Simple random walk on distance-regular graphs

TL;DR

This paper surveys known results on simple random walks on distance-regular graphs, highlighting explicit calculations of electric resistance, hitting times, and bounds on cover and mixing times to promote further research.

Contribution

It consolidates existing knowledge on random walks in distance-regular graphs and emphasizes their role as a natural setting for studying such processes.

Findings

Explicit formulas for electric resistance and hitting times

Bounds on cover times and mixing times

Discussion of harmonic functions and cutoff phenomenon

Abstract

A survey is presented of known results concerning simple random walk on the class of distance-regular graphs. One of the highlights is that electric resistance and hitting times between points can be explicitly calculated and given strong bounds for, which leads in turn to bounds on cover times, mixing times, etc. Also discussed are harmonic functions, moments of hitting and cover times, the Green's function, and the cutoff phenomenon. The main goal of the paper is to present these graphs as a natural setting in which to study simple random walk, and to stimulate further research in the field.