-x4req1
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-ycaaat
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-099nq4
Direct link to example
Out[1]=1
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-py0w5m
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-eh83m8
Direct link to example
Out[1]=1
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-7ov2rj
Direct link to example
Out[1]=1
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-ytzetc
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-lhobrq
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-rmqbaz
Direct link to example
Out[1]=1
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-y61bcx
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-qumm9f
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-hi26j6
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-kjocp9
Direct link to example
Out[2]=2
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-4aq73p
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-w2c6bh
Direct link to example
Spectral Classes (3)
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-wndcdi
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-p4ixrk
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-t51da8
Direct link to example
Classes Based on Forbidden Graphs (10)
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-j0gjct
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-37y8io
Direct link to example
Out[2]=2
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-h128hs
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-82nfbj
Direct link to example
Out[2]=2
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-mg7h0z
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-13n5fs
Direct link to example
Out[2]=2
Forbidden minors for pathwidth 1:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-j4vg85
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-ps4nyk
Direct link to example
Out[2]=2
Forbidden minors for pathwidth 2:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-ev3cjr
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-h77cu7
Direct link to example
Out[2]=2
Forbidden minors for linkless embeddabiity:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-ez9zdt
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-md5i0
Direct link to example
Out[2]=2
Projective planar forbidden minors:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-1njpbh
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-8d9k33
Direct link to example
Out[2]=2
Projective planar forbidden topological minors (homeomorphic subgraphs):
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-spi2cg
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-g9c53d
Direct link to example
Out[2]=2
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-7emksf
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-p1lv3m
Direct link to example
Out[2]=2
Unit-distance forbidden subgraphs:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-snhj64
Direct link to example
Special Classes (50)
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-otq4cq
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-v6zuhi
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-08wfnd
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-lz0z31
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-gw8gl1
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-9p9up3
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-3i0qqq
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-fnb84l
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-fgkst
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-jiyz10
Direct link to example
Chromatically unique graphs on six or fewer vertices:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-imp9lf
Direct link to example
Out[2]=2
The cubical graph is chromatically unique:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-rfhj6m
Direct link to example
Out[3]=3
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0e7p5nb2-c4osrj
Direct link to example
Out[4]=4
Find and show graphs that are cochromatic with the antenna graph:
Copy to clipboard.
In[5]:=5
✖
https://wolfram.com/xid/0e7p5nb2-1awhyg
Direct link to example
Out[5]=5
Copy to clipboard.
In[6]:=6
✖
https://wolfram.com/xid/0e7p5nb2-xhwlbg
Direct link to example
Out[6]=6
Copy to clipboard.
In[7]:=7
✖
https://wolfram.com/xid/0e7p5nb2-pn61y9
Direct link to example
Out[7]=7
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-ccb4et
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-eylu76
Direct link to example
Graph represents a configuration:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-yifl52
Direct link to example
Out[1]=1
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-yothlo
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-qfsqox
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-9h7x24
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-go3pr5
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-98ivx
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-wrhb71
Direct link to example
Out[1]=1
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-sq05xf
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-n0ojf6
Direct link to example
Out[2]=2
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-4zgn7z
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-1qcexm
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-4nzpdi
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-tg87sm
Direct link to example
LCF (regular Hamiltonian) graphs:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-8lkfup
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-tgaf9t
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-41auk2
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-hlvdgq
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-5xwdr7
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-d8lzja
Direct link to example
Out[2]=2
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-t7k3x6
Direct link to example
Out[1]=1
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-ryk8hw
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-yzy6qu
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-g5owk5
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-9rj738
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-qtwyte
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-o9za21
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-vwwsky
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-lm1hl0
Direct link to example
Quadratically embeddable graphs:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-1p5v2d
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-n9ngc6
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-jdwr2x
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-uvzlfj
Direct link to example
Out[1]=1
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-jww7wl
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-ljzx0a
Direct link to example
Out[2]=2
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-7hi509
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-228ouv
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-7olwxm
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-29mo9r
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-q6153x
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-3prdjs
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-wj01su
Direct link to example
Classes Associated with Polyhedra (12)
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-72e816
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-w21vz7
