Expansion in Distance Matrices

TL;DR

The paper studies the normalized distance Laplacian $\mathcal{D}^{\mathcal{L}}(G)$ of a connected graph, a distance-based generalization of the normalized Laplacian, and proves that both the Cheeger constant and the spectral gap are bounded away from zero independently of $G$ (and more generally in finite metric spaces). It derives a sharp lower bound on the Cheeger constant, with equality characterizations tied to complete bipartite structures, and proves a universal spectral gap $\partial_2 \ge \frac{9-4\sqrt{2}}{7} \approx 0.478$, with a conjectured strengthening to $\partial_2 \ge \frac{2}{3}$ and a stronger bound in Cayley graphs on abelian groups. The Cayley-graph analysis uses Fourier eigenspaces to obtain a stronger bound ($\partial_2 \ge \frac{2}{3}$, and $>0.718$ for odd order groups). The results illuminate the nonstandard expansion behavior of distance-based Laplacians and extend to arbitrary finite metric spaces, suggesting directions for higher-order spectral questions.

Abstract

The normalized distance Laplacian matrix $\mathcal{D}^{\mathcal{L}}(G)$ of a graph $G$ is a natural generalization of the normalized Laplacian matrix, arising from the matrix of pairwise distances between vertices rather than the adjacency matrix. Following the motif that this matrix behaves quite differently to the normalized Laplacian matrix, we show that both the spectral gap and Cheeger constant of $\mathcal{D}^{\mathcal{L}}(G)$ are bounded away from $0$ independently of the graph $G$. The spectral result holds more generally for finite metric spaces.

Theorems & Definitions (31)

• Theorem 1.1
• Theorem 1.2
• Conjecture 1.3
• Theorem 1.4
• Lemma 2.1
• proof
• Theorem 2.2
• Definition 2.3
• Lemma 2.4
• Lemma 3.1