Maximum walk entropy implies walk regularity

The notion of walk entropy SV(G,β) for a graph G at the inverse temperature β was put forward recently by Estrada et al. (2014) [7]. It was further proved by Benzi [1] that a graph is walk-regular if and only if its walk entropy is maximum for all temperatures β∈I, where I is a set of real numbers containing at least an accumulation point. Benzi [1] conjectured that walk regularity can be characterized by the walk entropy if and only if there is a β>0 such that SV(G,β) is maximum. Here we prove that a graph is walk regular if and only if the SV(G,β=1)=lnn. We also prove that if the graph is regular but not walk-regular SV(G,β)0 and limβ→0SV(G,β)=lnn=limβ→∞SV(G,β). If the graph is not regular then SV(G,β)≤lnn−ϵ for every β>0, for some ϵ>0.