The trace formula with respect to the Grover matrix of a graph
Norio Konno, Hideo Mitsuhashi, Hideaki Morita, Iwao Sato · Linear and Multilinear Algebra · 2020
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.