  The main goal of this note is to illustrate the advantage of analyzing the
non-backtracking spectrum of a regular graph rather than the ordinary spectrum.
We show that by switching to non-backtracking spectrum, the method of proof
used in [Puder 2015, arXiv::1212.5216] yields a bound of
$2\sqrt{d-1}+\frac{2}{\sqrt{d-1}}$ instead of the original $2\sqrt{d-1}+1$ on
the second largest eigenvalue of a random $d$-regular graph.
