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?*



Are there only finitely many distinct cubic walk-regular graphs that are neither vertex-transitive nor distance-regular?
A graph is called semisymmetric if it is regular, edge-transitive but not vertex-transitive. 


Semisymmetric graphs are walk-regular hence they provide example of graphs that are regular and walk-regular but not vertex-transitive. 


This answers your question, as it is known that there are infinitely many semisymmetric cubic graphs (I could probably dig up a reference for this if you need), and, as you note, only finitely many of these can be distance-regular.


Krystal Guo and I noticed this last year (and even e-mailed Brendan about it) but deemed it too minor to write it up.


