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 the non-backtracking spectrum, the method of proof used in [Pudl5] yields a bound of 2d−1+2d−1 instead of the original 2d−1+1 on the second largest eigenvalue of a random d-regular graph.