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Three different zeta functions are attached to a finite connected, possibly irregular graph X. They originate with a zeta function of Ihara which is an analogue of Riemann's as well as Selberg's zeta function. The three zeta functions are associated to one vertex variable, two variables for each edge, and 2r(2r&1) path variables, respectively. Here r is the number of generators of the fundamental group of X. We show how to specialize the variables of the last two zeta functions to obtain the first and we give elementary proofs of generalizations 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.

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We discuss zeta functions of finite irregular undirected connected graphs (which may be weighted) and apply them to obtain, for example an analog of the prime number theorem for cycles in graphs. We consider 3 types of zeta functions of graphs: vertex, edge, and path. Analogs of the Riemann hypothesis are also introduced.