Maximum walk entropy implies walk regularity

The notion of walk entropy $S^V(G,\beta)$ for a graph $G$ at the inverse temperature $\beta$ was put forward recently by Estrada et al. (2014) \cite{6}. It was further proved by Benzi \cite{1} that a graph is walk-regular if and only if its walk entropy is maximum for all temperatures $\beta \in I$, where $I$ is a set of real numbers containing at least an accumulation point. Benzi \cite{1} conjectured that walk regularity can be characterized by the walk entropy if and only if there is a $\beta>0$, such that $S^V(G,\beta)$ is maximum. Here we prove that a graph is walk regular if and only if the $S^V(G,\beta=1)=\ln n$. We also prove that if the graph is regular but not walk-regular $S^V(G,\beta)<\ln n$ for every $\beta >0$ and $\lim_{\beta \to 0} S^V(G,\beta)=\ln n=\lim_{\beta \to \infty} S^V(G,\beta)$. If the graph is not regular then $S^V(G,\beta) \leq \ln n-\epsilon$ for every $\beta>0$, for some $\epsilon>0$.