Abstract
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.