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arXiv · 2201.03973

A Generalized Grover/Zeta Correspondence

Takashi Komatsu·Norio Konno·Iwao Sato·Shunya Tamura
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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.

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Takashi Komatsu, Norio Konno, Iwao Sato, Shunya Tamura. 2022-01-09. A Generalized Grover/Zeta Correspondence. https://arxiv.org/abs/2201.03973

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