A Generalized Grover/Zeta Correspondence

TL;DR

This paper introduces a generalized Grover matrix for graphs, derives its spectral properties, and defines associated zeta functions, providing explicit formulas and limits for certain classes of graphs, extending the Ihara zeta function framework.

Contribution

It presents a new generalized Grover matrix, derives its characteristic polynomial and spectra, and defines generalized zeta functions with explicit formulas for vertex-transitive graphs.

Findings

Explicit formulas for the characteristic polynomial of the generalized Grover matrix.

Spectral analysis of the generalized Grover matrix for regular graphs.

Limit expressions for generalized zeta functions of vertex-transitive graphs and tori.

Abstract

We introduce a generalized Grover matrix of a graph and present an explicit formula for its characteristic polynomial. As a corollary, we give the spectra for the generalized Grover matrix of a regular graph. Next, we define a zeta function and a generalized zeta function of a graph $G$ with respect to its generalized Grover matrix as an analog of the Ihara zeta function and present explicit formulas for their zeta functions for a vertex-transitive graph. As applications, we express the limit on the generalized zeta functions of a family of finite vertex-transitive regular graphs by an integral. Furthermore, we give the limit on the generalized zeta functions of a family of finite tori as an integral expression.