On the Houdré-Tetali conjecture about an isoperimetric constant of graphs

TL;DR

The paper addresses the Houdré-Tetali conjecture relating the isoperimetric constant $φ_{1/2}$ to the second eigenvalue $λ_2$ of the normalized Laplacian, delivering both limitations and enhancements to the conjecture. It proves a universal spectral bound $(φ_p)^2 \le \frac{4}{2p-1} λ_2$ for $p>\tfrac{1}{2}$, matching and strengthening the hard direction of Cheeger-type inequalities, and shows the conjecture holds for all constant $p>\tfrac{1}{2}$, with extensions to directed graphs via Chung's directed Laplacian. The authors provide a counterexample family showing the extra logarithmic factor is necessary for $p=\tfrac{1}{2}$, while also reproducing Morris-Peres's log-factor bound using standard spectral methods without self-loop assumptions. They furthermore extend the framework to non-reversible chains and directed graphs, and supply detailed proofs of auxiliary lemmas underpinning the spectral arguments. Overall, the work clarifies the precise limits of the Houdré-Tetali conjecture and offers robust spectral tools for analyzing isoperimetric constants in both undirected and directed graphs.

Abstract

Houdré and Tetali defined a class of isoperimetric constants $\varphi_p$ of graphs for $0 \leq p \leq 1$, and conjectured a Cheeger-type inequality for $\varphi_\frac12$ of the form $$λ_2 \lesssim \varphi_\frac12 \lesssim \sqrt{λ_2}$$ where $λ_2$ is the second smallest eigenvalue of the normalized Laplacian matrix. If true, the conjecture would be a strengthening of the hard direction of the classical Cheeger's inequality. Morris and Peres proved Houdré and Tetali's conjecture up to an additional log factor, using techniques from evolving sets. We present the following related results on this conjecture. - We provide a family of counterexamples to the conjecture of Houdré and Tetali, showing that the logarithmic factor is needed. - We match Morris and Peres's bound using standard spectral arguments. - We prove that Houdré and Tetali's conjecture is true for any constant $p$ strictly bigger than $\frac12$, which is also a strengthening of the hard direction of Cheeger's inequality. Furthermore, our results can be extended to directed graphs using Chung's definition of eigenvalues for directed graphs.

Theorems & Definitions (24)

• Definition 1.1: Isoperimetric Constants for Markov Chains HT04
• Conjecture 1.2: HT04
• Theorem 1.3: MP05
• Theorem 1.4
• Theorem 1.5
• Remark 1.6: $\varphi_p$ for $p < \frac{1}{2}$
• Remark 1.7: $\varphi_p$ for Hypercubes
• Lemma 3.1
• Theorem 3.2
• Lemma 3.3