Walk entropy and walk-regularity

A graph is said to be walk-regular if, for each $\ell \geq 1$, every vertex is contained in the same number of closed walks of length $\ell$. We construct a $24$-vertex graph $H_4$ that is not walk-regular yet has maximized walk entropy, $S^V(H_4,\beta) = \log 24$, for some $\beta>0$. This graph is a counterexample to a conjecture of Benzi [Linear Algebra Appl.~443 (2014), 395--399, Conjecture 3.1]. We also show that there exist infinitely many temperatures $\beta_0>0$ so that $S^V(G,\beta_0)=\log n_G$ if and only if a graph $G$ is walk-regular.