Bartholdi zeta and L-functions of weighted digraphs, their coverings and products

• Autores: Young Bin Choe, Jin Ho Kwak, Yong Sung Park, Iwao Sato
• Localización: Advances in mathematics, ISSN 0001-8708, Vol. 213, Nº 2, 2007, págs. 865-886
• Idioma: inglés
• DOI: 10.1016/j.aim.2007.01.013
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• Resumen

  • Since a zeta function of a regular graph was introduced by Ihara [Y. Ihara, On discrete subgroups of the two by two projective linear group over p-adic fields, J. Math. Soc. Japan 19 (1966) 219¿235], many kinds of zeta functions and L-functions of a graph or a digraph have been defined and investigated. Most of the works concerning zeta and L-functions of a graph contain the following: (1) defining a zeta function, (2) defining an L-function associated with a (regular) graph covering, (3) providing their determinant expressions, and (4) computing the zeta function of a graph covering and obtaining its decomposition formula as a product of L-functions. As a continuation of those works, we introduce a zeta function of a weighted digraph and an L-function associated with a weighted digraph bundle. A graph bundle is a notion containing a cartesian product of graphs and a (regular or irregular) graph covering. Also we provide determinant expressions of the zeta function and the L-function. Moreover, we compute the zeta function of a weighted digraph bundle and obtain its decomposition formula as a product of the L-functions.