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The pentakis icosidodecahedral graph, illustrated above, is a graph on 42 vertices that is the skeleton of the pentakis icosidodecahedron.

The detour index omega(G) of a graph G is a graph invariant defined as half the sum of all off-diagonal matrix elements of the detour matrix of G. Unless otherwise stated, ...

The cricket graph is the 5-vertex graph illustrated above. It is implemented in the Wolfram Language as GraphData["CricketGraph"].

The net graph is the graph on 6 vertices illustrated above. It is implemented in the Wolfram Language as GraphData["NetGraph"]. The bipartite double graph of the net graph is ...

The domino graph is the graph on 6 vertices illustrated above. It is isomorphic to the 3-ladder graph and the (2,3)-grid graph. It is implemented in the Wolfram Language as ...

The fork graph, sometimes also called the chair graph, is the 5-vertex tree illustrated above. It could perhaps also be known as the 'h graph' (but not to be confused with ...

The Schläfli double sixes graph is the Levi graph of the double sixes configuration.

The paw graph is the 3-pan graph, which is also isomorphic to the (3,1)-tadpole graph. It is implemented in the Wolfram Language as GraphData["PawGraph"].

A graph G is the edge graph of a polyhedron iff G is a simple planar graph which is 3-connected.

The dart graph is the 5-vertex graph illustrated above. It is implemented in the Wolfram Language as GraphData["DartGraph"].