A Linear Cheeger Inequality using Eigenvector Norms

TL;DR

This paper introduces a new Cheeger Inequality that uses the eigenvector's infinity norm alongside the eigenvalue, providing tighter bounds on graph expansion and revealing different asymptotic behavior.

Contribution

The paper presents an alternative Cheeger Inequality incorporating eigenvector norms, improving bounds from quadratic to linear in relation to the eigenvalue.

Findings

Controls $h_G$ within a linear factor of $$-norm eigenvector

Shows $h_G$ approaches 1/2 as $$ approaches 1

Provides tighter bounds compared to classical Cheeger Inequality

Abstract

The Cheeger constant, $h_G$, is a measure of expansion within a graph. The classical Cheeger Inequality states: $\lambda_{1}/2 \le h_G \le \sqrt{2 \lambda_{1}}$ where $\lambda_1$ is the first nontrivial eigenvalue of the normalized Laplacian matrix. Hence, $h_G$ is tightly controlled by $\lambda_1$ to within a quadratic factor. We give an alternative Cheeger Inequality where we consider the $\infty$-norm of the corresponding eigenvector in addition to $\lambda_1$. This inequality controls $h_G$ to within a linear factor of $\lambda_1$ thereby providing an improvement to the previous quadratic bounds. An additional advantage of our result is that while the original Cheeger constant makes it clear that $h_G \to 0$ as $\lambda_1 \to 0$, our result shows that $h_G \to 1/2$ as $\lambda_1 \to 1$.