Zeta Functions of Finite Graphs and Representations of p-Adic Groups

Abstract

Dedicated to Prof. Friedrich Hirzehruch and Prof. lchiro Satake on their sixtieth birthdays Contents § 0. Introduction § 1. Graphs and multigraphs § 2. Zeta functions of finite graphs § 3. Spectrum of a finite multigraph § 4. Harmonic functions and the Hodge decomposition § 5. Representations of C[Ti, T2]; a proof of (3.14) § 6. Representations of p-adic groups § 7. Special values of Zx(u) § 8. Miscellaneous results § 9. Computations of Zx(u) for well known families of X § 10. Examples: list of graphs for n<6, m<1. § 11. References § 0. Introduction 0-1.In this paper we shall be concerned with the two different subjects, which have been developed separately. One is a combinatorial problem in algebraic graph theory, and the other is an arithmetic of discrete subgroups of p-adic groups and their representations.Suppose that Xis a finite (multi)graph, which is not a tree. We always assume that X is non-oriented. A closed path C in X is called reduced, if C and C 2 = C.C have no backtracking. Then obviously the set ~r•d(X) of reduced closed paths of length / is finite, and #(~['a(X)) -oo (l-oo) if Xis not homotopic to a circuit, i.e., S 1 • (See § I for

Selberg′s trace formula on the k‐regular tree and applications

“…Theorem 4.3 can be attributed to many people in the case of both regular and irregular finite graphs. Bass [2], Hashimoto [9], and Sunada [19] certainly should be mentioned. The proof we sketch is due to Ahumada [1].…”

“…It can also be viewed as an analogue of the Dedekind zeta function of an algebraic number field. References are Ahumada [1], Bass [2], Hashimoto [9], Ihara [12], Stark and Terras [17,18], Sunada [19], Venkov and Nikitin [24].…”

“…Some additional references for the subject are Ahumada [1], Brooks [3], Cartier [5], Figá-Talamanca and Nebbia [7], Hashimoto [9], Quenell [15], Stark and Terras [17,18], Sunada [19], Terras [21], Venkov and Nikitin [24].…”

A Balanced Signed Digraph

“…where [C] runs over all equivalence classes of prime, reduced cycles of G (see [1,3,[9][10][11]20,22,23]). Zeta functions of graphs started from zeta functions of regular graphs by Ihara [10].…”

Ihara Zeta Functions of Infinite Weighted Graphs

“…In the case of a finite graph with weight one this goes back to an idea of Hashimoto [Has89], which later was refined by Bass [Bas92].…”