GraphData

GraphData[name]

gives a graph with the specified name.

GraphData[entity]

gives the graph corresponding to the graph entity.

GraphData[entity,property]

gives the value of the property for the specified graph entity.

GraphData[class]

gives a list of available named graphs in the specified graph class.

GraphData[n]

gives a list of available named graphs with n vertices.

Details

  • 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 names that match the string pattern patt.
  • GraphData[] gives a list of all standard named graphs. GraphData[All] gives all available graphs.
  • GraphData[{n,i},] gives data for the i^(th) simple graph with n vertices.
  • GraphData[{"type",id},] gives data for the graph of the specified type with identifier id. The identifier is typically an integer or lists of integers.
  • 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,n] etc. gives a list of available graphs with n vertices etc. in the specified class.
  • GraphData["Classes"] gives a list of all supported classes.
  • GraphData["Properties"] gives a list of properties available for graphs.
  • Basic graph properties include:
  • "AdjacencyMatrix"adjacency matrix
    "AdjacencyMatrixCount"number of distinct adjacency matrices
    "DistanceMatrix"distance matrix
    "EdgeCount"total number of edges
    "Edges"edges
    "FaceCount"total number of faces (for a planar graph)
    "Faces"faces
    "IncidenceMatrix"incidence matrix
    "LaplacianMatrix"Laplacian matrix
    "NormalizedLaplacianMatrix"normalized Laplacian matrix
    "VertexCount"total number of vertices
    "Vertices"vertices
  • Properties related to graph connectivity include:
  • "AlgebraicConnectivity"second smallest eigenvalue of the Laplacian matrix
    "ArticulationVertexCount"number of articulation vertices
    "ArticulationVertices"list of vertices whose removal would disconnect the graph
    "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
    "Disconnected"disconnected
    "EdgeConnectivity"minimum edge deletions to disconnect the graph
    "SpanningTrees"spanning trees
    "Triangulated"triangulated (maximally planar)
    "VertexConnectivity"minimum vertex deletions to disconnect the graph
  • Properties related to graph display include:
  • "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
    "VertexCoordinates"vertex coordinates for the default layout
  • Properties returning Graph objects include:
  • "BipartiteDoubleGraph"bipartite double graph
    "CanonicalGraph"canonical version of graph that is isomorphic to the original
    "CochromaticGraphs"graphs with the same chromatic polynomial
    "ComplementGraph"graph complement
    "ConnectedComponents"connected components
    "CoresistanceGraphs"graphs with the same resistance multiset
    "CospectralGraphs"graphs with the same spectrum
    "DualGraph"dual graph
    "Graph"graph object
    "LineGraph"line graph
    "LocalGraph"local graph
  • Annotatable properties related to list-type output include:
  • "AdjacencyMatrix","outputtype"one or all adjacency matrices or count of all possible adjacency matrices
    "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:
  • "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
    "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
    "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
    "MinimalTotalDominationPolynomial"polynomial encoding the counts of minimal total dominating sets by size
    "MinimalVertexCoverPolynomial"polynomial encoding the counts of minimal vertex covers by size
    "OddChordlessCyclePolynomial"polynomial encoding the counts of odd chordless cycles by length
    "PathPolynomial"path 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"vertex cover polynomial
  • Coloringrelated graph properties include:
  • "ChromaticInvariant"chromatic invariant
    "ChromaticNumber"chromatic number
    "EdgeChromaticNumber"edge chromatic number
    "FractionalChromaticNumber"fractional chromatic number
    "FractionalEdgeChromaticNumber"fractional edge chromatic number
    "MinimumVertexColoring"minimum vertex coloring
    "MinimumEdgeColoring"minimum edge coloring
    "MinimumWeightFractionalColoring"minimum weight fractional coloring
  • Graph index properties include:
  • "BalabanIndex"Balaban index
    "CyclomaticNumber"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
    "StabilityIndex"stability index
    "TopologicalIndex"topological (first Schultz) index
    "WienerIndex"Wiener index
    "WienerSumIndex"Wiener sum index
  • Local graph properties include:
  • "ChordCount"number of chords
    "Chords"chords
    "IsolatedPointCount"number of isolated points
    "IsolatedPoints"vertices of degree 0
    "LeafCount"number of leaves
    "Leaves"vertices of degree 1
  • Global graph properties include:
  • "Anarboricity"maximum number of edgedisjoint nonacyclic subgraphs whose union is the original graph
    "Apices"list of vertices whose removal renders the graph planar
    "Arboricity"minimum number of edgedisjoint 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 linedisjoint nonplanar subgraphs
    "Corank"edge count minus vertex count plus connected component count
    "DegreeSequence"vertex degrees in monotonic nonincreasing order
    "DeterminedByResistance"no other graph shares the same multiset of resistances
    "DeterminedBySpectrum"no other graph shares the spectrum
    "DetourMatrix"matrix of longest path distances
    "Diameter"diameter of the graph
    "Eccentricities"eccentricities of each vertex
    "GracefulLabeling"graceful labeling
    "IntersectionArray"intersection array
    "LaplacianSpectrum"eigenvalues of the Laplacian matrix
    "MaximumLeafNumber"largest number of tree leaves in any spanning trees
    "MaximumVertexDegree"largest vertex degree
    "MeanDistance"mean distance between vertices
    "MinimumVertexDegree"smallest vertex degree
    "Periphery"indices of vertices with graph eccentricity equal to diameter
    "Rank"vertex count minus connected component count
    "RegularParameters"parameters describing common neighbor counts of vertices
    "ResistanceMatrix"resistances between pairs of vertices for unitresistance edges
    "Skewness"minimum number of edges whose removal would result in a planar graph
    "SpanningTreeCount"number of spanning trees
    "SpectralRadius"spectral radius
    "Spectrum"eigenvalues of the adjacency matrix
    "SpectrumSignature"tallies of adjacency matrix eigenvalues
    "Thickness"minimum number of planar subgraphs whose union is the original graph
    "Triameter"graph triameter (generalization of graph diameter)
    "VertexDegrees"vertex degeres
  • Topological graph properties include:
  • "CrossingNumber"minimum crossings in an embedding of the graph
    "Genus"minimum number of handles yielding a non-edge-crossing embedding
    "KleinBottleCrossigNumber"minimum crossings in a Klein bottle embedding
    "ProjectivePlaneCrossingNumber"minimum crossings in a projective plane embedding
    "RectilinearCrossingNumber"minimum crossings in a straightline embedding
    "ToroidalCrossingNumber"minimum crossings in a toroidal embedding
  • Cliquerelated properties include:
  • "CliquePolynomialclique polynomial
    "Cliquescliques
    "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
    "FractionalCliqueNumber"fractional clique number
    "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
  • Coverrelated properties include:
  • "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)
    "MinimumVertexCoverCount"number of minimum vertex covers
    "MinimumVertexCovers"minimum vertex covers
    "VertexCoverCount"number of vertex covers
    "VertexCoverNumber"size of a minimum vertex cover
    "VertexCoverPolynomial"vertex cover polynomial
  • Independent setrelated properties include:
  • "BipartiteDimension"bipartite dimension
    "EdgeIndependenceNumber"tallies of independent edge set sizes
    "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
    "MatchingGeneratingPolynomial"matchinggenerating polynomial
    "MatchingNumber"degree of the matchinggenerating polynomial
    "MatchingPolynomial"matchinggenerating 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 setrelated properties include:
  • "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
  • Dominating setrelated properties include:
  • "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
  • Symmetryrelated properties include:
  • "ArcTransitivity"maximal order s of an sarc-transitive graph
    "AutomorphismCount"order of the vertex automorphism group
    "AutomorphismGroup"graph automorphism permutation group
    "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
    "MinimumDistinguishingLabelingCount"number of minimum distinguishing labelings
    "MinimumDistinguishingLabelings"minimum distinguishing labelings
  • Informationrelated properties include:
  • "Bandwidth"graph bandwidth
    "BurningNumber"burning number
    "CheegerConstant"Cheeger constant
    "Conductance"graph conductance
    "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
    "ShannonCapacity"effective alphabet size in a graphrepresented communication model
    "TreeDepth"tree depth
    "Treewidth"graph treewidth
  • Path and cyclerelated properties include:
  • "ChordlessCycleCount"number of chordless cycles
    "ChordlessCyclePolynomial"polynomial encoding tallies of chordless cycle lengths
    "ChordlessCycles"cycles with no chords
    "ComplementOddChordlessCycleCount"number of odd chordless cycles in the graph complement
    "ComplementOddChordlessCyclePolynomial"polynomial encoiding tallies of lengths of odd chordless cycles of the graph complement
    "ComplementOddChordlessCycles"odd chordless cycles in the graph complement
    "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
    "OddChordlessCycleCount"number of odd chordless cycles
    "OddChordlessCyclePolynomial"polynomial encoding tallies of numbers of odd chordless cycles by length
    "OddChordlessCycles"chordless cycles of odd length
    "PathCount"number of distinct paths
    "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
    "TriangleCount"number of 3-cycles
  • Graph centralities include:
  • "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
  • Namingrelated properties include:
  • "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
  • Notationrelated properties include:
  • "LCFNotations"graph notations for embeddings based on Hamiltonian cycles
    "Notation"primary notation used for graph
    "NotationRules"rules for notations specifying the graph
  • 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:
  • "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 vertex degrees include:
  • "Cubic"each vertex is of degree 3
    "Octic"each vertex is of degree 8
    "Quartic"each vertex is of degree 4
    "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
    "TwoRegular"each vertex is of degree 2
  • Classes based on traversals include:
  • "Acyclic"free of cycles
    "AlmostHamiltonian"-node graph with Hamiltonian number
    "Bridged"contains at least one bridge
    "Bridgeless"free of bridges
    "Chordal"free of chordless cycles
    "Chordless"free of chords
    "Cyclic"contains at least one cycle
    "Eulerian"has a closed cycle containing every edge once
    "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"Hamiltonconnected with bipartitioned endpoints
    "HStarConnected"either Hamiltonconnected or Hamiltonlaceable
    "Hypohamiltonian"onevertexremoved graphs are Hamiltonian
    "Hypotraceable"onevertexremoved graphs are traceable
    "KempeCounterexample"counterexample to Kempe's 4coloring algorithm
    "MaximallyNonhamiltonian"maximally nonhamiltonian
    "Median"median graph
    "Noneulerian"not Eulerian
    "Nonhamiltonian"not Hamiltonian
    "Pancyclic"contains cycles of all lengths from 3 to vertex count
    "SquareFree"free of 4cycles
    "Traceable"contains a Hamiltonian path
    "TriangleFree"free of 3cycles
    "Unicyclic"possesses a single cycle
    "UniquelyPancyclic"uniquely pancyclic
    "Untraceable"not traceable
  • Classes based on chess boards include:
  • "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
    "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
  • Classes based on symmetry and regularity include:
  • "ArcTransitive"ordered pairs of adjacent vertices have identical environments
    "Asymmetric"not symmetric
    "Chang"strongly regular on 28 vertices
    "DistanceRegular"all vertices have identical distance sets
    "DistanceTransitive"all pairs of vertices have identical distance environments
    "EdgeTransitive"all edges have identical environments
    "Identity"automorphism group is of order unity
    "LocallyPetersen"locally Petersen
    "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
    "VertexTransitive"all vertices have identical environments
    "WeaklyRegular"regular, but not strongly regular
    "ZeroSymmetric"vertextransitive cubic with edges partitioned into three orbits
    "ZeroTwo"every two vertices have either 0 or 2 common neighbors
  • Classes based on forbidden graphs include:
  • "Beineke"Beineke graph (not a line graph if these are forbidden)
    "Kuratowski"Kuratowski graph ( or )
    "Metelsky"Metelsky graph (not a line graph if these are forbidden and )
    "UnitDistanceForbidden"minimal unit-distance forbidden
  • Special classes include:
  • "Apex"apex graph
    "Bicolorable"two or fewer vertex colors needed
    "Bicubic"bipartite and cubic
    "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
    "CriticalNonplanar"nonplanar and removal of any vertex gives a planar graph
    "DoubleToroidal"graph can be minimally emebdded on a double torus
    "Fullerene"planar cubic with all bounded faces pentagons or hexagons
    "Fusene"planar 2connected with all bounded faces hexagons
    "Graceful"graceful graph
    "Imperfect"imperfect (i.e. not perfect) graph
    "Incidence"incidence graph of a configuration
    "Integral"spectrum consists of integers
    "LCF"describable in LCF notation (regular Hamiltonian)
    "Line"line graph
    "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
    "Ore"Ore graph
    "Outerplanar"outerplanar graph
    "Perfect"perfect graph
    "PerfectMatching"has a matching with vertices
    "SelfComplementary"isomorphic to its complement
    "SelfDual"isomorphic to its dual
    "Snark"snark graph
    "StronglyPerfect"every induced subgraph has an independent set meeting all maximal cliques of
    "Toroidal"graph can be minimally embedded on a torus
    "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 fourdimensional solids
    "Wheel"skeleton of a pyramid
  • Special classes of trees and their generalization include:
  • "Cactus"connected graph in which any two graph 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")
    "Halin"Halin graph
    "Lobster"removal of leaves gives a caterpillar
    "Pseudoforest"contains at most one cycle per connected component
    "Pseudotree"a connected pseudoforest
    "Spider"one vertex of degree at most 3 and all others with degree at most 2
    "Tripod"a tree having exactly three tree leaves
  • Classes of graphs indexed by one or more integers include:
  • "Apollonian"connectivity graph of a 2D Apollonian gasket
    "BipartiteKneser"vertices represent k subsets and nk subsets of {1,,n}
    "Book"graph Cartesian product of a star and a twopath 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
    "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
    "Empty"n vertices with no edges
    "Fan"graph join of an empty graph with a path graph
    "FibonacciCube"a Fibonacci cube graph
    "FoldedCube"folded nhypercube graph
    "Gear"a wheel with vertices added between the vertices of the outer cycle
    "GeneralizedPolygon"an incidence plane based on a symmetric binary relation
    "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 nhypercube 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
    "HoneycombToroidal"honeycomb toroidal graph
    "Hypercube"an ndimensional hypercube
    "IGraph"generalization of a generalized Petersen graph
    "Johnson"graph describing adjacencies in the msubsets of an nset
    "JohnsonSkeleton"skeleton graph of the n^(th) Johnson solid
    "KayakPaddle"kayak paddle graph
    "Keller"Keller graph
    "KleinBottleTriangulation"a regular triangulation of a Klein bottle
    "Kneser"vertices represent ksubsets of {1,,n}
    "Ladder"a 2nvertex ladder graph
    "LadderRung"graph union of n twopaths
    "MoebiusLadder"an nsided prism graph with a halftwist
    "MengerSponge"connectivity graph of a Menger sponge
    "Mycielski"a trianglefree graph with chromatic number n
    "Odd"an odd graph
    "Paley"graph with vertices adjacent if their difference is a square in
    "Pan"an ncycle connected to a singleton graph by a bridge
    "Path"an nvertex tree with no branches
    "PathComplement"complement graph of the n-path graph
    "PermutationStar"a "star graph" on permutations of {1,,n} with edges at swaps
    "SierpinskiCarpet"connectivity graph of the Sierpiński carpet
    "SierpinskiSieve"connectivity graph of the Sierpiński sieve
    "SierpinskiTetrahedron"connectivity graph of the Sierpiński tetrahedron (tetrix)
    "StackedBook"graph Cartesian product of a star and a path graph
    "StackedPrism"a stacked prism graph
    "Star"a center vertex connected to n1 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
    "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", "MinimalPlanarIntegral", "MinimalRectilinearCrossing", "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:
  • "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
    "ComplementGraph"complement graph as a graph, entitiy, 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
    "DualGraph"dual graph as a graph, entitiy, entity name, graphics, or image
    "LineGraph"line graph as a graph, entitiy, entity name, graphics, or image
    "LocalGraph"local graph as a graph, entitiy, entity name, graphics, or image
    "PolyhedralEmbeddings"polyhedra having given graph as a skeleton as polyhedra, entities, or entity names
  • 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 all

