The class of walk-regular graphs contains the vertex-transitive graphs and the distance-regular graphs. However, there are walk-regular graphs that are neither vertex-transitive nor distance-regular. In particular, I believe that I found a first example of a cubic walk-regular graph that is neither vertex-transitive nor distance-regular on 20 vertices: its *Graph6* code is `SsP@@?OC?S@C@_@C?K?A_?AG?D??C_??]`

.

My question here is: are there only finitely many cubic walk-regular graphs (up to isomorphism) that are neither vertex-transitive nor distance-regular *(or is it e.g. possible to construct an infinite family of graphs of this kind)*?

*There are e.g. only finitely many distinct cubic distance-regular graphs - could it e.g. also be true that there are only finitely many distinct non-vertex-transitive cubic walk-regular graphs?*

connected finite simple graph, I suppose. – m4farrel Mar 8 '17 at 3:59