The trace formula with respect to the Grover matrix of a graph. Issue 10 (3rd July 2022)

• Record Type: Journal Article
• Authors: Konno, Norio; Mitsuhashi, Hideo; Morita, Hideaki; Sato, Iwao
• Is Part Of: Linear & multilinear algebra. Volume 70:Issue 10(2022)
• Page Start: 1963
• Page End: 1979
• Publication Date: 2022-07-03
• Subjects: Zeta function -- quantum walk -- Selberg trace formula -- regular graph; Algebras, Linear -- Periodicals; Multilinear algebra -- Periodicals; 512.505
• DOI: 10.1080/03081087.2020.1779173
• Languages: English
• ISSNs: 0308-1087

Abstract

ABSTRACT: As a first step of the study of the relation between the distribution of prime, reduced cycles of a graph and the limit theorem for the probability of a quantum walk on a graph, we present a trace formula with respect to the Grover matrix that is the transition matrix of the Grover walk on a graph. We define a zeta function with respect to the Grover matrix of a graph, and differentiate the logarithm of its zeta function of a regular graph. Furthermore, we obtain a trace formula with respect to the Grover matrix by a suitable complex integral of its logarithm differential.