A linear k-fold Cheeger inequality

TL;DR

This paper introduces a new linear higher-order Cheeger inequality that relates an average-case Cheeger constant to the average of eigenvalues, providing bounds applicable even when eigenvalues approach 1.

Contribution

It proposes an alternative higher-order Cheeger constant based on an average case, establishing linear inequalities involving eigenvector norms, extending previous quadratic bounds.

Findings

Relates average Cheeger constant to average of eigenvalues

Provides linear bounds using eigenvector norms

Applicable even when eigenvalues approach 1

Abstract

Given an undirected graph $G$, the classical Cheeger constant, $h_G$, measures the optimal partition of the vertices into 2 parts with relatively few edges between them based upon the sizes of the parts. The well-known Cheeger's inequality states that $2 \lambda_1 \le h_G \le \sqrt {2 \lambda_1}$ where $\lambda_1$ is the minimum nontrivial eigenvalue of the normalized Laplacian matrix. Recent work has generalized the concept of the Cheeger constant when partitioning the vertices of a graph into $k > 2$ parts. While there are several approaches, recent results have shown these higher-order Cheeger constants to be tightly controlled by $\lambda_{k-1}$, the $(k-1)$-th nontrivial eigenvalue, to within a quadratic factor. We present a new higher-order Cheeger inequality with several new perspectives. First, we use an alternative higher-order Cheeger constant which considers an "average case" approach. We show this measure is related to the average of the first $k-1$ nontrivial eigenvalues of the normalized Laplacian matrix. Further, using recent techniques, our results provide linear inequalities using the $\infty$-norms of the corresponding eigenvectors. Consequently, unlike previous results, this result is relevant even when $\lambda_{k-1} \to 1$.