A Note on the Trace Method for Random Regular Graphs

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.