Cheeger Inequalities for Vertex Expansion and Reweighted Eigenvalues
Tsz Chiu Kwok, Lap Chi Lau, Kam Chuen Tung

TL;DR
This paper improves Cheeger inequalities relating vertex expansion and reweighted eigenvalues, develops a new spectral theory for vertex expansion, and provides negative evidence for a conjecture on 0/1-polytope graph expansion.
Contribution
It refines Cheeger inequalities for vertex expansion, introduces a new spectral framework, and constructs examples with poor vertex expansion to challenge existing conjectures.
Findings
Improved the Cheeger inequality to depend on maximum degree d
Established analogs of Cheeger inequalities for vertex expansion
Constructed 0/1-polytopes with poor vertex expansion and near-linear mixing time
Abstract
The classical Cheeger's inequality relates the edge conductance of a graph and the second smallest eigenvalue of the Laplacian matrix. Recently, Olesker-Taylor and Zanetti discovered a Cheeger-type inequality connecting the vertex expansion of a graph and the maximum reweighted second smallest eigenvalue of the Laplacian matrix. In this work, we first improve their result to where is the maximum degree in , which is optimal assuming the small-set expansion conjecture. Also, the improved result holds for weighted vertex expansion, answering an open question by Olesker-Taylor and Zanetti. Building on this connection, we then develop a new spectral theory for vertex expansion. We discover that several interesting…
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsAdvanced biosensing and bioanalysis techniques · Markov Chains and Monte Carlo Methods · Graphene research and applications
