A linear k-fold Cheeger inequality

Abstract. Given an undirected graph G, the classical Cheeger constant, hG, measures the optimal partition of the vertices into 2 parts with relatively few edges between them based upon the sizes of the parts. The wellknown Cheeger’s inequality states that 2λ1 ≤ hG ≤ √ 2λ1 where λ1 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 λk−1, the (k−1) 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 ∞-norms of the corresponding eigenvectors. Consequently, unlike previous results, this result is relevant even when λk−1 → 1.