Basic Examples  (3)

Return the Pappus graph:

Show all available 3D embeddings of the graph:

Show the spectrum of the icosahedral graph:

Generate a list of named snark graphs:

Scope  (547)

Names and Classes  (7)

Obtain a list of all standard implemented graphs:

Obtain a list of all implemented graphs:

Find the English name of a graph:

A list of alternate names can also be found:

Additional names acceptable as input can be found:

Find the list of graph classes:

Find the list of named graphs belonging to a class:

Test whether a graph belongs to a class:

Properties and Annotations  (2)

Get a list of properties for a particular graph:

Get a short textual description of a property:

Get a longer textual description:

Property Values  (4)

The values of properties may take a variety of forms as Wolfram Language expressions:

A property that is not available for a graph has the value Missing["NotAvailable"]:

Some graph properties may be Missing but still include partial information:

A property whose value is too large to include has the value Missing["TooLarge"]:

Detailed Properties  (534)

Basic Graph Properties  (12)

Give the adjacency matrix, returned as a SparseArray object:

Convert to an explicit matrix:

Plot the matrix using ArrayPlot:

Verify that the positions of 1s in the adjacency matrix correspond to graph edges:

List the number of distinct adjacency matrices possible for the cubical graph:

Return the distance matrix of the octahedral graph:

Return the number of edges of the octahedral graph:

Compare with taking the length of the edge list:

Return the edges of the octahedral graph:

Give the face count of the octahedral graph:

Give the faces of the octahedral graph:

Give the incidence matrix of the octahedral graph:

Give it in expanded form:

Plot the matrix:

Give the Laplacian matrix of the octahedral graph:

Give it in expanded form:

Plot the matrix:

Give the normalized Laplacian matrix of the octahedral graph:

Give the matrix in expanded form:

Plot the matrix:

Give the number of vertices of the octahedral graph:

Return the vertices of the octahedral graph:

Properties Related to Graph Connectivity  (11)

Give the algebraic connectivity of the octahedral graph:

Algebraic connectivity is defined as the second smallest member of the Laplacian spectrum:

List all connected graphs:

List connected graphs on five vertices:

Check if the graph is connected:

Check if the graph is connected:

Return the number of connected components in the graph:

Do the same using the "Count" annotation:

Return the connected components of the graph:

List the names of the connected components:

List the indices of connected components:

Give the number of connected induced subgraphs of the diamond graph:

Compare with the value obtained from the connected induced subgraph polynomial:

