Zeta functions with respect to general coined quantum walk of periodic graphs

TL;DR

This paper introduces a new zeta function for graphs based on the time evolution matrix of general coined quantum walks, providing determinant formulas for finite and periodic infinite graphs.

Contribution

It defines a novel zeta function for graphs using quantum walk dynamics and derives explicit determinant expressions for both finite and infinite periodic graphs.

Findings

Determinant expression for the zeta function of finite graphs

Determinant expression for the zeta function of infinite periodic graphs

Extension of zeta function concept to quantum walk frameworks

Abstract

We define a zeta function of a graph by using the time evolution matrix of a general coined quantum walk on it, and give a determinant expression for the zeta function of a finite graph. Furthermore, we present a determinant expression for the zeta function of an (infinite) periodic graph.