Direct link to example
Out[2]=2
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-7rc604
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-q89s6g
Direct link to example
Out[2]=2
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-zrj979
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-2715l6
Direct link to example
Out[2]=2
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-j5sd8v
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-khxxei
Direct link to example
Out[2]=2
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-cmwstn
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-ujax7v
Direct link to example
Out[2]=2
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-9ny5qt
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-5e5nk2
Direct link to example
Out[2]=2
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-lhnk8f
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-7itbn0
Direct link to example
Out[1]=1
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-k13ba5
Direct link to example
Out[1]=1
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-17h9j4
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-0rezkm
Direct link to example
Out[2]=2
Uniform polyhedron skeleton graphs:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-p80gcq
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-7u1had
Direct link to example
Out[2]=2
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-oqjjwm
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-sh0l79
Direct link to example
Out[2]=2
Snark-Related Classes (4)
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-rm9wj7
Direct link to example
Out[1]=1
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-qdxd8j
Direct link to example
Out[1]=1
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-4114bt
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-7ejons
Direct link to example
Special Classes of Trees and Their Generalizations (12)
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-bvid74
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-ry7akw
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-pxd1wc
Direct link to example
Out[1]=1
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-5285v9
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-dcbh4b
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-r325rz
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-clm3lr
Direct link to example
Out[2]=2
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-tt0rpo
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-m14arz
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-ziwt2v
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-ps4ll5
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-eglxsu
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-bnvodi
Direct link to example
Out[1]=1
Classes Indexed by One or More Integers (90)
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-u29bb0
Direct link to example
Out[1]=1
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-nfargp
Direct link to example
Out[1]=1
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-onmv01
Direct link to example
Out[1]=1
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-smctfu
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-k3dm9h
Direct link to example
Out[1]=1
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-pnush3
Direct link to example
Out[1]=1
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-xzo26p
Direct link to example
Out[1]=1
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-mv23pp
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-p0qa9j
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-wrbi6c
Direct link to example
Out[1]=1
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-7t8058
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-lhhaj0
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-fta52o
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-28ebbt
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-zvhcsn
Direct link to example
Out[1]=1
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-ej176d
Direct link to example
Out[1]=1
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-dlsh3h
Direct link to example
Out[1]=1
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-789d6q
Direct link to example
Out[1]=1
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-1vq57c
Direct link to example
Out[1]=1
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-trtz9f
Direct link to example
Out[1]=1
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-tnzj5h
Direct link to example
Out[1]=1
Dorogovtsev–Goltsev–Mendes graphs:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-iu9zru
Direct link to example
Out[2]=2
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-lc4eqv
Direct link to example
Out[1]=1
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-jcq5og
Direct link to example
Out[1]=1
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-uxfehf
Direct link to example
Out[1]=1
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-pqffaa
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-c7pet8
Direct link to example
Out[1]=1
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-dibvdw
Direct link to example
Out[1]=1
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-qkj6ob
Direct link to example
Out[1]=1
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-tcy7f3
Direct link to example
Out[1]=1
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-0yk0bl
Direct link to example
Goethals–Seidel block design graphs:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-jalgkr
Direct link to example
Out[1]=1
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-kuqx76
Direct link to example
Out[1]=1
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-jl7atm
Direct link to example
Out[1]=1
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-jt76ou
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-ihsj0a
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-vxsdd5
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-gpvfuf
Direct link to example
Out[1]=1
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-b20xsu
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-gxmv1f
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-usnbbu
Direct link to example
Out[2]=2
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-7mfbzu
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-epfh4l
Direct link to example
Out[1]=1
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-qlo0bv
Direct link to example
Out[1]=1
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-5yey83
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-qm827r
Direct link to example
Out[1]=1
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-bx2bxa
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-bg11d6
Direct link to example
Out[1]=1
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-j2qriy
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-4qysh5
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-bvqrr3
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-ix9dlp
Direct link to example
Out[1]=1
Klein bottle triangulation graphs:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-y446bv
Direct link to example
Out[1]=1
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-1atdg5
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-cnw1iz
Direct link to example
Out[1]=1
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-k859yd
Direct link to example
Out[1]=1
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-clwoiw
