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.

We intorduce intriguing structures and behaviors of quantum walks through the survival probability on a connected graph with sinks in the long-time limit. The nonzero survival probability arises from the overlap of the initial state with the dark eigenstate, which is induced by specific graph structures such as fundamental cycles and the shortest path between pairs of self-loops. This phenomenon is derived from the spectrum mapping theorem from the underlying random walk to the induced quantum walk. We demonstrate this counterintuitive phenomenon on a ladder graph of length L, showing that as the length L increases, the survival probability decreases. This effect is called the hashigo-sake effect of the quantum walk. Furthermore, we explain the spectral structure responsible for inducing this behavior.