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http://dx.doi.org/10.4134/JKMS.2006.43.6.1269

ZETA FUNCTIONS OF GRAPH BUNDLES  

Feng, Rongquan (LMAM, School of Mathematical Sciences Peking University)
Kwak, Jin-Ho (Department of Mathematics Pohang University of Science and Technology)
Publication Information
Journal of the Korean Mathematical Society / v.43, no.6, 2006 , pp. 1269-1287 More about this Journal
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;
Citations & Related Records

Times Cited By Web Of Science : 2  (Related Records In Web of Science)
Times Cited By SCOPUS : 2
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1 K. Hashimoto, Zeta functions of finite graphs and representations of p-adic groups, Adv. Stud. Pure Math. 15 (1989), 211-280
2 Y. Ihara, On discrete subgroups of the two by two projective linear group over p-adic fields, J. Math. Soc. Japan 18 (1966), 219-235   DOI
3 J. H. Kwak and J. Lee, Isomorphism classes of graph bundles, Canad. J. Math. 42 (1990), no. 4, 747-761   DOI
4 J. H. Kwak and J. Lee,, Characteristic polynomials of some graph bundles II, Linear and. Multilinear Algebra 32 (1992), no. 1, 61-73   DOI
5 J. H. Kwak and Y. S. Kwon, Characteristic polynomials of graph bundles having voltages in a dihedral group, Linear Algebra Appl. 336 (2001), 99-118   DOI   ScienceOn
6 H. Mizuno and I. Sato, Zeta functions of graph coverings, J. Combin. Theory Ser, B 80 (2000), no. 2, 247-257   DOI   ScienceOn
7 H. Mizuno and I. Sato, , L-functions for images of graph coverings by some operations, Discrete Math. 256 (2002), no. 1-2, 335-347   DOI   ScienceOn
8 J.-P. Serre, Linear Representations of Finite Groups, Graduate Texts in Mathematics, Vol. 42, Springer-Verlag, New York, 1977
9 A. Terras, Fourier Analysis on Finite Groups and Applications, London Mathematical Society Student Texts, 43, Cambridge University Press, Cambridge, 1999
10 H. M. Stark and A. A. Terras, Zeta functions of finite graphs and coverings, Adv. Math. 121 (1996), no. 1, 124-165   DOI   ScienceOn
11 H. Bass, The Ihara-Selberg zeta function of a tree lattice, Internat. J. Math. 3 (1992), no. 6, 717-797   DOI
12 N. Biggs, Algebraic Graph Theory, 2nd ed., Cambridge University Press, Cambridge, 1993
13 J. L. Gross and T. W. Tucker, Generating all graph coverings by permutation voltage assignments, Discrete Math. 18 (1977), no. 3, 273-283   DOI   ScienceOn
14 S. Northshield, A note on the zeta function of a graph, J. Combin. Theory Ser. B 74 (1998), no. 2, 408-410   DOI   ScienceOn
15 T. Sunada, L-functions in geometry and some applications, Lecture Notes in Mathematics, Vol. 1201,266-284, Springer-Verlag, New York, 1986   DOI