Direct link to example
Out[1]=1
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-n1fsfd
Direct link to example
Out[1]=1
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-wop7o2
Direct link to example
Out[1]=1
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-1ftkoj
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-rhmagt
Direct link to example
Out[2]=2
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-3j0r24
Direct link to example
Out[1]=1
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-x0k5fu
Direct link to example
Out[1]=1
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-ts8ypw
Direct link to example
Out[1]=1
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-mdjdj6
Direct link to example
Out[1]=1
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-3eazph
Direct link to example
Out[1]=1
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-mxp4fd
Direct link to example
Out[1]=1
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-ms6gw2
Direct link to example
Out[1]=1
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-hwg0e
Direct link to example
Out[1]=1
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-g771c6
Direct link to example
Out[1]=1
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-f9cr9f
Direct link to example
Out[1]=1
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-nlaipf
Direct link to example
Out[1]=1
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-oeko4x
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-tvprrh
Direct link to example
Out[2]=2
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-k3qima
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-0u34zv
Direct link to example
Out[2]=2
Sierpiński tetrahedron graphs:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-uiingo
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-no0pln
Direct link to example
Out[2]=2
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-jovudd
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-yyhrd7
Direct link to example
Out[2]=2
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-cxu7n1
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-gzo658
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-yvs6rb
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-33s6h8
Direct link to example
Out[2]=2
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-oqqk70
Direct link to example
Out[1]=1
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-j953ex
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-bgf34g
Direct link to example
Out[2]=2
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-v5i0hc
Direct link to example
Out[1]=1
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-dvq0aa
Direct link to example
Out[1]=1
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-xd6lro
Direct link to example
Out[1]=1
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-vie70h
Direct link to example
Out[1]=1
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-xmfojw
Direct link to example
Out[1]=1
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-8pzrtz
Direct link to example
Out[1]=1
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-ugvu92
Direct link to example
Out[1]=1
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-stybl8
Direct link to example
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-udh7yd
Direct link to example
Out[1]=1
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-xuxkw4
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-mrtps7
Direct link to example
Out[2]=2
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-38ihso
Direct link to example
Out[1]=1
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-oz6nci
Direct link to example
Out[1]=1
Generalizations & Extensions (1)Generalized and extended use cases
Find the list of graph names matching a string wildcard expression:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-bkpbuz
Direct link to example
Out[1]=1
Find the list of graph names matching a string expression:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-gy5uza
Direct link to example
Out[2]=2
Find the list of graph names matching a regular expression:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-rt2yy5
Direct link to example
Out[3]=3
Applications (8)Sample problems that can be solved with this function
Generate a list of graphs on five nodes:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-cmgi6j
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-6mvlta
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-jsiwhk
Direct link to example
Out[3]=3
Generate a list of Hamiltonian graphs on five nodes:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-vtnvj6
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-pnpeb4
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-bwjdm2
Direct link to example
Out[3]=3
Generate a list of Hamiltonian planar graphs on five nodes:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-0s3cut
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-6v0hwo
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-zxr3rt
Direct link to example
Out[3]=3
Generate a list of graphs on five or fewer nodes:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-xlbbxa
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-n1rt3n
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-of4n6k
Direct link to example
Out[3]=3
Generate an array of Cayley graphs:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-epyr6x
Direct link to example
Out[1]=1
Visualize families of graphs by plotting edge count against vertex count:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-ujtt2m
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-dyrbdv
Direct link to example
Out[2]=2
Plot the numbers of graphs with different numbers of nodes available:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-h1cyf9
Direct link to example
Out[1]=1
Show the five known connected vertex-transitive nonhamiltonian graphs:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-3bgzyv
Direct link to example
Out[1]=1
Properties & Relations (10)Properties of the function, and connections to other functions
FromEntity can be used to make a graph from an entity:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-jcgdfu
Direct link to example
Out[1]=1
A graph can also be produced from an entity using the "Graph" property:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-ddue63
Direct link to example
Out[2]=2
ToEntity can be used to construct an entity from a graph:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-d19a0v
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-ctpsbc
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-wu2n6n
Direct link to example
An entity is also returned using the GraphData "Entity" property:
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0e7p5nb2-7xpaie
Direct link to example
Out[4]=4
Give the tesseract graph using GraphData directly:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-l3tels
Direct link to example
Out[1]=1
Do the same using the explicit "Graph" property:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-0gmkop
Direct link to example
Out[2]=2
Construct a graph using FromEntity and the "Entity" property:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-74yd7
Direct link to example
Out[3]=3
Construct a graph using an entity corresponding to its GraphData canonical name:
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0e7p5nb2-3ij3kk
Direct link to example
Out[4]=4
Use a member of an indexed graph family known to EntityValue:
Copy to clipboard.
