Graph Distance Graph

For a connected graph of graph diameter , the graph distance- graph for , ..., is a graph with the same vertex set and having edge set consisting of the pairs of vertices that lie a graph distance apart. It is therefore the case that (Brouwer et al. 1989, p. 437).

Graph distance graphs differ from Euclidean distance graphs in that the former are constructed based on graph distance between vertices, while the latter are constructed based on Euclidean distance.

Special classes of graphs isomorphic to their own graph -distance graph are summarized in the table below.

condition	distance- graph is isomorphic to itself
	-antiprism graph
odd	-caveman graph
odd	prism graph
odd	-sunlet graph
	torus grid graph
odd	hypercube graph

Notable graphs that are isomorphic to their own distance-k graph are summarized in the following table.

	graph and distance- graphs are isomorphic
2	generalized quadrangle , icosahedral graph, Paley graphs, Peisert graphs
3	Franklin graph, Hoffman graph, Pasch graph, rhombic dodecahedral graph, Wells graph
4	dodecahedral graph, Greenfield graph
5	rhombic triacontahedral graph, Schläfli double six graph, subdivided cubical graph, truncated octahedral graph
7	Foster graph
9	Foster graphs , , , , subdivided dodecahedral graph
11	Foster graphs , , , , , , ,
13	Foster graphs , , ,
17	Foster graphs