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

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 for the random walk to reach vertex $v$ given that it started at vertex $u$. The expected value of the hitting time is the average hitting time. In this paper we present an algebraic method for calculating the average hitting time between vertices of finite graphs exhibiting high regularity, along with its applications to multiple graph classes. Our work unifies and strengthens several previous results, such as the method by Rao [Statistics \& Probability Letters, 83(9) (2013)] to derive the hitting time from one vertex to a neighbor when the graph exhibits certain symmetry on the starting vertex.