Zeta functions of graphs with ℤ actions

2006/07/26 by Bryan Clair, Clair, Bryan
Mathematics · #11M41 #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT) #math.CO #math.NT #msc:11M41

paper · pdf · doi:10.48550/arxiv.math/0607689

16 pages, 3 figures

arxiv created 2006/07/26 · arxiv updated 2009/12/01

Abstract

Suppose Y is a regular covering of a graph X with covering transformation group π= ℤ. This paper gives an explicit formula for the L² zeta function of Y and computes examples. When π= ℤ, the L² zeta function is an algebraic function. As a consequence it extends to a meromorphic function on a Riemann surface. The meromorphic extension provides a setting to generalize known properties of zeta functions of regular graphs, such as the location of singularities and the functional equation.

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