  We consider the bulk eigenvalue statistics of Laplacian matrices of large
Erd\H{o}s-R\'enyi random graphs in the regime $p \geq N^{\delta}/N$ for any
fixed $\delta >0$. We prove a local law down to the optimal scale $\eta \gtrsim
N^{-1}$ which implies that the eigenvectors are delocalized. We consider the
local eigenvalue statistics and prove that both the gap statistics and averaged
correlation functions coincide with the GOE in the bulk.
