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. be more generally true that there are only finitely many distinct non-vertex-transitive cubic walk-regular graphs?

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?