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 ak-regular treeΞ(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 connectedk-regular graphX≅Γ\Ξare given along with two applications. The first application is to obtain an asymptotic formula for the number of closed paths of lengthrinX(without backtracking but possibly with tails). The second application is to deduce the chaotic properties of the induced geodesic flow onX(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 ofX, leading to a graph theoretic analogue of the prime number theorem.
The contest between the kernels in the Selberg trace formula for the (𝑞+1)-regular tree