  Let $G=G(n,p_n)$ be a homogeneous Erdos-R\'enyi graph, $A$ its adjacency
matrix with eigenvalues $\lambda_1(A) \geq \lambda_2(A) \geq ... \geq
\lambda_n(A).$ In such cases, it has been shown using local laws that
$\lambda_2(A)$ can exhibit different behaviors: real-valued Tracy-Widom for
$p_n \gg n^{-2/3},$ normal for $n^{-7/9} \ll p_n \ll n^{-2/3},$ or a mix of the
two for $p_n=cn^{-2/3}.$ Additionally, this technique renders the largest
eigenvalue $\lambda_1(A),$ separated from the rest of the spectrum for $p_n \gg
n^{-1},$ has Gaussian fluctuations when $p_n \gg n^{-1/3}.$ This paper extends
the range of the last convergence to $n^{\epsilon-1} \leq p_n \leq \frac{1}{2},
\epsilon \in (0,1):$ the tool behind this is a CLT for the eigenvalue
statistics of $A,$ which is justified by the method of moments.
