On the Houdr\'e-Tetali conjecture about an isoperimetric constant of graphs

TL;DR

This paper investigates the Houdré-Tetali conjecture on isoperimetric constants of graphs, providing counterexamples, confirming the conjecture for certain parameters, and extending results to directed graphs, thereby advancing understanding of graph spectral properties.

Contribution

The paper presents counterexamples to the conjecture, confirms it for p > 1/2, and extends findings to directed graphs, refining the theoretical landscape of graph isoperimetric inequalities.

Findings

Counterexamples show the logarithmic factor is necessary.

The conjecture holds for all p > 1/2.

Results extend to directed graphs using Chung's eigenvalues.

Abstract

Houdr\'e 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 $$\lambda_2 \lesssim \varphi_\frac12 \lesssim \sqrt{\lambda_2}$$ where $\lambda_2$ 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 Houdr\'e 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 Houdr\'e and Tetali, showing that the logarithmic factor is needed. - We match Morris and Peres's bound using standard spectral arguments. - We prove that Houdr\'e 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.