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On the Houdré-Tetali Conjecture About an Isoperimetric Constant of Graphs

Authors Lap Chi Lau, Dante Tjowasi

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Author Details

Lap Chi Lau
  • Cheriton School of Computer Science, University of Waterloo, Canada
Dante Tjowasi
  • Cheriton School of Computer Science, University of Waterloo, Canada

Funding

Supported by an NSERC Discovery Grant and a Math-funded Undergraduate Research Award from University of Waterloo.

Acknowledgements

We thank Christian Houdré and Prasad Tetali for their encouragement and the anonymous reviewers for their helpful comments.

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Lap Chi Lau and Dante Tjowasi. On the Houdré-Tetali Conjecture About an Isoperimetric Constant of Graphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 36:1-36:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2024.36

Abstract

Houdré and Tetali defined a class of isoperimetric constants φ_p of graphs for 0 ≤ p ≤ 1, and conjectured a Cheeger-type inequality for φ_(1/2) of the form λ₂ ≲ φ_(1/2) ≲ √λ₂, where λ₂ 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.  
1) We provide a family of counterexamples to the conjecture of Houdré and Tetali, showing that the logarithmic factor is needed. 
2) We match Morris and Peres’s bound using standard spectral arguments. 
3) We prove that Houdré and Tetali’s conjecture is true for any constant p strictly bigger than 1/2, 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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Random walks and Markov chains
Keywords
  • Isoperimetric constant
  • Markov chains
  • Cheeger’s inequality

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