Sharp Bounds on Eigenvalues via Spectral Embedding Based on Signless Laplacians

TL;DR

This paper develops bounds on eigenvalues of graph matrices using spectral embedding based on the signless Laplacian, impacting understanding of random walk mixing times and adjacency spectra.

Contribution

It introduces a novel spectral embedding approach to bound eigenvalues of graph matrices, extending previous methods to new spectral bounds.

Findings

Bounds on the spectrum of transition matrices

Bounds on return probabilities and mixing times

Bounds on the spectrum of adjacency matrices

Abstract

Using spectral embedding based on the signless Laplacian, we obtain bounds on the spectrum of transition matrices on graphs. As a consequence, we bound return probabilities and the uniform mixing time of simple random walk on graphs. In addition, spectral embedding is used in this article to bound the spectrum of graph adjacency matrices. Our method is adapted from [Lyons and Oveis Gharan, 2017].