Zeta Functions with Respect to General Coined Quantum Walk of Periodic Graphs

Takashi Komatsu,Iwao Sato,Norio Konno

doi:10.37236/9104

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 (inifinite) periodic graph.

Highlights

Starting from p-adic Selberg zeta functions, Ihara [12] introduced the Ihara zeta functions of graphs
The Ihara zeta function of a finite graph was extended to an infinite graph in [1, 3, 6, 7, 8, 9], and its determinant expressions were presented
Guido, Isola and Lapidus [8] presented a determinant expression for the Ihara zeta function of a periodic graph

Summary

Introduction

Starting from p-adic Selberg zeta functions, Ihara [12] introduced the Ihara zeta functions of graphs. The Ihara zeta function of a finite graph was extended to an infinite graph in [1, 3, 6, 7, 8, 9], and its determinant expressions were presented. Guido, Isola and Lapidus [8] presented a determinant expression for the Ihara zeta function of a periodic graph. We define a zeta function of a periodic graph by using the time evolution matrix of a general coined quantum walk on it, and present its determinant expression.

The Ihara zeta function of a graph

The Grover walk on a graph

Periodic graphs

An analytic determinant for von Neumann algebras with a finite trace