Entropy and the growth rate of universal covering trees

TL;DR

This paper investigates the relationship between two graph parameters, establishing a condition for their equality and exploring implications for entropy growth and random walk behavior.

Contribution

It provides a necessary and sufficient condition for the equality of the parameters and analyzes the variance of random bits in non-backtracking random walks.

Findings

Derived an easy-to-check condition for $ ho = \Lambda$

Showed variance of random bits is bounded if $ ho = \\Lambda$

Identified infinitely many graphs with $ ho = \\Lambda$

Abstract

This work studies the relation between two graph parameters, $\rho$ and $\Lambda$. For an undirected graph $G$, $\rho(G)$ is the growth rate of its universal covering tree, while $\Lambda(G)$ is a weighted geometric average of the vertex degree minus one, corresponding to the rate of entropy growth for the non-backtracking random walk (NBRW). It is well known that $\rho(G) \geq \Lambda(G)$ for all graphs, and that graphs with $\rho=\Lambda$ exhibit some special properties. In this work we derive an easy to check, necessary and sufficient condition for the equality to hold. Furthermore, we show that the variance of the number of random bits used by a length $\ell$ NBRW is $O(1)$ if $\rho = \Lambda$ and $\Omega(\ell)$ if $\rho > \Lambda$. As a consequence we exhibit infinitely many non-trivial examples of graphs with $\rho = \Lambda$.