A weighted graph zeta function involved in the Szegedy walk

TL;DR

This paper introduces a new weighted zeta function for finite graphs, deriving its determinant expression, which directly relates to the characteristic polynomial of the Szegedy walk's transition matrix.

Contribution

The paper presents a novel weighted zeta function for graphs and connects it to the Szegedy walk's transition matrix, providing new analytical tools.

Findings

Derived the determinant expression of the weighted zeta function.

Established the connection to the characteristic polynomial of the Szegedy walk.

Provided a new analytical framework for studying quantum walks on graphs.

Abstract

We define a new weighted zeta function for a finite graph and obtain its determinant expression. This result gives the characteristic polynomial of the transition matrix of the Szegedy walk on a graph.