A graph is called semisymmetric if it is regular, edge-transitive but not vertex-transitive. I think that semisymmetric graphs are walk-regular hence they provide example of graphs that are regular and walk-regular but not vertex-transitive. Krystal Guo and I proved this last year (and even e-mailed Brendan about it) but deemed it too minor to write it up.

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.

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.

A graph is called semisymmetric if it is regular, edge-transitive but not vertex-transitive. I think that semisymmetric graphs are walk-regular hence they provide example of graphs that are regular and walk-regular but not vertex-transitive. In fact, I think Krystal Guo and I proved this last year (and even e-mailed Brendan about it) but deemed it too minor to write it up. I'll try to see if we still have some notes on this.

If this is right, 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.

A graph is called semisymmetric if it is regular, edge-transitive but not vertex-transitive. I think that semisymmetric graphs are walk-regular hence they provide example of graphs that are regular and walk-regular but not vertex-transitive. Krystal Guo and I proved this last year (and even e-mailed Brendan about it) but deemed it too minor to write it up.

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.