Smallest Graph that is Regular but not Vertex-Transitive?

Question

I'm trying to find the smallest graph that is regular but not vertex-transitive, where by smallest I mean "least number of vertices", and if two graphs have the same number of vertices, then the smaller is the one with the lower number of edges.

I currently have that the smallest such graph is the disjoint union of the three-cycle and the four-cycle.

Are there any smaller graphs?

Answer 1

Note that a graph is vertex transitive if and only if its complement is, and a regular graph on at most six vertices is the complement of a regular graph with valency at most two. It is easy to write down all graphs with valency at most two on at most six vertices.