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 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 number of vertices of the octahedral graph:

Return the vertices of the octahedral graph:

Give the block count of the butterfly graph:

Show the blocks:

Give the blocks of the butterfly graph:

Show the original graph:

Show the bridges of the 3-barbell graph:

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:

Give the cyclic edge connectivity of the Petersen graph:

Show the two cyclic components of a edge cut set of size 5:

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:

Return the number of edge cuts in the triangle graph:

Do the same with the "Count" annotation:

Compare with the length of edge cuts:

Show the edge cuts in the triangle graph:

Visualize how the cuts disconnect the graph:

Give the incidence lines that generate the Pappus graph:

Show the Levi graph generated by these incidences:

Identify the resulting graph:

Give a realization of the Pappus configuration showing the incidences:

Give the lambda components of the domino graph:

Give the Luccio–Sami components of the domino graph:

Return the number of minimal edge cuts in the triangle graph:

Do the same with the "Count" annotation:

Compare with the length of edge cuts:

Show the minimal edge cuts in the triangle graph:

Visualize how the cuts disconnect the graph:

Return the number of minimal vertex cuts in the butterfly graph:

Do the same with the "Count" annotation:

Compare with the length of edge cuts:

Show the minimal edge cuts in the butterfly graph:

Visualize how the cuts disconnect the graph:

Return the number of minimum vertex cuts of the square graph with triangles erected on two opposite sides:

Return as an annotation:

Compare with the cuts:

Give the number of minimum cyclic edge cuts of the Petersen graph:

Do the same as an annotation to "MinimumCyclicEdgeCuts":

Compare with the number of minimum cyclic edge cuts:

Return the minimum cyclic edge cuts of the Petersen graph:

Compare with the cyclic edge connectivity:

Visualize the cuts to reveal the resulting two cyclic components:

Give the minimum vertex cuts of the square graph with triangles erected on two opposite sides:

Show the disconnected graphs resulting from these cuts:

Give spanning trees for the tetrahedral graph:

Give the strength of the butterfly graph:

Compute using edge cuts:

Give the toughness of the butterfly graph:

Compute using edge cuts:

Give the vertex connectivity of Tietze's graph:

Return the number of vertex cuts in the butterfly graph:

Do the same with the "Count" annotation:

Compare with the length of edge cuts:

Show the vertex cuts of the butterfly graph:

Visualize how the cuts disconnect the graph:

Show the number of graph minors the octahedral graph:

Give the Hadwiger number of a wheel complement graph:

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:

Show the icosahedral graph as a polyhedron:

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

Plot the vertices:

Show vertex coordinates for all cataloged embeddings:

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 co-degree sequence graphs of the 5-path graph:

Since this graph shares a degree sequence, it is not unigraphic:

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 Conway–Smith graph:

Return the root graph of the icosahedral line graph:

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:

Do the same using the "Graph" annotation:

Return the indices of the connected components:

Return the names of the connected components:

Return the number of connected components:

Do the same using the "Count" annotation:

Give the vertex counts of the connected components:

Do the same using the "List" annotation:

Do the same using the "VertexCount" annotation:

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 Conway–Smith 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:

Return the root graph of the icosahedral line graph:

Return the name of the line graph:

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:

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:

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 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 chordless cycle lengths:

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:

As a function of a variable x:

Compare to the explicit covers:

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

As a function of a variable x:

Compare to the explicit cuts:

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 edge cut polynomial of the utility graph as a pure function:

As a function of a variable x:

Compare to the explicit cuts:

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 minimal vertex cut polynomial of the utility graph as a pure function:

As a function of a variable x:

Compare to the explicit cuts:

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 Q-chromatic polynomial of the Chvátal 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 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:

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

As a function of a variable x:

Compare to the explicit cuts:

Give the chromatic invariant cubical graph:

Give the chromatic number of the cubical graph:

Compare with the built-in function:

Visualize the chromatic number:

Give the edge chromatic number of the cubical graph:

Compare with the built-in function:

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 the number of minimum vertex colorings of the octahedral graph:

Do the same using an annotation:

Compare with the number of actual colorings:

Give the minimum vertex colorings of the octahedral graph:

Verify the colorings are consistent with the chromatic number:

Visualize the colorings:

Show a minimum-weight fractional coloring of the Petersen graph:

Compute the fractional chromatic number:

Compare with the direct property:

Give the Q-chromatic polynomial of the Chvátal graph:

The Q-chromatic polynomial has smaller degree and coefficients than the usual chromatic polynomial for graphs with chromatic number at least 3:

The Q-chromatic polynomial is not defined for graphs with chromatic number less than 3:

Give the Weisfeiler–Leman dimension of the tetrahedral graph:

Give the ABC index of the butane graph:

Compare with the sum of ABC matrix elements:

Give the arithmetic-geometric index of the propane graph:

Compare with the sum of arithmetic-geometric matrix elements:

Display the Balaban index of the Coxeter graph:

Give the Balaban index of the isobutane graph:

Give the circuit rank of the Coxeter graph:

Display the circuit rank 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 Randić index of the octahedral graph:

Compare with the sum of Randić matrix elements:

Give the Sombor index of the octahedral graph:

Compare with the sum of Sombor matrix elements:

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 first Zagreb index of the cubic graph:

The first Zagreb index is defined as the sum of squared vertex degrees:

Give the second Zagreb index of the cubic graph:

The second Zagreb index is defined as the sum of squared vertex degrees:

Give the ABC matrix of the propane graph:

Give the arithmetic-geometric matrix of the propane graph:

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:

Give the detour matrix of the cubical graph:

Return the distance matrix 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:

Show the maximum flow matrix for the E graph:

Show the minimum cost flow matrix for the E graph:

Give the normalized Laplacian matrix of the octahedral graph:

Give the matrix in expanded form:

Plot the matrix:

Give the Randić matrix of the propane graph:

Give the resistance matrix of the cubical graph:

Give the Sombor matrix of the propane graph:

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:

Display the curvatures of the A graph:

Show isolated points:

Find leaves:

Show number of leaves directly:

Do the same using the "Count" annotation:

Visualize: