Walk entropy and walk-regularity

A graph is said to be walk-regular if, for each ℓ≥1, every vertex is contained in the same number of closed walks of length ℓ. We construct a 24-vertex graph H4 that is not walk-regular yet has maximized walk entropy, SV(H4,β)=log⁡24, for some β>0. This graph is a counterexample to a conjecture of Benzi (2014) [1, Conjecture 3.1]. We also show that there exist infinitely many temperatures β0>0 so that SV(G,β0)=log⁡nG if and only if a graph G is walk-regular.