Journal of the
Korean Mathematical Society
JKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

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J. Korean Math. Soc. 2006; 43(6): 1269-1287

Printed November 1, 2006

Copyright © The Korean Mathematical Society.

Zeta functions of graph bundles

Rongquan Feng and Jin Ho Kwak

Peking University, Pohang University of Science and Technology

Abstract

As a continuation of computing the zeta function of a regular covering graph by Mizuno and Sato in [9], we derive in this paper computational formulae for the zeta functions of a graph bundle and of any (regular or irregular) covering of a graph. If the voltages to derive them lie in an abelian or dihedral group and its fibre is a regular graph, those formulae can be simplified. As a by-product, the zeta function of the cartesian product of a graph and a regular graph is obtained. The same work is also done for a discrete torus and for a discrete Klein bottle.

Keywords: zeta function, graph bundle, voltage assignment, discrete torus or Klein bottle

MSC numbers: 05C50, 05C25, 15A15, 15A18

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