Zeta Functions of Finite Graphs and Representations of p-Adic Groups

This chapter focuses on zeta functions of finite graphs and representations of p-adic groups. It discusses two different subjects: first is a combinatorial problem in algebraic graph theory, and the other is arithmetic of discrete subgroups of p-adic groups and their representations. The chapter presents the notation and basic definitions in graph theory. It also presents a generalization of the definition of zeta function. Spectrum of a finite multigraph is analyzed in the chapter. Moreover, the chapter also describes harmonic functions and the Hodge decomposition. The chapter also presents the computation of zeta functions Z_x(u) for some well known families of graphs. These computations give many examples of graphs that are not Ramanujan graphs.