ABSTRACTWe show that for an Erdős–Rényi graph on vertices with expected degree satisfying , the largest eigenvalues can be precisely determined by small neighborhoods around vertices of close to maximal degree. Moreover, under the added condition that , 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 satisfying and answer a question of Guionnet. To achieve high accuracy in the constant degree regime, instead of comparing the true eigenvector to that of a tree with more regularity, we examine the eigenvector equation at each vertex in a ball to deduce localization of the eigenvector and derive a continued fraction formula for the eigenvalue. This formula can be applied to any tree and could be of independent interest, especially for rooted trees with large central degree.