28 research outputs found
A remark on zeta functions of finite graphs via quantum walks
From the viewpoint of quantum walks, the Ihara zeta function of a finite
graph can be said to be closely related to its evolution matrix. In this note
we introduce another kind of zeta function of a graph, which is closely related
to, as to say, the square of the evolution matrix of a quantum walk. Then we
give to such a function two types of determinant expressions and derive from it
some geometric properties of a finite graph. As an application, we illustrate
the distribution of poles of this function comparing with those of the usual
Ihara zeta function.Comment: 14 pages, 1 figur
On eigenvalue distribution of random matrices of Ihara zeta function of large random graphs
We consider the ensemble of real symmetric random matrices
obtained from the determinant form of the Ihara zeta function of random graphs
that have vertices with the edge probability . We prove that the
normalized eigenvalue counting function of weakly converges in
average as and for any to a
shift of the Wigner semi-circle distribution. Our results support a conjecture
that the large Erdos-R\'enyi random graphs satisfy in average the weak graph
theory Riemann Hypothesis.Comment: version 5: slightly corrected with respect to version
An analogue of the Riemann Hypothesis via quantum walks
We consider an analogue of the well-known Riemann Hypothesis based on quantum
walks on graphs with the help of the Konno-Sato theorem. Furthermore, we give
some examples for complete, cycle, and star graphs.Comment: 14 page