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.
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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)
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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
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DOI: https://doi.org/10.1007/978-3-540-89550-3_15
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