Graph theory zetas first appeared in work of Ihara on p-adic groups in the 1960s (see [62]). Serre [113] made the connection with graph theory. The main authors on the subject in the 1980s and 1990s were Sunada [128–130], Hashimoto [50], [51], and Bass [12]. Other references are Venkov and Nikitin [138] and Northshield's paper in the volume of Hejhal et al. [53]. The main properties of the Riemann zeta function have graph theory analogs, at least for regular graphs. For irregular graphs there is no known functional equation and it is difficult to formulate the Riemann hypothesis, but we will try.

Much of our discussion can be found in the papers of the author and Harold Stark [119], [120], [122]. Part II will include the story of Ihara zetas of regular graphs, the connection between the Riemann hypothesis and expanders, and the graph theory prime number theorem.

We do not consider zeta functions of infinite graphs here; such zeta functions are discussed by, for example, Clair and Mokhtari-Sharghi [31], Grigorchuk and Zuk [46], and Guido, Isola, and Lapidus [48]. Nor do we consider directed graphs; zeta functions for such graphs are discussed by, for example, Horton [57], [58]. There are also extensions to hypergraphs (see Storm [124]) and buildings (see Ming-Hsuan Kang, Wen-Ching Winnie Li, and Chian-Jen Wang [66]).

Throughout Part II we will assume Theorem 2.5, Ihara's theorem. It will be proved in Part III, where we will also consider the multivariable zeta functions known as edge and path zeta functions of graphs. We will show how to specialize the path zeta to the edge zeta and the edge zeta to the original one-variable Ihara (vertex) zeta.