On the Houdré-Tetali Conjecture About an Isoperimetric Constant of Graphs
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.
Isoperimetric constant
Markov chains
Cheeger’s inequality
Theory of computation~Random walks and Markov chains
36:1-36:18
RANDOM
Supported by an NSERC Discovery Grant and a Math-funded Undergraduate Research Award from University of Waterloo.
We thank Christian Houdré and Prasad Tetali for their encouragement and the anonymous reviewers for their helpful comments.
Lap Chi
Lau
Lap Chi Lau
Cheriton School of Computer Science, University of Waterloo, Canada
Dante
Tjowasi
Dante Tjowasi
Cheriton School of Computer Science, University of Waterloo, Canada
10.4230/LIPIcs.APPROX/RANDOM.2024.36
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Lap Chi Lau and Dante Tjowasi
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