Display the diamond graph and compare the above counts with those computed from scratch:

Give the connected induced subgraph polynomial of the diamond graph:

Compare to the actual subgraphs:

List disconnected graphs on five vertices:

Check if the graph is disconnected:

Check if the graph is disconnected:

Find the edge connectivity of a complete binary tree of order 4:

Give spanning trees for the tetrahedral graph:

List triangulated graphs known to GraphData:

Give the vertex connectivity of Tietze's graph:

Properties Related to Graph Display  (11)

Show the default embedding of the octahedral graph:

Show classes of tabulated 2D embeddings of the cubical graph:

Show classes of known 3D embeddings of the cubical graph:

List the known 2D embeddings of the octahedral graph:

List the known 3D embeddings of the octahedral graph:

Show the default embedding of the octahedral graph:

Show the default 3D embedding of the octahedral graph:

Show a labeled version:

Give all cataloged 3D embeddings of the cubical graph:

Show a graphic of the octahedral graph:

Show a labeled version:

Give all cataloged embeddings of the octahedral graph as graphics:

Show a 3D graphic of the octahedral graph:

Show a labeled version:

Give all cataloged embeddings of the cubical graph as 3D graphics:

Show an image of the default embedding of the octahedral graph:

Show the three-dimensional embedding of the cubical graph as an image:

Show the Nauru graph as a mesh region:

Label the mesh:

Show all available embeddings as mesh regions:

Show a mesh region embedded in 3D:

Return the vertex coordinates for the default embedding of the octahedral graph:

Plot the vertices:

Show vertex coordinates for all cataloged embeddings:

Properties Returning Graph Objects  (11)

Return the bipartite double graph of the Clebsch graph:

Give the standard name of the bipartite double graph:

Return the cubical graph in canonical form:

The edge lists are not necessarily the same:

Verify the graphs are isomorphic:

Return the cochromatic graphs of the claw graph:

Return the graph complement of the icosahedral graph:

Return the connected components graph of the graph:

Show coresistance graphs:

Return a graph object for the dual graph of the icosahedral graph:

Return a graph object for the icosahedral graph:

This is also the default property for graphs:

Return a 3D graph object for the icosahedral graph:

Return the line graph of the icosahedral graph:

Return the local graph of the ConwaySmith graph:

Annotatable Properties Related to Graph Output  (11)

Return groups whose Cayley graphs generate the cubical graph using the default output type:

Return them explicitly as groups:

Return FiniteGroupData standard names whose Cayley graphs generate the cubical graph:

Return FiniteGroupData entities whose Cayley graphs generate the cubical graph:

Give cochromatic graphs of the claw graph:

Return the names of the cochromatic graphs:

Verify these share the same chromatic polynomial:

Show graph names for graphs that are cochromatic with the 5-star graph:

Give the names of graphs cochromatic with the bull graph:

Return the graph complement of the icosahedral graph:

Do the same using the annotation "Graph":

Return as an Image:

Return as a Graphics object:

Give the name of the complement graph:

Return the entity corresponding to the complement graph:

Show the graph names for the complement of graphs on four or fewer vertices:

The complement graph name of a self-complementary graph is identical to the StandardName:

Show graph and complement names (should be the same) for "interesting" small self-complementary graphs:

Return the connected components graph of the graph:

Return the indices of the connected components:

Return the names of the connected components:

Show coresistance graphs:

Do the same using the "Graph" annotation:

Get the names of the coresistance graphs:

Show the graph and its coresistance graph together:

Verify that the graphs in question have the same resistance sets:

Show graphs that share a resistance multiset with at least one other distinct graph:

List the names of graphs sharing the same multiset of resistances with a given graph:

Display these graphs:

Show graph names for graphs that are equivalent with a particular 20-vertex graph:

Return the cospectral graphs of the tesseract graph using the default output type:

Return the cospectral graphs of the tesseract graph explicitly as a graph:

Return the standard names:

Return graphics:

Return entities:

Return images:

Give the names of graphs cospectral with the Shrikhande graph:

Show graph names for graphs that are cospectral with the tesseract graph:

Return a graph object for the dual graph of the icosahedral graph:

Return the name:

Not all graphs have unique duals:

Show the graph name for the graph that is dual to the icosahedral graph:

It is in turn dual to the icosahedral graph:

List graphs with a tabulated graph dual:

Display the Johnson solid J8 skeleton:

Verify that it is dual to itself:

Tabulate cataloged self-dual graphs:

Return the line graph of the icosahedral graph:

Return the name of the line graph:

Give the name of the line graph for the Petersen graph:

Give the names of the line graphs of the Platonic graphs:

Show the Platonic graphs and their line graphs:

Show the graph names for the line graphs of non-empty graphs on four or fewer vertices:

Taking the line graph twice does not in general give back the original graph:

The line graph of a graph is isomorphic to itself only for cycle graphs or unions of identical cycle graphs:

Return the local graph of the ConwaySmith graph:

Return the name of the local graph:

Return polyhedra whose skeletons are isomorphic to the cubical graph using the default output type:

Return explicitly as PolyhedronData standard names:

Return them as polyhedra:

Return them as entities:

Annotatiable Properties Related to List-Type Output  (10)

Give the adjacency matrix of the default embedding for the square graph:

Give the number of possible adjacency matrices:

Return the same using the "Count" annotation:

Give all possible adjacency matrices:

Give (undirected) cycles of the house graph:

Do the same using the explicit "Undirected" annotation:

Give directed cycles:

Give count of undirected cycles:

Compare with the direct property:

Return the edges of the octahedral graph:

Verify the above agrees with the "EdgeCount" property:

Return the edges as a set of rules, suitable for plotting in GraphPlot:

Return a graphic of the edges:

Give (undirected) Eulerian cycles of the butterfly graph:

Compare with the output of FindEulerianCycle:

Do the same using the explicit "Undirected" annotation:

Give directed cycles:

Give count of undirected cycles:

Compare with the direct property:

Give the faces of the octahedral graph:

Verify the above agrees with the "FaceCount" property:

Show the faces of the octahedral graph:

Give (undirected) Hamiltonian cycles of the house X graph:

Do the same using the explicit "Undirected" annotation:

Give directed Hamiltonian cycles:

Give count of undirected Hamiltonian cycles:

Compare with the direct property:

