Maximum walk entropy implies walk regularity

TL;DR

This paper establishes a precise characterization of walk-regular graphs through the maximum walk entropy at specific inverse temperatures, confirming conjectures and clarifying the behavior of walk entropy across different graph regularities.

Contribution

It proves that a graph is walk-regular if and only if its walk entropy at $eta=1$ equals $ ext{ln} n$, and clarifies the entropy behavior for regular and non-regular graphs.

Findings

Walk-regular graphs have maximum walk entropy at $eta=1$ equal to $ ext{ln} n$.

Regular but not walk-regular graphs have strictly lower walk entropy for all $eta>0$.

Non-regular graphs have walk entropy bounded away from $ ext{ln} n$ by a positive epsilon.

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