Zeta functions of finite graphs and coverings
This paper deals with zeta functions \(Z_X(u)\) attached to a finite connected, possibly regular, graph \(X\). They originate with a zeta function of Ihara and are generalized by the authors. There are three types of generalizations of zeta functions; zeta functions with vertex variables, zeta functions with edge variables, zeta functions with path variables, the latter appearing here for the first time. The authors show how to specialize the edge variables and the path variables to obtain zeta functions with vertex variables. In addition, they give elementary proofs of Ihara's formula, which says that the zeta function for a regular graph is the reciprocal of a polynomial. Many examples of covering graphs are also considered in this paper.
scientific article; zbMATH DE number 934294