Cheeger Inequalities for Vertex Expansion and Reweighted Eigenvalues

TL;DR

This paper improves Cheeger inequalities relating vertex expansion and reweighted eigenvalues, develops a new spectral theory for vertex expansion, and provides negative evidence for a conjecture on 0/1-polytope graph expansion.

Contribution

It refines Cheeger inequalities for vertex expansion, introduces a new spectral framework, and constructs examples with poor vertex expansion to challenge existing conjectures.

Findings

Improved the Cheeger inequality to depend on maximum degree d

Established analogs of Cheeger inequalities for vertex expansion

Constructed 0/1-polytopes with poor vertex expansion and near-linear mixing time

Abstract

The classical Cheeger's inequality relates the edge conductance $\phi$ of a graph and the second smallest eigenvalue $\lambda_2$ of the Laplacian matrix. Recently, Olesker-Taylor and Zanetti discovered a Cheeger-type inequality $\psi^2 / \log |V| \lesssim \lambda_2^* \lesssim \psi$ connecting the vertex expansion $\psi$ of a graph $G=(V,E)$ and the maximum reweighted second smallest eigenvalue $\lambda_2^*$ of the Laplacian matrix. In this work, we first improve their result to $\psi^2 / \log d \lesssim \lambda_2^* \lesssim \psi$ where $d$ is the maximum degree in $G$, which is optimal assuming the small-set expansion conjecture. Also, the improved result holds for weighted vertex expansion, answering an open question by Olesker-Taylor and Zanetti. Building on this connection, we then develop a new spectral theory for vertex expansion. We discover that several interesting generalizations of Cheeger inequalities relating edge conductances and eigenvalues have a close analog in relating vertex expansions and reweighted eigenvalues. These include an analog of Trevisan's result on bipartiteness, an analog of higher order Cheeger's inequality, and an analog of improved Cheeger's inequality. Finally, inspired by this connection, we present negative evidence to the $0/1$-polytope edge expansion conjecture by Mihail and Vazirani. We construct $0/1$-polytopes whose graphs have very poor vertex expansion. This implies that the fastest mixing time to the uniform distribution on the vertices of these $0/1$-polytopes is almost linear in the graph size. This does not provide a counterexample to the conjecture, but this is in contrast with known positive results which proved poly-logarithmic mixing time to the uniform distribution on the vertices of subclasses of $0/1$-polytopes.