In this chapter we explore a theory which gives an alternative approach to some of the diffusion processes presented in the introduction (namely the random walk on the discrete circle, the Ehrenfest and the Bernoulli––Laplace models). In some sense, this can be regarded as a theory of (finite) Gelfand pairs without group theory. Thus, Sections 5.1, 5.2, and 5.3 (as well as Sections 6.1 and 6.3 in the next chapter) do not rely on group representation theory and can be read independently of Chapters 3 and 4. The connection with group theory will be presented in the final part of Section 5.4 and in Section 6.2.

Harmonic analysis on distance-regular graphs

In this section we focus our attention on a remarkable class of finite graphs for which it is possible to develop a nice harmonic analysis. Our exposition is inspired to the monographs by Bailey and by Bannai and Ito. We would like to mention that during our preparation of this book we attended a minicourse by Rosemary Bailey on association schemes which undoubtedly turned out to be very useful and stimulating for us.

We shall denote by X a finite, connected (undirected) graph without self-loops. Recall that given two vertices x, y ∈ X, their distance d(x, y) is the length of the shortest path joining x and y. This way, (X, d) becomes a metric space.