BUILT-IN MATHEMATICA SYMBOL

# GraphData

GraphData[name]
gives a graph with the specified name.

GraphData[name, "property"]
gives the value for the specified property for a named graph.

GraphData["class"]
gives a list of named graphs in the specified class.

GraphData[n]
gives a list of named graphs with n vertices.

## DetailsDetails

• Graphs can be specified by standard names such as and .
• GraphData[patt] gives a list of all graph names that match the string pattern patt.
• gives a list of all standard named graphs. gives all available graphs.
• GraphData[name] gives a Graph object.
• 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 identifier is typically an integer or lists of integers.
• GraphData[;;n] gives a list of standard named graphs with n vertices.
• GraphData[m;;n] gives a list of standard named graphs with m through n vertices.
• GraphData["class", n] etc. gives a list of 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 "DistanceMatrix" distance matrix "EdgeCount" total number of edges "EdgeIndices" pairs of vertex indices for each edge "EdgeList" edges specified using undirected edges () "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
• Properties related to graph connectivity include:
•  "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 "Triangulated" triangulated (maximally planar) "VertexConnectivity" minimum vertex deletions to disconnect the graph
• Properties related to graph display include:
•  "AllImages" list of images of all available layouts for the graph "AllVertexCoordinates" vertex coordinates for all available layouts "EmbeddingClasses" list of embedding class tags, one to an embedding "EmbeddingClasses3D" list of 3D embedding class tags, one to a 3D embedding "Embeddings" alternate name for "AllVertexCoordinates" "Embeddings3D" vertex coordinates for all available 3D 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
• Properties returning Graph objects include:
•  "ComplementGraph" graph complement "ConnectedComponentGraphs" 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
• GraphData[name, "property", "type"] gives a set of specific graphs, images, or embeddings, where in 2D may include , , , , , , , , , , , , , , , and ; and in 3D may include , , , and .
• Annotations related to graph display include:
•  "Embeddings","type" embeddings of a given type "Embeddings3D","type" 3D embeddings of a given type "Graph","type" graph of a given type "Graphs","type" graphs of a given type "Images","type" images of a given type "Images3D","type" 3D images of a given type
• Properties giving pure functions representing graph polynomials include:
•  "CharacteristicPolynomial" characteristic polynomial of the adjacency matrix "ChromaticPolynomial" chromatic polynomial "DetourPolynomial" characteristic polynomial of the detour matrix "DistancePolynomial" distance polynomial "FlowPolynomial" flow polynomial "IdiosyncraticPolynomial" Tutte's idiosyncratic polynomial "IndependencePolynomial" independence polynomial "LaplacianPolynomial" Laplacian polynomial "MatchingGeneratingPolynomial" matching generating polynomial "MatchingPolynomial" matching polynomial "RankPolynomial" rank polynomial "ReliabilityPolynomial" reliability polynomial "SigmaPolynomial" chromatic polynomial in falling factorial basis "TuttePolynomial" Tutte polynomial
• Coloring-related graph properties include:
•  "ChromaticallyUnique" no other graph shares the chromatic polynomial "ChromaticInvariant" chromatic invariant "ChromaticNumber" chromatic number "EdgeChromaticNumber" edge chromatic number "FractionalChromaticNumber" fractional chromatic number "FractionalEdgeChromaticNumber" fractional edge chromatic number "MimimumVertexColoring" 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 "ZIndex" Z index
• Global graph properties include:
•  "ArticulationVertices" list of vertices whose removal would disconnect the graph "Bridges" list of edges whose removal would disconnect the graph "Center" indices of vertices with graph eccentrity equal to radius "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 "DetourMatrix" matrix of longest path distances "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 "MeanDistance" mean distance between vertices "Periphery" indices of vertices with graph eccentricity equal to diameter "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 "SpanningTreeCount" number of spanning trees "Spectrum" eigenvalues of the adjacency matrix "ToroidalCrossingNumber" minimum crossings in a toroidal embedding
• Matching-, clique-, and cover-related properties include:
•  "CliqueNumber" number of vertices in a maximum clique "EdgeCoverNumber" size of the minimum edge cover "FractionalCliqueNumber" fractional clique number "IndependenceNumber" size of the largest independent set "MatchingNumber" degree of the matching-generating polynomial "MaximalCliqueCount" number of distinct maximal cliques "MaximalCliques" maximal cliques "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 "MaximumCliqueCount" number of maximum cliques "MaximumCliques" maximum cliques "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 "MinimumEdgeCoverCount" number of minimum edge covers (matchings) "MinimumEdgeCovers" minimum edge covers (matchings) "MinimumVertexCoverCount" number of minimum vertex covers "MinimumVertexCovers" minimum vertex covers "VertexCoverNumber" size of a minimum vertex cover
• Symmetry-related properties include:
