Metzler/Zeta Correspondence

TL;DR

This paper derives an explicit determinant formula for Metzler matrices of digraphs and introduces a walk-type zeta function for symmetric digraphs of finite tori, providing a limit expression via integrals.

Contribution

It offers a novel explicit determinant formula for Metzler matrices and introduces a new walk-type zeta function with a limit expression for symmetric digraphs.

Findings

Explicit determinant formula for Metzler matrices.

Introduction of a walk-type zeta function for symmetric digraphs.

Limit formula expressed through integrals.

Abstract

We present an explicit formula for the determinant on the Metzler matrix of a digraph $D$. Furthermore, we introduce a walk-type zeta function with respect to this Metzler matrix of the symmetric digraph of a finite torus, and express its limit formula by using the integral expression.