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

Abstract

Dedicated to Prof. Friedrich Hirzehruch and Prof. lchiro Satake on their sixtieth birthdaysGraphs 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

Nonbacktracking spectrum of random graphs: Community detection and nonregular Ramanujan graphs

“…Hashimoto introduced the matrix B in the context of the Ihara zeta function [12]. We have the identity…”

“…It is indexed by the set E = {(u, v) : {u, v} ∈ E} of oriented edges in E and defined by B ef = 1 I(e 2 = f 1 )1 I(e 1 = f 2 ) = 1 I(e 2 = f 1 )1 I(e = f −1 ), where for any e = (u, v) ∈ E, we set e 1 = u, e 2 = v, e −1 = (v, u). This matrix was introduced by Hashimoto [12]. A non-backtracking walk is a directed path of directed edges of G such that no edge is the inverse of its preceding edge.…”

On the K-Theory of Graph C *-Algebras

“…This operator is represented by a 2m × 2m-matrix A, whose entries are in {0, 1}. The operator T was considered by Hashimoto [13] and Bass [2] in connection with their study of the Ihara zeta function of a graph (see also: Stark and Terras [21]). Although not strictly necessary for the main argument of this paper, we will comment upon this relation in 2.12 below.…”

“…The operator T was considered by Hashimoto [13] and Bass [2] in connection with their study of the Ihara zeta function of a graph (see also: Stark and Terras [21]). …”

Répartition asymptotique des valeurs propres de l’opérateur de Hecke 𝑇_𝑝

“…Supposons E connexe, et soit E son revêtement universel, relativement a un point-base x ∈ som E. Le graphe E est un arbre régulier de valence q + 1, et l'on peutécrire E sous la forme E/Γ E , où Γ E = π 1 (E, x) est un sous-groupe discret sans torsion de Aut( E). Les définitions ci-dessus (ainsi que celles données plus loin) peuvent se traduire en termes de Γ E ; c'est le point de vue de Ihara [14]; voir aussi [11] et [16].…”