  We complete the analysis of the extremal eigenvalues of the the adjacency
matrix $A$ of the Erd\H{o}s-R\'enyi graph $G(N,d/N)$ in the critical regime $d
\asymp \log N$ of the transition uncovered in
[arXiv:1704.02953,arXiv:1704.02945], where the regimes $d \gg \log N$ and $d
\ll \log N$ were studied. We establish a one-to-one correspondence between
vertices of degree at least $2d$ and nontrivial (excluding the trivial top
eigenvalue) eigenvalues of $A / \sqrt{d}$ outside of the asymptotic bulk
$[-2,2]$. This correspondence implies that the transition characterized by the
appearance of the eigenvalues outside of the asymptotic bulk takes place at the
critical value $d = d_* = \frac{1}{\log 4 - 1} \log N$. For $d < d_*$ we obtain
rigidity bounds on the locations of all eigenvalues outside the interval
$[-2,2]$, and for $d > d_*$ we show that no such eigenvalues exist. All of our
estimates are quantitative with polynomial error probabilities.
  Our proof is based on a tridiagonal representation of the adjacency matrix
and on a detailed analysis of the geometry of the neighbourhood of the large
degree vertices. An important ingredient in our estimates is a matrix
inequality obtained via the associated nonbacktracking matrix and an Ihara-Bass
formula [arXiv:1704.02945]. Our argument also applies to sparse Wigner
matrices, defined as the Hadamard product of $A$ and a Wigner matrix, in which
case the role of the degrees is replaced by the squares of the $\ell^2$-norms
of the rows.
