Lecture 2. Three symmetric spaces
- Zeta Functions of Graphs
First let us consider the graph theoretic analogue of Selberg's zeta function. This
function was first considered by Ihara [40J in a paper that considers p-adic groups
rather than graphs. Then Serre [70] explained the connection with graphs. Later
authors extended Ihara's results to non-regular graphs and zeta functions of many
variables. References are Bass [8], Hashimoto [34], Stark and Terras [73], [7 4],
[75], [76], Sunada [80], Venkov and Nikitin [89].
Let X be a connected finite (not necessarily regular) graph with undirected edge
set E. For examples look at Figure 2. We orient the edges of X arbitrarily and label
them ei , e2, ... , e1e 1, elEl+l = e11, ... , e 2 1e 1 = e ji- 1. Here the inverse of an edge is the
edge taken with the opposite orientation. A prime [CJ in Xis an equivalence class
of tailless backtrackless primitive paths in X. Here write C = a 1 a 2 ···as, where
aj is an oriented edge of X. The length v(C) = s. Backtrackless means that
ai+ 1 #-ai^1 , for all i. Tailless means that as #-a!^1. The equivalence class of C
is [CJ= {a 1 a2 ···as , a2 · · · asa1 , ... , asa1a2 · · · as-1}; i.e., the same path with all
possible starting points. We call the equivalence class primitive if C #-Dm , for
all integers m 2 2, and a ll paths Din X.
Definition 1. The !hara zeta function of X is defined for u E C with f uf
sufficiently small by
(x(u) = IT (1-uv(C))-1.
[CJ prime
in X
Theorem 1. (!hara). If A denotes the adjacency matrix of X and Q the diagonal
matrix with jth entry qj = (d egree of the jth vertex -1), then
(x(u)-^1 = (1 - u^2 r-^1 det(J - Au+ Qu^2 ).
Here r denotes the rank of the fundamental group of X. That is, r = fEf - fVI + 1.
For regula r graphs, when Q = ql is a scalar matrix, you can prove this theorem
using the Selberg trace formula (discussed below) for the (q + 1)-regular tree T.
See Terras [82J or Venkov and Nikitin [89J. Here X = T/ r , where r denotes the
fundamental group of X. In the general case there are m any proofs. See Stark and
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