  We consider the ensemble of adjacency matrices of Erd\H{o}s-R\'{e}nyi random
graphs, that is, graphs on $N$ vertices where every edge is chosen
independently and with probability $p\equiv p(N)$. We rescale the matrix so
that its bulk eigenvalues are of order one. We prove that, as long as
$pN\to\infty$ (with a speed at least logarithmic in $N$), the density of
eigenvalues of the Erd\H{o}s-R\'{e}nyi ensemble is given by the Wigner
semicircle law for spectral windows of length larger than $N^{-1}$ (up to
logarithmic corrections). As a consequence, all eigenvectors are proved to be
completely delocalized in the sense that the $\ell^{\infty}$-norms of the
$\ell^2$-normalized eigenvectors are at most of order $N^{-1/2}$ with a very
high probability. The estimates in this paper will be used in the companion
paper [Spectral statistics of Erd\H{o}s-R\'{e}nyi graphs II: Eigenvalue spacing
and the extreme eigenvalues (2011) Preprint] to prove the universality of
eigenvalue distributions both in the bulk and at the spectral edges under the
further restriction that $pN\gg N^{2/3}$.
