On Formal Graph-Theoretic Definitions of the Ihara Zeta Function

First defined in 1966, the Ihara zeta function has been an important tool in the study of graphs for over half a century. During that time, two prominent equivalent definitions of the zeta function have emerged. The primary purpose of this paper is to provide a useful survey of the literature concerning these definitions of the Ihara zeta function and to compile them into one framework. In the context of this framework, we then introduce a new definition which is potentially more useful to certain applications, and demonstrate its utility by giving an example of computing the Ihara zeta function via the new definition, and showing that it can be applied to give fairly simple proofs of existing results. This paper contains little original mathematics. Its aim is rather to provide a useful new tool for those working with the Ihara zeta function