TITLE: Does the clique-coclique bound hold for all walk-regular graphs?
QUESTION [8 upvotes]: The clique-coclique bound is said to hold for a simple graph $G$ on $n$ vertices if $\lvert \omega(G) \rvert \lvert \alpha(G) \lvert \leq n$, letting $\omega(G)$ and $\alpha(G)$ denote its clique and coclique (independent set) numbers respectively.
It is known, in particular, that the clique-coclique bound holds for all vertex-transitive graphs and distance-regular graphs - two families of walk-regular graphs.
The clique-coclique also appears to hold for all of the examples of walk-regular graphs that I know of that are neither vertex-transitive nor distance-regular. It is also apparent that the clique-coclique bound holds for some other families of walk-regular graphs, namely semi-symmetric graphs.
Could it be possible that the clique-coclique bound actually holds for all (connected) walk-regular graphs?
By informal reasoning in head, it feels plausible to me that this could be the case? I wonder what would might be a good approach to take to try to prove or disprove this?

REPLY [4 votes]: This is an old thread and already answered, but in case it helps anyone, here is a smaller walk-regular graph that does not satisfy the clique/coclique bound. (It may not be the smallest.)

It has 18 vertices, and its automorphism group has two orbits on vertices (the pink and the blue in the picture). 
Graph6: QGEENDpMGwpppowomEHp`FBEKMG

It has six maximum cliques of size $5$, with representative $\{0, 6, 7, 12, 13\}$ and six maximum cocliques of size $4$, with representative $\{0, 1, 9, 16\}$.
As $\alpha \times \omega = 4 \times 5 = 20 > 18$, this violates the clique-coclique condition.