On the spectral radius of the non-backtracking matrix of the configuration model

We prove a concentration result for the leading eigenvalue of the non--backtracking matrix of the configuration model under the assumption of uniformly bounded degrees. Let $P$ denote the limiting degree distribution. Assuming polynomial approximation, we show that as the number of vertices tends to infinity, the leading eigenvalue of the non--backtracking matrix concentrates around \[ \frac{\mathbb{E}[P(P-1)]}{\mathbb{E}[P]}. \] This quantity corresponds to the mean offspring number of the excess--degree branching process associated with the local limit of the configuration model. As a byproduct of our work we explain how this result can be applied to prove the density of the growth rates of the subgroups of the free group.