Abstract
In this paper, we consider branched (di)graph coverings or graphs with semi-free action. A (di)graph with semi-free action of a group Γ is a (di)graph such that a sub(di)graph is fixed by Γ while its complement carries a free action. A branched regular covering of a (di)graph is a (di)graph, where vertices are either regular (free orbits) or totally ramified (fixed vertices). Deng, Sato and Wu treated the characteristic polynomial of a branched covering of digraph, where a subdigraph is an irregular covering of some digraph and its complement is totally ramified.
We give a decompostion formula for the Bartholdi zeta function of a branched covering of a digraph D which treated by Deng, Sato and Wu. As a corollary, we obtain a decomposition formula for the Bartholdi zeta function of a graph having a semi-free action.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Bartholdi, L.: Counting paths in graphs. Enseign. Math. 45, 83–131 (1999)
Bass, H.: The Ihara-Selberg zeta function of a tree lattice. Internat. J. Math. 3, 717–797 (1992)
Deng, A., Wu, Y.: Characteristic polynomials of digraphs having a semi-free action. Linear Algebra Appl. 408, 189–206 (2005)
Deng, A., Sato, I., Wu, Y.: Homomorphisms, representations and characteristic polynomials of digraphs. Linear Algebra Appl. 423, 386–407 (2007)
Foata, D., Zeilberger, D.: A combinatorial proof of Bass’s evaluations of the Ihara-Selberg zeta function for graphs. Trans. Amer. Math. Soc. 351, 2257–2274 (1999)
Gross, J.L., Tucker, T.W.: Topological Graph Theory. Wiley-Interscience, New York (1987)
Hashimoto, K.: Zeta Functions of Finite Graphs and Representations of p-Adic Groups. In: Adv. Stud. Pure Math., vol. 15, pp. 211–280. Academic Press, New York (1989)
Ihara, Y.: On discrete subgroups of the two by two projective linear group over p-adic fields. J. Math. Soc. Japan 18, 219–235 (1966)
Lee, J., Kim, H.K.: Characteristic polynomials of graphs having a semi-free action. Linear Algebra Appl. 307, 35–46 (2000)
Kotani, M., Sunada, T.: Zeta functions of finite graphs. J. Math. Sci. U. Tokyo 7, 7–25 (2000)
Mizuno, H., Sato, I.: Zeta functions of graph coverings. J. Combin. Theory Ser. B 80, 247–257 (2000)
Mizuno, H., Sato, I.: Bartholdi zeta functions of graph coverings. J. Combin. Theory Ser. B 89, 27–41 (2003)
Mizuno, H., Sato, I.: Bartholdi zeta functions of digraphs. European J. Combin. 24, 947–954 (2003)
Sato, I.: Bartholdi zeta functions of group coverings of digraphs. Far East J. Math. Sci. 18, 321–339 (2005)
Serre, J.-P.: Linear Representations of Finite Group. Springer, New York (1977)
Stark, H.M., Terras, A.A.: Zeta functions of finite graphs and coverings. Adv. Math. 121, 124–165 (1996)
Sunada, T.: L-Functions in Geometry and Some Applications. Lecture Notes in Math, vol. 1201, pp. 266–284. Springer, New York (1986)
Sunada, T.: Fundamental Groups and Laplacians. Kinokuniya, Tokyo (1988) (in Japanese)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Mizuno, H., Sato, I. (2008). Bartholdi Zeta Functions of Branched Coverings of Digraphs. In: Ito, H., Kano, M., Katoh, N., Uno, Y. (eds) Computational Geometry and Graph Theory. KyotoCGGT 2007. Lecture Notes in Computer Science, vol 4535. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-89550-3_15
Download citation
DOI: https://doi.org/10.1007/978-3-540-89550-3_15
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-89549-7
Online ISBN: 978-3-540-89550-3
eBook Packages: Computer ScienceComputer Science (R0)