Selberg's trace formula on the -regular tree and applications

We survey graph theoretic analogues of the Selberg trace and pretrace formulas along with some applications. This paper includes a review of the basic geometry of a k-regular tree Ξ (symmetry group, geodesics, horocycles, and the analogue of the Laplace operator). A detailed discussion of the spherical functions is given. The spherical and horocycle transforms are considered (along with three basic examples, which may be viewed as a short table of these transforms). Two versions of the pretrace formula for a finite connected k-regular graph X≅Γ\Ξ are given along with two applications. The first application is to obtain an asymptotic formula for the number of closed paths of length r in X (without backtracking but possibly with tails). The second application is to deduce the chaotic properties of the induced geodesic flow on X (which is analogous to a result of Wallace for a compact quotient of the Poincaré upper half plane). Finally, the Selberg trace formula is deduced and applied to the Ihara zeta function of X, leading to a graph theoretic analogue of the prime number theorem.