International Journal of Mathematics and Mathematical Sciences
Volume 2003 (2003), Issue 8, Pages 501-526
doi:10.1155/S016117120311126X
Selberg's trace formula on the k-regular tree and applications
1Department of Mathematics, University of California (UCSD), San Diego, La Jolla 92093-0112, CA, USA
2Department of Mathematics, Dartmouth College, Hanover 03755, NH, USA
Received 7 November 2001
Copyright © 2003 Audrey Terras and Dorothy Wallace. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.