A cubic vertex-transitive graph is a cubic graph that is vertex transitive. Read and Wilson (1998, pp. 161-163) enumerate all connected cubic vertex-transitive graphs on 34 and fewer nodes, some of which are illustrated above. The numbers of such graphs on , 4, 6, ... nodes are 0, 1, 2, 2, 3, 4, 3, 4, 5, 7, 3, 11, 5, 6, 10, 10, 5, ... (Sloane's A032355).

The cubic symmetric graphs are a special case of the cubic vertex-transitive graphs (i.e., those that are also edge-transitive).

Classes of connected cubic vertex-transitive graphs include the prism graphs, even Möbius ladders, and crossed prism graphs. Specific cases are summarized in the following table.

vertices	id	graph
4	Ct1	tetrahedral graph
8	Ct5	cubical graph
10	Ct8	Petersen graph
12	Ct11	truncated tetrahedral graph
12	Ct12	Franklin graph
14	Ct15	Heawood graph
18	Ct24	Pappus graph
20	Ct26	Desargues graph
20	Ct27	dodecahedral graph
24	Ct39	cubic symmetric graph
24	Ct40	truncated octahedral graph
24	Ct45	truncated cubical graph
28	Ct53	Coxeter graph
30	Ct63	Levi graph
32	Ct71	Dyck graph

Read, R. C. and Wilson, R. J. An Atlas of Graphs. Oxford, England: Oxford University Press, pp. 161-163, 1998.

Sloane, N. J. A. Sequences A032355 in "The On-Line Encyclopedia of Integer Sequences."

CITE THIS AS:

Weisstein, Eric W. "Cubic Vertex-Transitive Graph." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/CubicVertex-TransitiveGraph.html