Give (undirected) Hamiltonian paths of the butterfly graph:

Do the same using the explicit "Undirected" annotation:

Give directed Hamiltonian paths:

Give count of undirected Hamiltonian paths:

Compare with the direct property:

Give (undirected) Hamiltonian walks of the butterfly graph:

Do the same using the explicit "Undirected" annotation:

Give directed Hamiltonian paths:

Give count of undirected Hamiltonian paths:

Compare with the direct property:

Give the vertex coordinates of the default embedding of the octahedral graph:

Specify the "All" annotation:

This is equivalent to the "Embeddings" property:

Plot the vertex positions:

Return the vertices of the octahedral graph:

Verify the above agrees with the "VertexCount" property:

Annotatable Properties Related to Graph Display  (7)

Give all tabulated embeddings of the cubical graph:

Show classes for each embedding:

Display all embeddings:

Return the primary embedding of a graph:

Return all tabulated planar embeddings of the graph:

Return all tabulated LCF embeddings of the graph:

Give all tabulated 3D embeddings of the cubical graph:

Show classes for each 3D embedding:

Return all tabulated Polyhedron embeddings of the graph:

Return the cubical graph as a graph object:

Show a labeled version:

Return tabulated planar embeddings:

Return the cubical graph as a 3D graph object:

Show a labeled version:

Return tabulated polyhedron embeddings:

Return the cubical graph as a graphics object:

Show a labeled version:

Return tabulated planar embeddings:

Return the cubical graph as a 3D graph object:

Show a labeled version:

Return tabulated polyhedron embeddings as graphics objects:

Return an image of the primary embedding of a graph:

Show a labeled version:

Give all cataloged embeddings of the octahedral graph:

Show the three-dimensional embedding of the cubical graph as an image:

Show a labeled version:

Give all cataloged 3D embeddings of the cubical graph as images:

Properties Representing Graph Polynomials  (37)

Display the characteristic polynomial of the Coxeter graph as a pure function:

As a function of a variable x:

Compare with the directly computed value:

Give the chordless cycle polynomial of the cubical graph as a pure function:

Extract tallies by cycle length:

Extract tallies by cycle length a different way:

Compare with the tallies of chordless cycle lengths:

Give the chromatic polynomial of the cubical graph as a pure function:

As a function of a variable x:

The chromatic polynomial is a special case of the rank polynomial:

Give the chromatic polynomial of the icosahedral graph in terms of a variable x:

Give the clique polynomial of the utility graph as a pure function:

Compare to the explicit cliques:

Give the complement odd chordless cycle polynomial of the circulant graph as a pure function:

Extract tallies by cycle length:

Extract tallies by cycle length a different way:

Compare with the tallies of complement odd chordless cycle lengths:

Give the coboundary polynomial of the complete graph K5 as a pure function:

As a function of a variables q and t:

The coboundary polynomial is a special case of the Tutte polynomial:

Give the connected domination polynomial of the fish graph:

Compare with the connected dominating set count:

Compare with connected dominating set tallies:

Give the connected induced subgraph polynomial of the Eiffel Tower graph:

Compare with the count of connected induced subgraphs:

Give the cycle polynomial of the cubical graph as a pure function:

Compare to cycle tallies:

Give the detour polynomial of the cubical graph as a pure function:

As a function of a variable x:

Give the distance polynomial of the cubical graph as a pure function:

As a function of a variable x:

Compute from distance matrix:

Give the domination polynomial of the Coxeter graph as a pure function:

As a function of a variable x:

Give the edge cover polynomial of the utility graph as a pure function:

Compare to the explicit covers:

Give the flow polynomial of the cubical graph as a function of a variable u:

The flow polynomial is a special case of the rank polynomial:

Give the idiosyncratic polynomial of the cubical graph:

Compare with a direct computation:

Give the independence polynomial of the cubical graph:

Give the irredundance polynomial of the cubical graph:

Compute from the irredundant sets:

Give the Laplacian polynomial of the cubical graph as a pure function:

As a function of a variable x:

Compute from Laplacian polynomial:

Give the matching generating polynomial of the cubical graph:

Give the matching polynomial of the cubical graph:

Give the maximal clique polynomial of the house graph as a pure function:

Extract tallies by maximal clique size:

Extract tallies by maximal clique size a different way:

Compare with the tallies of maximal clique sizes:

Give the maximal independence polynomial of the cricket graph as a pure function:

Extract tallies by maximal independent vertex set size:

Extract tallies by maximal independent vertex set size a different way:

Compare with the tallies of maximal independent edge sets:

Give the maximal irredundance polynomial of the cricket graph as a pure function:

Extract tallies by maximal irredundant set size:

Extract tallies by maximal irredundant set size a different way:

Compare with the tallies of maximal irredundant sets:

Give the maximal matching generating polynomial of the cricket graph as a pure function:

Extract tallies by maximal independent edge set size:

Extract tallies by maximal independent edge set size a different way:

Compare with the tallies of maximal independent edge sets:

Give the minimal connected domination polynomial of the 5-wheel graph as a pure function:

Extract tallies by minimal connected dominating set size:

Extract tallies by minimal connected dominating set size a different way:

Compare with the tallies of minimal connected dominating sets:

Give the minimal domination polynomial of the cricket graph as a pure function:

Extract tallies by minimal dominating set size:

Extract tallies by minimal dominating set size a different way:

Compare with the tallies of minimal dominating sets:

Give the minimal edge cover polynomial of the cricket graph as a pure function:

Extract tallies by minimal edge cover size:

Extract tallies by minimal edge cover size a different way:

Compare with the tallies of minimal edge covers:

Give the minimal total domination polynomial of the butterfly graph as a pure function:

Extract tallies by total dominating set size:

Extract tallies by total dominating set size a different way:

Compare with the tallies of total dominating sets:

Give the minimal vertex cover polynomial of the cricket graph as a pure function:

Extract tallies by minimal vertex cover size:

Extract tallies by minimal vertex cover size a different way:

Compare with the tallies of minimal vertex covers:

Give the odd chordless cycle polynomial of the Golomb graph as a pure function:

Extract tallies by odd chordless cycle lengths:

Extract tallies by odd chordless cycle lengths in a different way:

Compare with the tallies of odd chordless cycle lengths:

Give the path polynomial of the cubical graph as a pure function:

