Journal of Mathematical Physics, Analysis, Geometry 2017, vol. 13, No 3, pp. 268-282   https://doi.org/10.15407/mag13.03.268     ( to contents , go back )

### On Eigenvalue Distribution of Random Matrices of Ihara Zeta Function of Large Random Graphs

O. Khorunzhiy

Université de Versailles Saint-Quentin-en-Yvelines 45 Avenue des Etats-Unis, 78035 Versailles, France
E-mail: oleksiy.khorunzhiy@uvsq.fr

Received September 29, 2015, revised October 11, 2016

Abstract

We consider the ensemble of real symmetric random matrices H(n,ρ) obtained from the determinant form of the Ihara zeta function of random graphs that have n vertices with the edge probability ρ/n. We prove that the normalized eigenvalue counting function of H(n,ρ) converges weakly in average as n, ρ→∞ and ρ = o(nα) for any α > 0 to a shift of the Wigner semi-circle distribution. Our results support a conjecture that the large Erdős-Rényi random graphs satisfy in average the weak graph theory Riemann Hypothesis.

Mathematics Subject Classification 2000: 05C50, 05C80, 15B52, 60F99.
Key words: random graphs, random matrices, Ihara zeta function, eigenvalue distribution.