Asymptotic absence of poles of Ihara zeta function of large Erdos-Renyi random graphs

TL;DR

This paper demonstrates that for certain large Erdős-Rényi random graphs, the Ihara zeta function exhibits an asymptotic absence of poles, supporting a version of the graph theory Riemann Hypothesis.

Contribution

It establishes a connection between eigenvalue concentration results and the absence of poles of the Ihara zeta function in large random graphs.

Findings

Poles of Ihara zeta function are asymptotically absent in large Erdős-Rényi graphs.

Supports a version of the graph theory Riemann Hypothesis for these graphs.

Shows the relation between eigenvalue concentration and zeta function properties.

Abstract

Using recent results on the concentration of the largest eigenvalue and maximal vertex degree of large random graphs, we show that the infinite sequence of Erd\H os-R\'enyi random graphs $G(n,\rho_n/n)$ such that $\rho_n/\log n$ infinitely increases as $n\to\infty$ verifies a version of the graph theory Riemann Hypothesis.