  In this note, we give a precise description of the limiting empirical
spectral distribution (ESD) for the non-backtracking matrices for an
Erd\H{o}s-R\'{e}nyi graph assuming $np/\log n$ tends to infinity. We show that
derandomizing part of the non-backtracking random matrix simplifies the
spectrum considerably, and then we use Tao and Vu's replacement principle and
the Bauer-Fike theorem to show that the partly derandomized spectrum is, in
fact, very close to the original spectrum.
