Zeta Functions of Finite Graphs and Coverings, Part II

Abstract

Galois theory for normal unramified coverings of finite irregular graphs (which may have multiedges and loops) is developed. Using Galois theory we provide a construction of intermediate coverings which generalizes the classical Cayley and Schreier graph constructions. Three different analogues of Artin L-functions are attached to these coverings. These three types are based on vertex variables, edge variables, and path variables. Analogues of all the standard Artin L-functions results for number fields are proved here for all three types of L-functions. In particular, we obtain factorization formulas for the zeta functions introduced in Part I as a product of L-functions. It is shown that the path L-functions, which depend only on the rank of the graph, can be specialized to give the edge L-functions, and these in turn can be specialized to give the vertex L-functions. The method of Bass is used to show that Ihara type quadratic formulas hold for vertex L-functions. Finally, we use the theory to give examples of two regular graphs (without multiple edges or loops) having the same vertex zeta functions. These graphs are also isospectral but not isomorphic.