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| Built-in Mathematica Symbol | See Also »|More About » |
GraphData |  |
simple graph with n vertices.| "AdjacencyMatrix" | adjacency matrix | |
| "DistanceMatrix" | distance matrix | |
| "EdgeCount" | total number of edges | |
| "EdgeIndices" | pairs of vertex indices for each edge | |
| "EdgeRules" | edges specified as vertex connectivity rules | |
| "FaceCount" | total number of faces (for a planar graph) | |
| "FaceIndices" | indices of faces (for a planar graph) | |
| "IncidenceMatrix" | incidence matrix | |
| "LaplacianMatrix" | Laplacian matrix | |
| "NormalizedLaplacianMatrix" | normalized Laplacian matrix | |
| "VertexCount" | total number of vertices |
| "Connected" | connected | |
| "ConnectedComponentCount" | number of connected components | |
| "ConnectedComponentGraphNames" | names of graphs induced on connected components | |
| "ConnectedComponentIndices" | indices of connected components | |
| "Disconnected" | disconnected | |
| "EdgeConnectivity" | minimum edge deletions to disconnect the graph | |
| "VertexConnectivity" | minimum vertex deletions to disconnect the graph |
| "AllImages" | list of images of all available layouts for the graph | |
| "AllVertexCoordinates" | vertex coordinates for all available layouts | |
| "Image" | image of the default layout | |
| "Image3D" | image embedded in 3D | |
| "LabeledImage" | image of the default layout with vertex numbers included | |
| "VertexCoordinates" | vertex coordinates for the default layout |
| "CharacteristicPolynomial" | characteristic polynomial of the adjacency matrix | |
| "ChromaticPolynomial" | chromatic polynomial | |
| "FlowPolynomial" | flow polynomial | |
| "IdiosyncraticPolynomial" | Tutte's idiosyncratic polynomial | |
| "IndependencePolynomial" | independence polynomial | |
| "MatchingPolynomial" | matching polynomial | |
| "RankPolynomial" | rank polynomial | |
| "ReliabilityPolynomial" | reliability polynomial | |
| "SigmaPolynomial" | chromatic polynomial in falling factorial basis | |
| "TuttePolynomial" | Tutte polynomial |
| "ChromaticallyUnique" | no other graph shares the chromatic polynomial | |
| "ChromaticInvariant" | chromatic invariant | |
| "ChromaticNumber" | chromatic number | |
| "EdgeChromaticNumber" | edge chromatic number |
| "BalabanIndex" | Balaban index | |
| "CyclomaticNumber" | minimum number of edges to remove to turn acylic | |
| "HosoyaIndex" | Hosoya index | |
| "KirchhoffIndex" | Kirchhoff index | |
| "KirchhoffSumIndex" | Kirchhoff sum index | |
| "StabilityIndex" | stability index | |
| "WeinerIndex" | Wiener index | |
| "WeinerSumIndex" | Wiener sum index |
| "ArcTransitivity" | maximal order s of an s-arc-transitive graph | |
| "ArticulationVertices" | list of vertices whose removal would disconnect the graph | |
| "AutomorphismCount" | order of the vertex automorphism group | |
| "Automorphisms" | vertex permutations corresponding to automorphisms | |
| "Bridges" | list of edges whose removal would disconnect the graph | |
| "CliqueNumber" | number of vertices in the largest clique | |
| "Corank" | edge count minus vertex count plus connected component count | |
| "CrossingNumber" | minimum crossings in an embedding of the graph | |
| "Degrees" | degrees for each vertex | |
| "DeterminedByResistance" | no other graph shares the same multiset of resistances | |
| "DeterminedBySpectrum" | no other graph shares the spectrum | |
| "Diameter" | the diameter of the graph | |
| "Eccentricities" | eccentricities of each vertex | |
| "Genus" | minimum number of handles to get a planar embedding | |
| "Girth" | length of the shortest cycle | |
| "HamiltonianCycleCount" | number of distinct Hamiltonian cycles | |
| "HamiltonianCycles" | lists of Hamiltonian cycles | |
| "HamiltonianPathCount" | number of distinct Hamiltonian paths | |
| "HamiltonianPaths" | lists of Hamiltonian paths | |
| "IndependenceNumber" | size of the largest independent set | |
| "LovaszNumber" | Lovász number (estimate of Shannon capacity) | |
| "Rank" | vertex count minus connected component count | |
| "RectilinearCrossingNumber" | minimum crossings in a straight-line embedding | |
| "ResistanceMatrix" | resistances between pairs of vertices for unit-resistance edges | |
| "ShannonCapacity" | effective alphabet size in a graph-represented communication model | |
| "SpanningTreeCount" | number of spanning trees | |
| "Spectrum" | eigenvalues of the adjacency matrix | |
| "ToroidalCrossingNumber" | minimum crossings in a toroidal embedding | |
| "Unitransitivity" | maximal order s of an s-unitransitive graph |
| "AlternateNames" | alternate English names | |
| "AlternateStandardNames" | alternate standard Mathematica names | |
| "CochromaticGraphNames" | graphs sharing the same chromatic polynomial | |
| "ComplementGraphName" | name of the complement graph | |
| "CoresistanceGraphNames" | graphs sharing the same resistance distance multiset | |
| "CospectralGraphNames" | graphs sharing the same spectrum | |
| "DualGraphName" | name of the graph dual | |
| "LineGraphName" | name of the line graph | |
| "Name" | English name | |
| "NotationRules" | rules for notations specifying the graph | |
| "StandardName" | standard Mathematica name |
| "Bipartite" | bipartite (two components connected by every edge) | |
| "Nonplanar" | nonplanar (requires crossings) | |
| "Planar" | planar (no crossings) | |
| "Tree" | tree (no cycles) |