•  "ArcTransitivity" maximal order s of an s-arc-transitive graph "AutomorphismCount" order of the vertex automorphism group "AutomorphismGroup" graph automorphism permutation group "Automorphisms" vertex permutations corresponding to automorphisms "CayleyGraphGeneratingGroupNames" names of groups that generate the graph as a Cayley graph "CayleyGraphGeneratingGroups" groups that generate the graph as a Cayley graph "Unitransitivity" maximal order s of an s-unitransitive graph
• Information-related properties include:
•  "Bandwidth" graph bandwidth "LovaszNumber" Lovász number (estimate of Shannon capacity) "Pathwidth" graph pagewidth "ShannonCapacity" effective alphabet size in a graph-represented communication model "Treewidth" graph treewidth
• Path- and cycle-related properties include:
•  "DirectedCycleCount" number of distinct directed cycles "DirectedCycles" lists of directed cycles "DirectedEulerianCycleCount" number of distinct directed Eulerian cycles "DirectedEulerianCycles" lists of directed Eulerian cycles "DirectedHamiltonianCycleCount" number of distinct directed Hamiltonian cycles "DirectedHamiltonianCycles" lists of directed Hamiltonian cycles "DirectedHamiltonianPathCount" number of distinct directed Hamiltonian paths "DirectedHamiltonianPaths" lists of directed Hamiltonian paths "UndirectedCycleCount" number of distinct undirected (simple) cycles "UndirectedCycles" lists of undirected (simple) cycles "UndirectedEulerianCycleCount" number of distinct undirected (simple) Eulerian cycles "UndirectedEulerianCycles" lists of undirected (simple) Eulerian cycles "UndirectedHamiltonianCycleCount" number of distinct undirected (simple) Hamiltonian cycles "UndirectedHamiltonianCycles" lists of undirected (simple) Hamiltonian cycles "UndirectedHamiltonianPathCount" number of distinct undirected (simple) Hamiltonian paths "UndirectedHamiltonianPaths" lists of undirected (simple) Hamiltonian paths
• Naming-related properties include:
•  "AlternateNames" alternate English names "AlternateStandardNames" alternate standard Mathematica names "CochromaticGraphNames" graphs sharing the same chromatic polynomial "ComplementGraphName" name of the complement graph "ConnectedComponentGraphNames" graphs making up the connected component "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 "LocalGraphName" name of the local graph "Name" English name "Names" English name and alternate names "StandardName" standard Mathematica name "StandardNames" standard and alternate Mathematica names
• Notation-related 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) "Planar" planar (no crossings) "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 of of degree 2
• Classes based on traversals include:
•  "Acyclic" free of cycles "Antelope" moves of an antelope generalized chess piece "Bishop" moves of two (white and black) chess bishops "BlackBishop" moves of a black chess bishop "Bridged" contains at least one bridge "Bridgeless" free of bridges "Chordal" free of chordless cycles "Cyclic" contains at least one cycle "Eulerian" has a closed cycle containing every edge once "Fiveleaper" moves of a fiveleaper generalized chess piece "HamiltonConnected" every pair of vertices bounds a Hamiltonian path "Hamiltonian" has a closed cycle containing every vertex once "HamiltonLaceable" Hamilton-connected with bipartitioned endpoints "Hypohamiltonian" one-vertex-removed graphs are Hamiltonian "Hypotraceable" one-vertex-removed graphs are traceable "KempeCounterexample" counterexample to Kempe's 4-coloring algorithm "King" moves of a chess king "Knight" moves of a chess knight "Noneulerian" not Eulerian "Nonhamiltonian" not Hamiltonian "Queen" moves of a chess queen "SquareFree" free of 4-cycles "Traceable" contains a Hamiltonian path "TriangleFree" free of 3-cycles "Untraceable" not traceable "WhiteBishop" moves of a white chess 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" vertex-transitive cubic with edges partitioned into three orbits "ZeroTwo" every two vertices have either 0 or 2 common neighbors
• Special classes include:
•  "Bicolorable" two or fewer vertex colors needed "Bicubic" bipartite and cubic "Cage" smallest graph of a given girth "Cayley" Cayley graph "ClawFree" free of the claw graph "Conference" conference graph "CriticalNonplanar" nonplanar and removal of any vertex gives a planar graph "Fullerene" planar cubic with all bounded faces pentagons or hexagons "Fusene" planar 2-connected with all bounded faces hexagons "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 "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 "Toroidal" graph can be embedded on a torus "UnitDistance" embeddable with edges of unit length
• Graph centralities include:
•  "ClosenessCentralities" closeness centrailitities "DegreeCentralities" vertex degrees "EccentricityCentralities" reciprocal of vertex eccentricities "EdgeBetweennessCentralities" edge betweenness centralities "EigenvectorCentralities" eigenvector centrailitities "HITSCentrailities" hub centrailitities "KatzCentralities" Katz centrailitities "PageRankCentralities" page rank centrailitities "RadialityCentralities" radiality centralities "StatusCentralities" status centralities
• 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 "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
• 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:
• 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.

## ExamplesExamplesopen allclose all

### Basic Examples (3)Basic Examples (3)

Return the Pappus graph:

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Show all available images of the graph:

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Show the spectrum of the icosahedral graph:

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Generate a list of named snark graphs:

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