Corrections to and improvements on results from "The Laplacian spectrum of large graphs sampled from graphons"

In this note we correct the proof of Proposition 4 in our paper "The Laplacian Spectrum of Large Graphs Sampled from Graphons" (arXiv:2004.09177) and we improve several results therein. To this end, we prove a new concentration lemma about degrees and Laplacian eigenvalues. This lemma allows us to improve several bounds and to dispense from assuming that the graphon is bounded away from zero in several results. This extension leads, in particular, to correct the proof of Proposition 4. Additionally, we extend Proposition 4 to graphs that are sampled from graphons by using deterministic latent variables.