In[5]:=5
✖
https://wolfram.com/xid/0e7p5nb2-m8ag7f
Direct link to example
Out[5]=5
Convert undirected graphs to graph entities using ToEntity:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-g9dgsb
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-hqrfoz
Direct link to example
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-kte1xs
Direct link to example
Out[3]=3
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0e7p5nb2-lg2owl
Direct link to example
Out[4]=4
Convert to the corresponding GraphData entity using CanonicalName:
Copy to clipboard.
In[5]:=5
✖
https://wolfram.com/xid/0e7p5nb2-3b88yx
Direct link to example
Out[5]=5
Use GraphPlot and GraphPlot3D to construct graph drawings from connectivity:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-dbrkup
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-j6s9lp
Direct link to example
Out[2]=2
Use the embedding provided by GraphData:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-63i692
Direct link to example
Out[3]=3
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0e7p5nb2-ffix2i
Direct link to example
Out[4]=4
Show that the integral graphs have integer-valued spectra:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-i8rlra
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-h3s06o
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-8tfrt4
Direct link to example
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0e7p5nb2-faurwh
Direct link to example
Out[4]=4
Show that the graphs classified as snarks satisfy their defining properties:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-l0qq7t
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-31vxhe
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-v1vftu
Direct link to example
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0e7p5nb2-eqqb6f
Direct link to example
Out[4]=4
Construct an attractive symmetric embedding of the Gray graph from its LCF notation:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-o97pjp
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-gtylgn
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-z2a3e5
Direct link to example
Out[3]=3
Verify that an antiprism graph is the skeleton of an antiprism:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-bspgyw
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-m0yq4v
Direct link to example
Out[2]=2
Display the corresponding PolyhedronData object:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-p7irz0
Direct link to example
Out[3]=3
Get the skeleton graph from the polyhedron object:
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0e7p5nb2-6tzj7p
Direct link to example
Out[4]=4
Copy to clipboard.
In[5]:=5
✖
https://wolfram.com/xid/0e7p5nb2-7hdymd
Direct link to example
Out[5]=5
Show that the automorphism group of the complete graph 
is the symmetric group 
:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-goq2rl
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-z5o9x4
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-7euwgc
Direct link to example
Out[3]=3
Possible Issues (4)Common pitfalls and unexpected behavior
GraphData results are complete when the number of graphs is not too large:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-5hqjrh
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-v5fe14
Direct link to example
Out[2]=2
For queries corresponding to a large number of graphs, the list returned by GraphData may not be exhaustive:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-5u4k39
Direct link to example
Out[3]=3
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0e7p5nb2-ytn5g4
Direct link to example
Out[4]=4
Using nonstandard graph names will not work:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-1yyoe
Direct link to example
Out[1]=1
Use string patterns directly in GraphData:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-dmreqx
Direct link to example
Out[2]=2
Or use general string-matching capabilities:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-f8khg
Direct link to example
Out[3]=3
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0e7p5nb2-bbeyd7
Direct link to example
Out[4]=4
Using nonstandard property names will not work:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-dy0con
Direct link to example
Out[1]=1
Use general string patterns to locate standard property names:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-4otl7
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-kiuzee
Direct link to example
Out[3]=3
Arithmetical operations cannot be carried out on Missing entries:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-v6837q
Direct link to example
Out[1]=1
Remove the Missing entries before performing operations:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-3m2lt8
Direct link to example
Out[2]=2
Display the data as a formatted table using TextGrid:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-jv4nu
Direct link to example
Out[3]=3
Interactive Examples (1)Examples with interactive outputs
Neat Examples (3)Surprising or curious use cases
Display planar embeddings of the heptahedral graphs:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-iinb5g
Direct link to example
Out[1]=1
See the toric structure of some of the cubic symmetric graphs:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-y0f3k3
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-bm6wrv
Direct link to example
Out[2]=2
Construct the connectivity graph of the contiguous US states plus Washington, DC:
Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0e7p5nb2-8w4bn5
Direct link to example
Out[1]=1
Make a list of rules representing adjacent states:
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0e7p5nb2-5c8j8m
Direct link to example
Visualize the resulting graph:
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0e7p5nb2-78b232
Direct link to example
Out[3]=3
Check the vertex and edge counts:
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0e7p5nb2-84sm8q
Direct link to example
Out[4]=4
Compare with the counts for the "ContiguousUSAGraph":
Copy to clipboard.
