TOPICS

# Cubic Vertex-Transitive Graph

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).

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 Tutte 8-cage 32 Ct71 Dyck graph

Crossed Prism Graph, Cubic Graph, Cubic Symmetric Graph, Quartic Vertex-Transitive Graph, Vertex-Transitive Graph

## Explore with Wolfram|Alpha

More things to try:

## References

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."

## Referenced on Wolfram|Alpha

Cubic Vertex-Transitive Graph

## Cite this as:

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