  We show that for an Erd\H{o}s-R\'{e}nyi graph on $N$ vertices with expected
degree $d$ satisfying $\log^{-1/9}N\leq d\leq \log^{1/40}N$, the largest
eigenvalues can be precisely determined by small neighborhoods around vertices
of close to maximal degree. Moreover, under the added condition that
$d\geq\log^{-1/15}N$, the corresponding eigenvectors are localized, in that the
mass of the eigenvector decays exponentially away from the high degree vertex.
This dependence on local neighborhoods implies that the edge eigenvalues
converge to a Poisson point process. These theorems extend a result of Alt,
Ducatez, and Knowles, who showed the same behavior for $d$ satisfying
$(\log\log N)^4\ll d\leq (1-o_{N}(1))\frac{1}{\log 4-1}\log N$.
  To achieve high accuracy in the constant degree regime, instead of attempting
to guess an approximate eigenvector of a local neighborhood, we analyze the
true eigenvector of a local neighborhood, and show it must be localized and
depend on local geometry.
