Vertex-Face/Zeta correspondence

TL;DR

This paper derives the spectra of vertex-face walks on graphs and tori, introduces a new zeta function related to these walks, and provides explicit formulas for these spectral properties.

Contribution

It introduces a new walk-type zeta function for vertex-face walks on the 2D torus and derives explicit spectral formulas for these walks.

Findings

Spectral formulas for vertex-face walk transition matrices

Explicit characteristic polynomial expressions for 2D torus

A new zeta function associated with vertex-face walks

Abstract

We present the characteristic polynomial for the transition matrix of a vertex-face walk on a graph, and obtain its spectra. Furthermore, we express the characteristic polynomial for the transition matrix of a vertex-face walk on the 2-dimensional torus by using its adjacency matrix, and obtain its spectra. As an application, we define a new walk-type zeta function with respect to the transition matrix of a vertex-face walk on the 2-dimensional torus, and present its explicit formula.