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, ... (OEIS
A032355).

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."