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:

Get a list of properties for a particular graph:

Get a short textual description of a property:

Get a longer textual description:

A property value can be any valid *Mathematica* expression:

A property that is not available for a graph has the value

Missing:

Some graph properties may be

Missing but still include partial information:

A property whose value is too large to include has the value

Missing:

Give the adjacency matrix, returned as a

SparseArray object:

Convert to an explicit matrix:

Plot the matrix using

ArrayPlot:

Return the distance matrix of the octahedral graph:

Return the number of edges of the octahedral graph:

List the indices of edges of the octahedral graph:

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

GraphPlot:

Show 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 it in expanded form:

Plot the matrix:

Show the faces of the octahedral graph:

Give the vertex count:

List all connected graphs:

List connected graphs on five vertices:

Check if the graph

is connected:

Check if the graph

is connected:

List the indices of the connected components of the

graph:

List the number of connected components:

List the names of the connected components:

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:

List triangulated graphs:

Give the vertex connectivity of Tietze's graph:

Show all available images of the octahedral graph:

Return the vertex coordinates for all embeddings:

Show the default embedding of the octahedral graph:

Show the three-dimensional embedding of the octahedral graph returned by

GraphPlot3D:

Show a labeled version of the default embedding of the octahedral graph:

Return the vertex coordinates for the default embedding:

Return the graph complement of the icosahedral graph:

Return the dual graph of the icosahedral graph:

Not all graphs have duals:

Return a graph object for the icosahedral graph:

This is also the default property for graphs:

Return the line graph of the icosahedral graph:

Return the primary embedding of a graph:

Return all tabulated embeddings of the graph:

Equivalent annotated syntax:

Return all tabulated planar embeddings of the graph:

Return all tabulated LCF embeddings of the graph:

Return an image of the primary embedding of a graph:

Return all tabulated images of the primary embedding of a graph:

Equivalent annotated syntax:

Return a 3D image of the primary embedding of a graph:

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 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 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 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 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 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 Tutte polynomial of the cubical graph:

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

Chromatically unique graphs:

The cubical graph is chromatically unique:

The antenna graph is not:

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:

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:

Give the Z index of the cubical graph:

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:

Find graphs having articulation vertices:

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

Explicitly give the automorphisms of the octahedral graph:

Find graphs having bridges:

List the chromatically unique graphs on 6 or fewer vertices:

Check if the square graph is chromatically unique:

Show the chromatic number of the icosahedral graph:

Display the clique number of the icosahedral graph:

Display the corank of the icosahedral graph:

Compute the corank from other graph properties:

Display the crossing number of the icosahedral graph:

The crossing number is 0 since the graph is planar:

Show the vertex degrees of the claw graph:

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:

Return the edge chromatic number of the 120-cell graph:

Give the genus of the cubical graph:

Display the girth of the Petersen graph:

Return the number of directed Hamiltonian cycles of the cubical graph:

List the directed Hamiltonian cycles of the cubical graph:

Return the number of Hamiltonian paths of the tetrahedral graph:

Return the Hamiltonian paths of the cubical graph:

Return the independence number of the Heawood graph:

Return the Lovász number of the 5-cycle graph:

Display the rank of the icosahedral graph:

Compute the corank from other graph properties:

Give the rectilinear crossing numbers for complete graphs:

Give the resistance matrix of the cubical graph:

Display the Shannon capacity of the cubical graph:

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

Display the spectrum of the 600-cell graph:

Display a nicely formatted version:

Give the toroidal crossing numbers for complete graphs:

List the alternate English names of the tesseract graph:

Show the alternate standard names for the tesseract graph:

Show graph names for graphs that are cochromatic with the claw graph:

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

Give the names of graphs cochromatic with the bull graph:

Give the name of the graph complement of the cubical 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

:

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:

Give the names of graphs cospectral with the Shrikhande graph:

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

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

It is in turn dual to the tesseract graph:

List graphs with a tabulated graph dual:

Show the name of the graph dual of the 24-cell graph:

Show that the 24-cell graph and tesseract graph are dual to one another:

Display the quartic transitive graph

:

Verify that it is dual to itself:

Self-dual graphs are dual to themselves:

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 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:

Give the textual name of the octahedral graph:

Give the name of the complete graph

:

Verify the standard name for this 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 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:

Bipartite graphs:

Nonplanar graphs:

Planar graphs:

Trees:

Regular graphs:

Cubic graphs:

Quartic graphs:

Quintic graphs:

Sextic graphs:

Septic graphs:

Octic graphs:

Acyclic graphs:

Bridged graphs:

Bridgeless graphs:

Cyclic graphs:

Eulerian graphs:

Hamilton-connected graphs:

Hamiltonian graphs:

Hamilton-laceable graphs:

Hypohamiltonian graphs:

Hypotraceable graphs:

Graphs that provide counterexamples to Kempe's purported proof of the four-color theorem:

King's tour graphs:

Knight's tour graphs:

Noneulerian graphs:

Nonhamiltonian graphs:

Queen's tour graphs:

Square-free graphs:

Traceable graphs:

Triangle-free graphs:

Untraceable graphs:

Arc-transitive graphs:

Asymmetric graphs:

Chang graphs:

Distance-regular graphs:

Distance-transitive graphs:

Edge-transitive graphs:

Identity graphs:

Locally Petersen graphs:

Paulus graphs:

Semisymmetric graphs:

Strongly regular graphs:

Symmetric graphs:

Taylor graphs:

Vertex-transitive graphs:

Weakly regular graphs:

Zero-symmetric graphs:

Bicolorable graphs:

Bicubic graphs:

Cage graphs:

Cayley graphs:

Claw-free graphs:

Conference graphs:

Critical nonplanar graphs:

Fullerenes:

Fusenes:

Incidence graphs:

Integral graphs:

LCF (regular Hamiltonian) graphs:

Line graphs:

Moore graphs:

Perfect graphs:

Perfect matching graphs:

Self-complementary graphs:

Self-dual graphs:

Named snarks:

Triangulated graphs:

Unit-distance graphs:

Antiprism graphs:

Archimedean graphs:

Archimedean dual graphs:

Platonic graphs:

Polyhedral graphs:

(Generalized) prism graphs:

Prism graphs:

Regular polychoron graphs:

Caterpillar trees:

Centipede trees:

Lobster trees:

Spider trees:

Apollonian graphs:

Bipartite Kneser graphs:

Book graphs:

Circulant graphs:

Complete graphs:

Complete bipartite graphs:

Complete tripartite graphs:

Cone graphs:

Crown graphs:

Cycle graphs:

Cyclotomic graphs:

Doob graphs:

Empty graphs:

Fan graphs:

Folded cube graphs:

Gear graphs:

Generalized polygon graphs:

Grid graphs:

Haar graphs:

Hadamard graphs:

Halved cube graphs:

Hamming graphs:

Hanoi graphs:

Helm graphs:

Hypercube graphs:

I-graphs:

Johnson graphs:

Kneser graphs:

Ladder graphs:

Ladder rung graphs:

Lattice graphs:

Möbius ladder graphs:

Mycielski graphs:

Odd graphs:

Paley graphs:

Pan graphs:

Path graphs:

Permutation star graphs:

Sierpinski graphs:

Square graphs:

Stacked book graphs:

Star graphs:

Sun graphs:

Sunlet graphs:

Tetrahedral graphs:

Torus grid graphs:

Triangular graphs:

Turán graphs:

Wheel graphs:

Windmill graphs: