  We establish bounds on the spectral radii for a large class of sparse random
matrices, which includes the adjacency matrices of inhomogeneous
Erd\H{o}s-R\'enyi graphs. Our error bounds are sharp for a large class of
sparse random matrices. In particular, for the Erd\H{o}s-R\'enyi graph
$G(n,d/n)$, our results imply that the smallest and second-largest eigenvalues
of the adjacency matrix converge to the edges of the support of the asymptotic
eigenvalue distribution provided that $d \gg \log n$. Together with the
companion paper [3], where we analyse the extreme eigenvalues in the
complementary regime $d \ll \log n$, this establishes a crossover in the
behaviour of the extreme eigenvalues around $d \sim \log n$. Our results also
apply to non-Hermitian sparse random matrices, corresponding to adjacency
matrices of directed graphs. The proof combines (i) a new inequality between
the spectral radius of a matrix and the spectral radius of its nonbacktracking
version together with (ii) a new application of the method of moments for
nonbacktracking matrices.
