The combinatorial zeta function is a type of zeta function defined for discrete structures, originating from the study of expressions for the Ihara zeta function of graphs. The Ihara zeta function is expressed in four forms: exponential, Euler, Hashimoto, and Ihara. However, the combinatorial zeta function supports only three of these representations, excluding the Ihara expression. This paper provides an overview of these expressions from the perspective of combinatorial zeta function. Finally, it examines the challenges associated with the Ihara representation and outlines the current progress toward achieving its anticipated conclusions.
