  We consider the statistics of the extreme eigenvalues of sparse random
matrices, a class of random matrices that includes the normalized adjacency
matrices of the Erd\H{o}s-R\'enyi graph $G(N,p)$. Tracy-Widom fluctuations of
the extreme eigenvalues for $p\gg N^{-2/3}$ was proved in [17,46]. We prove
that there is a crossover in the behavior of the extreme eigenvalues at $p\sim
N^{-2/3}$. In the case that $N^{-7/9}\ll p\ll N^{-2/3}$, we prove that the
extreme eigenvalues have asymptotically Gaussian fluctuations. Under a mean
zero condition and when $p=CN^{-2/3}$, we find that the fluctuations of the
extreme eigenvalues are given by a combination of the Gaussian and the
Tracy-Widom distribution. These results show that the eigenvalues at the edge
of the spectrum of sparse Erd\H{o}s-R\'enyi graphs are less rigid than those of
random $d$-regular graphs [4] of the same average degree.