| "Regular" | each vertex is of the same degree | |
| "Cubic" | each vertex is of degree 3 | |
| "Quartic" | each vertex is of degree 4 | |
| "Quintic" | each vertex is of degree 5 | |
| "Sextic" | each vertex is of degree 6 | |
| "Septic" | each vertex is of degree 7 | |
| "Octic" | each vertex is of degree 8 |
| "Acyclic" | free of cycles | |
| "Bridged" | contains at least one bridge | |
| "Bridgeless" | free of bridges | |
| "Cyclic" | contains at least one cycle | |
| "Eulerian" | has a closed cycle containing every edge once | |
| "HamiltonConnected" | every pair of vertices bounds a Hamiltonian path | |
| "Hamiltonian" | has a closed cycle containing every vertex once | |
| "Hypohamiltonian" | one-vertex-removed graphs are Hamiltonian | |
| "Hypotraceable" | one-vertex-removed graphs are traceable | |
| "KempeCounterexample" | counterexample to Kempe's 4-coloring algorithm | |
| "KingsTour" | tours of a chess king | |
| "KnightsTour" | tours of a chess knight | |
| "Noneulerian" | not Eulerian | |
| "Nonhamiltonian" | not Hamiltonian | |
| "QueensTour" | tours of a chess queen | |
| "SquareFree" | free of 4-cycles | |
| "Traceable" | contains a Hamiltonian path | |
| "TriangleFree" | free of 3-cycles | |
| "Untraceable" | not traceable |
| "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 | |
| "Paulus" | strongly regular on 25 or 26 vertices | |
| "Semisymmetric" | regular and edge- but not vertex-transitive | |
| "StronglyRegular" | strongly regular | |
| "Symmetric" | both edge- and vertex-transitive | |
| "Taylor" | distance-regular with intersection array of form  | |
| "VertexTransitive" | all vertices have identical environments | |
| "WeaklyRegular" | regular, but not strongly regular | |
| "ZeroSymmetric" | vertex-transitive cubic with edges partitioned into three orbits |
| "Bicolorable" | two or fewer vertex colors needed | |
| "Bicubic" | bipartite and cubic | |
| "Cage" | smallest graph of a given girth | |
| "CayleyGraph" | Cayley graph | |
| "ClawFree" | free of the claw graph | |
| "Conference" | conference graph | |
| "Fullerene" | planar cubic with all bounded faces pentagons or hexagons | |
| "Fusene" | planar 2-connected with all bounded faces hexagons | |
| "Incidence" | incidence graph of a configuration | |
| "Integral" | spectrum consists of integers | |
| "LCF" | describable in LCF notation (regular Hamiltonian) | |
| "LineGraph" | line graph | |
| "Moore" | graphs with the Moore property | |
| "Perfect" | perfect graph | |
| "PerfectMatching" | has a matching with n/2 vertices | |
| "SelfComplementary" | isomorphic to its complement | |
| "SelfDual" | isomorphic to its dual | |
| "Snark" | snark graph | |
| "UnitDistance" | embeddable with edges of unit length |
| "Antiprism" | skeleton of an antiprism | |
| "Archimedean" | skeleton of one of the 13 Archimedean solids | |
| "ArchimedeanDual" | skeleton of one of the 13 Archimedean duals | |
| "Platonic" | skeleton of one of the 5 Platonic solids | |
| "Polyhedral" | skeleton of a polyhedron | |
| "Prism" | skeleton of a prism | |
| "RegularPolychoron" | skeleton of one of the 6 regular 4-dimensional solids |
| "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 | |
| "Lobster" | removal of leaves gives a caterpillar | |
| "Spider" | one vertex of degree at most 3 and all others with degree at most 2 |
| "Apollonian" | connectivity graph of a 2D Apollonian gasket | |
| "BipartiteKneser" | vertices represent k subsets and (n-k) subsets of {1 ,..., n} | |
| "Book" | graph Cartesian product of a star and a 2-path 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 | |
| "CompleteTripartite" | all neighboring pairs connected across 3 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  (n) | |
| "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 | |
| "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 | |
| "Grid" | an array of points with grid connectivity | |
| "Haar" | Haar (regular bipartite) graph of index n | |
| "Hadamard" | graph corresponding to a (-1, 1) matrix satisfying  | |
| "HalvedCube" | halved n-hypercube graph | |
| "Hamming" | direct product of m complete graphs of size n | |
| "Hanoi" | a Hanoi graph | |
| "Helm" | a wheel with a pendant edge adjoined at each cycle vertex | |
| "Hypercube" | an n-dimensional hypercube | |
| "Johnson" | graph describing adjacencies in the m-subsets of an n-set | |
| "Kneser" | vertices represent k-subsets of {1 ,..., n} | |
| "Ladder" | a 2n-vertex ladder graph | |
| "LadderRung" | graph union of n 2-paths | |
| "Lattice" | a line graph of the complete bipartite 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 GF(n) | |
| "Pan" | an n-cycle connected to a singleton graph by a bridge | |
| "Path" | an n-vertex tree with no branches | |
| "PermutationStar" | a "star graph" on permutations of {1 ,..., n} with edges at swaps | |
| "Sierpinski" | a Sierpinski graph | |
| "Square" | vertices represent the n2 ordered pairs of {1 ,..., n} | |
| "StackedBook" | graph Cartesian product of a star and an n-path 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 (n, 3) Johnson graph | |
| "TorusGrid" | grid graph on a torus | |
| "Triangular" | an (n, 2) Johnson graph | |
| "Wheel" | a cycle with all vertices connected to a center | |
| "Windmill" | m copies of the complete graph  with a vertex in common |
| "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 |
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