In[5]:=5
✖
https://wolfram.com/xid/0e7p5nb2-era7hm
Direct link to example
Out[5]=5
Exclude four pairs of states that either share only a single point in common or have a sea border but no land border:
Copy to clipboard.
In[6]:=6
✖
https://wolfram.com/xid/0e7p5nb2-l46gor
Direct link to example
Visualize the excluded borders:
Copy to clipboard.
In[7]:=7
✖
https://wolfram.com/xid/0e7p5nb2-htgqeq
Direct link to example
Out[7]=7
Construct connectivity rules excluding these four edges:
Copy to clipboard.
In[8]:=8
✖
https://wolfram.com/xid/0e7p5nb2-v2f3uw
Direct link to example
Copy to clipboard.
In[9]:=9
✖
https://wolfram.com/xid/0e7p5nb2-blky0w
Direct link to example
Out[9]=9
Check the vertex and edge counts:
Copy to clipboard.
In[10]:=10
✖
https://wolfram.com/xid/0e7p5nb2-n8hb3x
Direct link to example
Out[10]=10
Verify this graph is isomorphic to the "ContiguousUSAGraph" using ToEntity:
Copy to clipboard.
In[11]:=11
✖
https://wolfram.com/xid/0e7p5nb2-0q066r
Direct link to example
Out[11]=11
Copy to clipboard.
In[12]:=12
✖
https://wolfram.com/xid/0e7p5nb2-zfvhq5
Direct link to example
Out[12]=12
Related Links
GraphData Source Information
Wolfram Research (2007), GraphData, Wolfram Language function, https://reference.wolfram.com/language/ref/GraphData.html (updated 2025).
Copy to clipboard.
✖
Wolfram Research (2007), GraphData, Wolfram Language function, https://reference.wolfram.com/language/ref/GraphData.html (updated 2025).Text
Wolfram Research (2007), GraphData, Wolfram Language function, https://reference.wolfram.com/language/ref/GraphData.html (updated 2025).
Copy to clipboard.
✖
Wolfram Research (2007), GraphData, Wolfram Language function, https://reference.wolfram.com/language/ref/GraphData.html (updated 2025).CMS
Wolfram Language. 2007. "GraphData." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2025. https://reference.wolfram.com/language/ref/GraphData.html.
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✖
Wolfram Language. 2007. "GraphData." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2025. https://reference.wolfram.com/language/ref/GraphData.html.APA
Wolfram Language. (2007). GraphData. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/GraphData.html
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✖
Wolfram Language. (2007). GraphData. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/GraphData.htmlBibTeX
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✖
@misc{reference.wolfram_2025_graphdata, author="Wolfram Research", title="{GraphData}", year="2025", howpublished="\url{https://reference.wolfram.com/language/ref/GraphData.html}", note=[Accessed: 15-April-2025
]}BibLaTeX
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✖
@online{reference.wolfram_2025_graphdata, organization={Wolfram Research}, title={GraphData}, year={2025}, url={https://reference.wolfram.com/language/ref/GraphData.html}, note=[Accessed: 15-April-2025
]}