Compare to cycle tallies:

Give the rank polynomial of the cubical graph:

Give the reliability polynomial of the cubical graph:

The reliability polynomial is a special case of the Tutte polynomial:

Give the sigma polynomial of the cubical graph:

Give the total domination polynomial of the 5-wheel graph as a pure function:

Extract tallies by total dominating set size:

Extract tallies by total dominating set size a different way:

Compare with the tallies of total dominating sets:

Give the Tutte polynomial of the cubical graph:

The Tutte polynomial is a special case of the rank polynomial:

Give the vertex cover polynomial of the utility graph as a pure function:

Compare to the explicit covers:

Coloring-Related Graph Properties  (8)

Give the chromatic invariant cubical graph:

Give the chromatic number of the cubical graph:

Visualize the chromatic number:

Give the edge chromatic number of the cubical graph:

Visualize the edge chromatic number:

Give the fractional chromatic numbers of the Mycielski graphs:

Compare with the closed form known for this fractional chromatic number:

Give the fractional edge chromatic numbers of the flower snark :

Give a minimum edge coloring of the octahedral graph:

Visualize the coloring:

Give a minimum vertex coloring of the octahedral graph:

Visualize the coloring:

Show a minimum weight fractional coloring of the Petersen graph:

Compute the fractional chromatic number:

Compare with the direct property:

Graph Index Properties  (12)

Display the Balaban index of the Coxeter graph:

Give the Balaban index of the isobutane graph:

Give the cyclomatic number (i.e. circuit rank) of the Coxeter graph:

Display the cyclomatic number of the icosahedral graph:

Compare with the value obtained from other properties:

Give the detour index of the cubical graph:

Give the Harary index of the cubical graph:

Give the Hosoya index of the cubical graph:

The Hosoya index is identical to the independent edge set count:

Give the Kirchhoff index of the cubical graph:

Give the Kirchhoff index of the isobutane graph:

Give the Kirchhoff sum index of the cubical graph:

Give the Kirchhoff sum index of the isobutane graph:

Give the molecular topological index of the cubical graph:

Give the stability index of the cubical graph:

Give the topological index of the cubical graph:

Give the Wiener index of the cubical graph:

Give the Wiener index of the isobutane graph:

Give the Wiener sum index of the cubical graph:

Give the Wiener sum index of the isobutane graph:

Local Graph Properties  (5)

Find graphs having bridges:

List the bridges of the Walther graph:

Number of bridges:

Do the same using the "Count" annotation:

Give the number of chords in the house graph:

Give the chord:

List the chords of the house graph:

Number of chords:

Do the same using the "Count" annotation:

Show isolated points:

Find leaves:

Show number of leaves directly:

Do the same using the "Count" annotation:

Visualize:

Global Graph Properties  (30)

List named connected (nonplanar) apex graphs:

Find all apices of the Wagner graph:

Delete apices and verify planarity of the resulting graphs:

Show the arboricity of the complete graphs:

Compare with the value form theory:

Give the number of articulation vertices in the A-graph:

Compare with the list of articulation vertices:

Find graphs having articulation vertices:

Give the indices of the articulation vertices of the Walther graph:

Visualize the articulation vertices:

Display the center of a tree:

Visualize the center:

Display the circumference of the icosahedral graph:

Verify that this corresponds to the length of the longest cycle:

Display the corank of the icosahedral graph:

Compute the corank from other graph properties:

Give the graphs on four or fewer vertices that are determined by resistance:

Check if the cubical graph is determined by spectrum:

Check if the tesseract graph is determined by spectrum:

Give the names of the graphs with the same spectrum as the tesseract graph:

Give the detour matrix of the cubical graph:

Give the diameter of the Pappus graph:

Give the eccentricities of the Pappus graph:

Show a graceful labeling of the Petersen graph:

Visualize the labeling:

Verify the labeling is graceful:

Give the intersection array of the cubical graph:

Display the Laplacian spectrum of the 600-cell graph:

Display a nicely formatted version:

Display the maximum leaf number of the A graph:

Give the maximum vertex degree of the E graph:

Display the mean distance of the Petersen graph:

Give the minimum vertex degree of the E graph:

Display the periphery of a tree:

Visualize the periphery:

Display the rank of the icosahedral graph:

Compute the rank from other graph properties:

Display the regular parameters of the cubical graph:

Give the resistance matrix of the cubical graph:

Display the skewness of the tesseract graph:

Compare with the result from theory:

Display the number of spanning trees in the 120-cell graph:

Give the spectral radius of the 600-cell graph:

Compare with the graph spectrum:

Display the spectrum of the 600-cell graph:

Display a nicely formatted version:

Show the triameter of the Coxeter graph:

Compare with the directly computed value:

Show the vertex degrees of the claw graph:

Give the toroidal crossing numbers for complete graphs:

Topological Graph Properties  (6)

Display the crossing number of the icosahedral graph:

The crossing number is 0 since the graph is planar:

Give the genus of the cubical graph:

Give the genus of the unique graph that triangulates the torus and Klein bottle:

Give the Klein bottle crossing number of the unique graph that triangulates the torus and Klein bottle:

Give the projective plane crossing numbers of the complete bipartite graphs :

Compare with the known closed form values:

Give the rectilinear crossing numbers for complete graphs:

Give the toroidal crossing number of the unique graph that triangulates the torus and Klein bottle:

CliqueRelated Graph Properties  (11)

Return the clique count of the octahedral graph:

Compare with the explicit listing of cliques:

Return the clique number of the icosahedral graph:

Compare with the lengths of the maximum cliques:

Give the clique polynomial of the utility graph:

Compare to the explicit cliques:

Return the cliques of the octahedral graph:

Return the fractional clique number of the Heawood graph:

Return the number of maximal cliques of the cubical graph:

Compare with the number of maximal cliques:

Return the maximal clique polynomial of the cubical graph:

Return the maximal cliques of the bull graph:

Return the maximum cliques of the bull graph:

Give the smallest covers of the cubical graph by maximal cliques:

Give the number of smallest covers of the cubical graph by maximal cliques:

CoverRelated Graph Properties  (17)

Give the biclique cover of the cubical graph:

Return the clique covering number of the icosahedral graph:

Verify it is the chromatic number of the graph complement:

Give the number of edge covers of the cubical graph:

Compare to the length of the list of edge covers:

Return the edge cover number of the icosahedral graph:

Compare with the length of a minimum edge cover:

Give the edge cover polynomial of the utility graph:

Compare to the explicit covers:

Give the edge covers of the utility graph:

Return the number of minimal edge covers for the cubical graph:

Compare to the length of the list:

Use the "Count" annotation:

Return the minimal edge cover polynomial of the tetrahedral graph:

Compute from the minimal edge covers:

Return the minimal edge covers of the tetrahedral graph:

Return the number of minimal vertex covers for the cubical graph:

Compare to the length of the list:

Use the "Count" annotation:

Return the minimal vertex cover polynomial of the tetrahedral graph:

Compute from the minimal vertex covers:

Return the minimal vertex covers of the tetrahedral graph:

Return the number of minimum clique coverings for the cubical graph:

Compare to the length of the list:

Give the number of vertex covers of the cubical graph:

Compare to the length of the list of edge covers:

Return the vertex cover number:

Compare with the size of a minimum vertex cover:

Give the vertex cover polynomial of the utility graph:

Compare to the explicit covers:

Give the vertex covers of the cubical graph:

Independent SetRelated Graph Properties  (23)

Give the biclique cover of the cubical graph:

Give the bipartite dimension of the cubical graph:

Return the independence number of the Heawood graph:

Compare with the size of a maximum independent vertex set:

Return the independence polynomial of the utility graph:

Compare to the explicit independent sets:

Return the independence ratio of the Petersen graph:

Compare to the direct ratio of properties:

Return the number of independent edge sets of the Heawood graph:

Compare to the explicit independent sets:

Return the independent edge sets of the utility graph:

Return the number of independent vertex sets of the cubical graph:

Compare to the explicit independent sets:

Return the independent vertex sets of the Heawood graph:

Return the matching-generating polynomial of the utility graph:

Compare to the explicit independent edge sets:

Return the matching number of the cubical graph:

Compare with the matching-generating polynomial:

Return the maximal independence polynomial for the Heawood graph:

Compare with the numbers of maximal independent vertex sets:

Return the number of maximal independent edge sets for the Heawood graph:

Compare with the number of maximal independent edge sets:

Return the maximal independent edge sets for the Heawood graph:

Return the maximal matching-generating polynomial for the Heawood graph:

Compare with the explicit edge sets:

Return the number of maximal independent vertex sets for the Heawood graph:

Compare with the number of maximal independent vertex sets:

Return the maximal independent vertex sets for the cubical graph:

Return the maximal independence polynomial for the cubical graph:

Compare with the explicit edge sets:

Return the maximal matching generating polynomial for the Heawood graph:

Compare with the numbers of maximal independent edge sets:

Return the number of maximum independent edge sets for the Heawood graph:

Do the same with the "Count" annotation:

Return the maximum independent edge sets for the Heawood graph:

Return the number of maximum independent vertex sets for the Heawood graph:

Do the same with the "Count" annotation:

Return the maximum independent vertex sets for the Heawood graph:

Irredundant SetRelated Graph Properties  (7)

Give the number of irredundant sets in the cubical graph:

Compare with the length of the list of sets:

Use the "Count" annotation:

Give the irredundant sets in the cubical graph:

Give the maximal irredundant polynomial of the cubical graph:

Compute from the maximal irredundant sets:

Give the number of maximal irredundant sets in the cubical graph:

Compare with the length of the list of sets:

Use the "Count" annotation:

Give the maximal irredundant sets in the cubical graph:

Give the number of maximum irredundant sets in the cubical graph:

Compare with the length of the list of sets:

Use the "Count" annotation:

Give the maximum irredundant sets in the cubical graph:

Dominating SetRelated Graph Properties  (28)

Return the connected dominating set count of the cubical graph:

Compare with the connected domination polynomial:

Compare with the connected dominating sets:

Give the connected dominating sets of the square graph:

Return the connected domination number of the cubical graph:

Compare with the connected domination polynomial:

Compare with the connected dominating sets:

Return the connected domination polynomial of the cubical graph:

Compare with the sizes of connected dominating sets:

Give the domatic numbers of the first several hypercube graphs:

Return the number of dominating sets of the cubical graph:

Compare with the domination polynomial:

Compare with the dominating sets:

Return the dominating sets of the cubical graph:

Return the domination number of the cubical graph:

Return the domination polynomial of the cubical graph:

Compare with the sizes of dominating sets:

Give the number of minimal connected dominating sets in the Nauru graph:

Compute from the minimal connected dominating sets:

Use the "Count" annotation:

Return the minimal connected dominating sets in the Nauru graph:

Give the minimal connected domination polynomial of the Nauru graph:

Compute from the minimal connected dominating sets:

Give the number of minimal dominating sets in the cubical graph:

Compute from the minimal dominating sets:

Use the "Count" annotation:

Return the minimal dominating sets in the cubical graph:

Give the minimal domination polynomial of the cubical graph:

Compute from the minimal dominating sets:

Give the minimal total dominating set count of the cubical graph:

Compute from the minimal total dominating sets:

Use the "Count" annotation:

Give the minimal total dominating sets in the cubical graph:

Give the minimal total domination polynomial of the cubical graph:

Compute from the minimal total dominating sets:

Return the minimum connected dominating set count of the cubical graph:

Compare with the connected domination polynomial:

Compare with the minimum connected dominating sets:

Give the minimum connected dominating sets of the square graph:

Return the number of minimum dominating sets for the cubical graph:

Do the same using the "Count" annotation:

Compare to the value computed from the domination polynomial:

Return the minimum dominating sets for the cubical graph:

Compare to the number from the direct property:

Return the number of minimum total dominating sets for the cubical graph:

Do the same using the "Count" annotation:

Compare to the value computed from the total domination polynomial:

Return the minimum total dominating sets for the cubical graph:

Compare to the number from the direct property:

Return the number of total dominating sets of the cubical graph:

Compare with the total domination polynomial:

Compare with the total dominating sets:

Return the total dominating sets of the cubical graph:

Return the total domination number of the cubical graph:

Return the total domination polynomial of the cubical graph:

Compare with the sizes of total dominating sets:

SymmetryRelated Graph Properties  (7)

Display the arc transitivity of the Coxeter graph:

