  We consider inhomogeneous Erd\H{o}s-R\'enyi graphs. We suppose that the
maximal mean degree $d$ satisfies $d \ll \log n$. We characterize the
asymptotic behavior of the $n^{1 - o(1)}$ largest eigenvalues of the adjacency
matrix and its centred version. We prove that these extreme eigenvalues are
governed at first order by the largest degrees and, for the adjacency matrix,
by the nonzero eigenvalues of the expectation matrix. Our results show that the
extreme eigenvalues exhibit a novel behaviour which in particular rules out
their convergence to a nondegenerate point process. Together with the companion
paper [3], where we analyse the extreme eigenvalues in the complementary regime
$d \gg \log n$, this establishes a crossover in the behaviour of the extreme
eigenvalues around $d \sim \log n$. Our proof relies on a new tail estimate for
the Poisson approximation of an inhomogeneous sum of independent Bernoulli
random variables, as well as on an estimate on the operator norm of a pruned
graph due to Le, Levina, and Vershynin.
