  Let $\Xi$ be the adjacency matrix of an Erd\H{o}s-R\'enyi graph on $n$
vertices and with parameter $p$ and consider $A$ a $n\times n$ centered random
symmetric matrix with bounded i.i.d. entries above the diagonal. When the mean
degree $np$ diverges, the empirical spectral measure of the normalized Hadamard
product $(A \circ \Xi)/\sqrt{np}$ converges weakly in probability to the
semicircle law. In the regime where $p\ll 1$ and $ np \gg \log n$, we prove a
large deviations principle for the empirical spectral measure with speed $n^2p$
and with a good rate function solution of a certain variational problem. The
rate function reveals in particular that the only possible deviations at the
exponential scale $n^2p$ are around measures coming from Quadratic Vector
Equations. As a byproduct, we obtain a large deviations principle for the
empirical spectral measure of supercritical Erd\H{o}s-R\'enyi graphs.