List arc-transitive graphs:

Produce a table of the arc transitivities of some small arc-transitive graphs:

Give the order of the automorphism group of the octahedral graph:

Directly compute from the group:

List the names of groups generating the cubical graph as an (undirected) Cayley graph:

Visualize the (directed) Cayley graphs:

Verify the undirected graphs are isomorphic with the cubical graph:

List group representations (not necessarily corresponding to distinct groups) that generate the cubical graph as an (undirected) Cayley graph:

Verify these all generate the cubical graph:

Give the distinguishing number of the Petersen graph:

Give the number of minimum distinguishing labelings of the Petersen graph:

Do the same using the "Count" annotation:

Give minimum distinguishing labelings of the Petersen graph:

Compare with the value from the direct property:

InformationRelated Graph Properties  (9)

Give the bandwidth of the 5-hypercube graph:

Compare to the known closed form:

Give the burning number of the 5-star graph:

Give the gonality of the 5-star graph:

Return the likelihood of the Petersen graph:

Likelihoods are normalized for fixed node count:

Return the Lovász number of the 5-cycle graph:

Give the pathwidths of the hypercube graphs:

Compare to the known closed form:

Display the Shannon capacity of the cubical graph:

Give the tree depth of a complete graph:

Give the treewidth of a complete graph:

Give the treewidth of a complete bipartite graph:

Path and CycleRelated Properties  (37)

Return the number of chordless cycles in the Petersen graph:

Verify against the chordless cycle polynomial:

Verify against the chordless cycles:

Return the chordless cycle polynomial of the Petersen graph:

Give the chordless cycles of the Petersen graph:

Visually verify the returned cycles are chordless:

Return the number of off chordless cycles in the complement of the Petersen graph:

Verify against the odd chordless cycle polynomial:

Verify against the chordless cycles:

Return the chordless cycle polynomial in the complement of the Petersen graph:

Compare with the property computed from the complement:

Return the chordless cycles of the complement of the Petersen graph:

Return the number of cycles of the cubical graph:

Do the same using the "Count" annotation:

Return the cycles:

Give the cycle polynomial of the cubical graph as a pure function:

Compare to cycle tallies:

Return the cycles in the cubical graph:

Return the number of Eulerian cycles of the butterfly graph:

Return the cycles:

Visualize the Eulerian cycles:

Give the tally of cubical graph face sizes:

Compute directly from faces:

Display the girth of the Petersen graph:

Return the number of Hamilton decompositions of the octahedral graph:

Do the same using the "Count" annotation:

Return Hamilton decompositions of the octahedral graph:

Return the number of Hamiltonian cycles of the cubical graph:

Do the same using the "Count" annotation:

List the Hamiltonian cycles of the cubical graph:

Display the Hamiltonian cycles:

Return the Hamiltonian number of the octahedral graph:

Return the number of Hamiltonian paths of the tetrahedral graph:

Do the same using the "Count" annotation:

Return the Hamiltonian paths of the cubical graph:

Display the Hamiltonian paths:

Return the number of Hamiltonian walks of the cubical graph:

Do the same using the "Count" annotation:

List the Hamiltonian walks of the cubical graph:

Give the hexagon count of the truncated tetrahedral graph:

Compare with the value from the cycle polynomial:

Return the k-cyclic indices of the butterfly graph:

Give the longest cycle count of the (3,4)-cone graph:

Do the same using the "Count" annotation:

Give the longest cycles of the (3,4)-cone graph:

Compare to the value from the direct property:

Compare to the number of hexagons:

Give the longest path count of the (3,4)-cone graph:

Do the same using the "Count" annotation:

Give the longest path length of the (3,4)-cone graph:

Compare to the value obtained from the paths themselves (adjusting for the fact that a path specified by vertices has length ):

Give the longest paths of the (3,4)-cone graph:

Compare to the value from the direct property:

Return the number of odd chordless cycles in the Petersen graph:

Verify against the chordless cycle polynomial:

Verify against the chordless cycles:

Give the odd chordless cycle polynomial of the 9-antiprism graph:

Compare with the length tallies of all odd chordless cycles:

Give the odd chordless cycles of the Petersen graph:

Compare with the set of cycles extracted from all chordless cycles:

Return the number of paths of the butterfly graph:

Do the same using the "Count" annotation:

Return the cycles:

Give the path polynomial of the butterfly graph as a pure function:

Compare to path length tallies:

Return the paths in the butterfly graph:

Give the pentagon count of the snub cubical graph:

Compare with the value from the cycle polynomial:

Give the square count of the snub cubical graph:

Compare with the value from the cycle polynomial:

Graphs with nonzero square count should not be square-free:

Give the square count of the snub cubical graph:

Compare with the value from the cycle polynomial:

Graphs with nonzero triangle count should not be triangle-free:

Graph Centralities  (11)

Closeness centralities:

Degree centralities:

Identical to degrees:

Eccentricity centralities:

Identical to reciprocal of vertex eccentricities for a connected graph:

Edge betweenness centralities:

Eigenvector centralities:

HITS centralities:

Katz centralities:

Link rank centralities:

Page rank centralities:

Return using explicit variables:

Radiality centralities:

Status centralities:

NamingRelated Properties  (7)

List the alternate English names of the tesseract graph:

Show the alternate standard names for the tesseract graph:

Give the entity for the Petersen graph:

Give the textual name of the octahedral graph:

Give the name of the complete graph :

Verify the standard name for this graph:

Give all textual names of the octahedral graph:

Query the standard name of the 4-hypercube graph:

Show other alternate standard names corresponding to this standard name:

Give the standard name of the complete graph :

Give all standard names of the octahedral graph:

NotationRelated Properties  (3)

Give LCF notations for the octahedral graph (sorted by exponents):

Tally the LCF notation exponents:

Display the nontrivial LCF embeddings:

Give the primary notation of the cubical graph:

Display the notation with traditional typesetting:

Give a list of rules for notations associated with the complete graph :

Give rules for various notations for the octahedral graph:

Basic Classes  (4)

Bipartite graphs:

Nonplanar graphs:

Planar graphs:

Trees:

Classes Based on Vertex Degrees  (8)

Cubic graphs:

Octic graphs:

Quartic graphs:

Quintic graphs:

Regular graphs:

Septic graphs: