Advanced studies in pure mathematics· 2018Q3
Zeta Functions of Finite Graphs and Representations of $p$-Adic Groups
- 275citations
- Q3SCImago
- 2018year
Short summary
This paper introduces a novel connection between combinatorial graph theory and the arithmetic of p-adic groups by defining and analyzing zeta functions for finite graphs.
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Key points
- Defines zeta functions for finite, non-oriented graphs, extending concepts from algebraic graph theory.
- Investigates the connection between graph spectra, harmonic functions, and Hodge decomposition.
- Establishes a link between graph zeta functions and representations of p-adic groups.
- Provides examples and computations for specific families of graphs.
AI-generated from the title and abstract; the full text is not read.
Abstract
§ 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
The authors' abstract, as published at the source. Advanced studies in pure mathematics, 2018 · DOI ↗
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Field: Mathematical Physics
Mathematical PhysicsMathematics