Quantum walks, Ihara zeta functions and cospectrality in regular graphs

In this paper we explore an interesting relationship between discrete-time quantum walks and the Ihara zeta function of a graph. The paper commences by reviewing the related literature on the discrete-time quantum walks and the Ihara zeta function. Mathematical definitions of the two concepts are then provided, followed by analyzing the relationship between them. Based on this analysis we are able to account for why the Ihara zeta function can not distinguish cospectral regular graphs. This analysis suggests a means by which to develop zeta functions that have potential in distinguishing such structures.