Quantum graph walks II: Quantum walks on graph coverings

TL;DR

This paper introduces new determinant formulas for quantum graph scattering matrices, explores their properties through graph coverings, and defines an L-function to analyze quantum walks on graphs.

Contribution

It provides novel determinant expressions for characteristic polynomials and introduces an L-function, advancing the theoretical understanding of quantum walks on graph coverings.

Findings

Determinant expression for the bond scattering matrix's characteristic polynomial

Decomposition formula for regular coverings of graphs

Definition and determinant expression of an L-function for graphs

Abstract

We give a new determinant expression for the characteristic polynomial of the bond scattering matrix of a quantum graph G. Also, we give a decomposition formula for the characteristic polynomial of the bond scattering matrix of a regular covering of G. Furthermore, we define an L-function of G, and give a determinant expression of it. As a corollary, we express the characteristic polynomial of the bond scattering matrix of a regular covering of G by means of its L-functions. As an application, we introduce three types of quantum graph walks, and treat their relation.