Eigenvalues of the non-backtracking operator detached from the bulk

We describe the non-backtracking spectrum of a stochastic block model with connection probabilities $p_{mathrm{in}}, p_{mathrm{out}} = omega(log n)/n$. In this regime we answer a question posed in DallAmico and al. (2019) regarding the existence of a real eigenvalue `inside the bulk, close to the location $frac{p_{mathrm{in}}+ p_{mathrm{out}}}{p_{mathrm{in}}- p_{mathrm{out}}}$. We also introduce a variant of the Bauer-Fike theorem well suited for perturbations of quadratic eigenvalue problems, and which could be of independent interest.