Pure mathematics

Discrete mathematics

Graph theory

Arithmetic zeta function

Computer science

Riemann zeta function

Ramanujan's sum

Graph

Mathematics

Notation

Mathematical analysis

Algebraic graph theory

Combinatorics

Generalization

Algebra over a field

Arithmetic

Advanced studies in pure mathematics

§ 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

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