  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.
