Chapter 7 Symmetric Graphs and Chains

In this Chapter we show how general results in Chapters 3, 4 and 6 can sometimes be strengthened when symmetry is present. Many of the ideas are just simple observations. Since the topic has a \discrete math" avor our default convention is to work in discrete time, though as always the continuous-time case is similar. Note that we use the word \symmetry" in the sense of spatial symmetry (which is the customary use in mathematics as a whole) and not as a synonym for time-reversibility. Note also our use of \random ight" for what is usually called \random walk" on a group. Biggs 6] contains an introductory account of symmetry properties for graphs, but we use little more than the deenitions. I have deliberately not been overly fussy about giving weakest possible hypotheses. For instance many results for symmetric reversible chains depend only of the symmetry of mean hitting times (7), but I haven't spelt this out. Otherwise one can end up with more deenitions than serious results! Instead, we focus on three diierent strengths of symmetry condition. Starting with the weakest, section 1 deals with symmetric reversible chains, a minor generalization of what is usually called \symmetric random walk on a nite group". In the graph setting, this specializes to random walk on a Cayley or vertex-transitive graph. Section 2 deals with random walk on an arc-transitive graph, encompassing what is usually called \random walk on a nite group with steps uniform on a conjugacy class". Section 3 deals with random walk on a distance-regular graph, which roughly corresponds to nearest-neighbor isotropic random walk on a discrete Gelfand pair. This book focuses on inequalities rather than exact calculations, and the 1