On eigenvalue distribution of random matrices of Ihara zeta function of large random graphs

TL;DR

This paper studies the eigenvalue distribution of matrices derived from the Ihara zeta function of large random graphs, showing convergence to a shifted Wigner semi-circle distribution, which supports a conjecture relating to the Riemann Hypothesis for random graphs.

Contribution

It proves the weak convergence of eigenvalue distributions of Ihara zeta-based matrices for large random graphs to a shifted semi-circle law, linking graph theory and random matrix theory.

Findings

Eigenvalue distribution converges to a shifted Wigner semi-circle law.

Results support the conjecture relating large random graphs to the Riemann Hypothesis.

Convergence holds as graph size and edge probability grow under specified conditions.

Abstract

We consider the ensemble of real symmetric random matrices $H^{(n,\rho)}$ obtained from the determinant form of the Ihara zeta function of random graphs that have $n$ vertices with the edge probability $\rho/n$. We prove that the normalized eigenvalue counting function of $H^{(n,\rho)}$ weakly converges in average as $n,\rho\to\infty$ and $\rho=o(n^\alpha)$ for any $\alpha>0$ 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.