An algebraic-combinatorial framework for finding the average hitting times in graphs with high regularity

Abstract

For any given vertices $u$ and $v$ in a graph, the hitting time of a random walk on a finite graph is the number of steps it takes for a random walk to reach vertex $v$ starting at vertex $u$. The expected value of the hitting time is the average hitting time. In this paper, we present an algebraic-combinatorial method for calculating the average hitting time between vertices of finite graphs exhibiting high regularity, along with its applications to multiple graph classes. Our approach exploits a novel connection between maximal-entropy random walks and weight-equitable partitions, providing a unifying framework that strengthens and extends several known results, including Rao's method [Statistics \& Probability Letters, 2013] for computing the hitting time from a vertex to a neighbor under certain symmetries of the starting vertex.