Maximum walk entropy implies walk regularity

Abstract

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 $S^V(G,\beta )\le \ln n-\epsilon$ for every $\beta >0$, for some $\epsilon >0$.

The proof of the main result contains a flaw. It has been corrected in the paper: arXiv:1708.09700, Kloster K, Sullivan BD. Walk entropy and walk-regularity. Linear Algebra and its Applications. 2018 Jun 1;546:115-21

References in corpus (1)

• The Physics of Communicability in Complex Networks