Zeta functions of finite graphs and coverings. II
type: Item
instance of: scholarly article
title: Zeta functions of finite graphs and coverings. II
zbMATH Open document ID: 0972.11086
author: Audrey A. Terras
author: Harold M. Stark
published in: Advances in Mathematics
publication date: 2001-11-22T00:00:00Z
review text: Part I, cf. Adv. Math. 121, 124-165 (1996; Zbl 0874.11064). This is the second part of the authors' paper on zeta functions associated to finite graphs. They developed a Galois theory for normal unramified coverings of finite irregular graphs and established analogues between the graph zeta-functions and the classical Artin's \(L\)-functions. They defined three kinds of \(L\)-functions attached to finite graphs as an analogy of Artin's \(L\)-functions for algebraic number fields. These three types are based on vertex variables, edge variables, and path variables. Analogues of all the standard Artin's \(L\)-functions results for number fields are proved here for all three types of \(L\)-functions. In particular, they obtain factorization formulas for the zeta functions introduced in Part I as a product of \(L\)-functions. It is shown that the path \(L\)-functions, which depend only on the rank of the graph, can be specialized to give the edge \(L\)-functions, and these in turn can be specialized to give the vertex \(L\)-functions. In order to prove that Ihara type quadratic formulas hold for vertex \(L\)-functions, the method of Bass is used. Finally, they use the theory to give examples of two regular graphs without multiple edges or loops having the same vertex zeta functions. These graphs are also isospectral but not isomorphic.
reviewed by: Masakazu Muro
Mathematics Subject Classification ID: 11M41
Mathematics Subject Classification ID: 05C25
Mathematics Subject Classification ID: 11T99
zbMATH DE Number: 1525153
zbMATH Keywords: Ihara type quadratic formulas
zbMATH Keywords: zeta functions associated to finite graphs
zbMATH Keywords: \(L\)-functions attached to finite graphs
zbMATH Keywords: vertex variables
zbMATH Keywords: edge variables
zbMATH Keywords: path variables
zbMATH Keywords: factorization formulas
zbMATH Keywords: product of \(L\)-functions
zbMATH Keywords: path \(L\)-functions
zbMATH Keywords: edge \(L\)-functions
zbMATH Keywords: vertex \(L\)-functions
zbMATH Keywords: Ihara type
MaRDI profile type: Publication
full work available at URL: aima.2000.1917
OpenAlex ID: W4213217062
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DOI: 10.1006/AIMA.2000.1917
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description: scientific article; zbMATH DE number 1525153