Bipartite noisy hypercubes have large higher-order Cheeger separation

TL;DR

This paper constructs the Bipartite Noisy Hypercube to demonstrate the sharpness of the gap between spectral bipartite expansion and bipartite edge expansion, highlighting limitations of higher-order Cheeger's inequalities.

Contribution

It introduces the Bipartite Noisy Hypercube and proves the sharpness of the spectral bipartite expansion gap in the dual higher-order Cheeger's inequality.

Findings

Bipartite Noisy Hypercube constructed to test spectral bipartite expansion

Sharpness of the gap between spectral and edge bipartite expansion demonstrated

Highlights limitations of higher-order Cheeger's inequalities in bipartite graphs

Abstract

The expansion of a graph is typically associated with its spectral properties - testing whether a graph is an expander is usually done using Cheeger's inequality. One can also use multiple eigenvalues in a higher-order Cheeger's inequality to test a deeper set of properties on the graph. However Cheeger's inquality, and the higher-order Cheeger's inequality, can be imprecise tools. Recently Lee, Gharan, and Trevisan constructed the Noisy Hypercube to prove the sharpness of the gap between spectral expansion and edge expansion in the higher-order Cheeger's inequality. We are concerned with the dual problem: using the upper end of the Laplacian spectrum to test a graph's bipartite nature. This has been shown to have several applications, and recently a dual version of Cheeger's inequality and a dual version of the higher-order Cheeger's inequality have been presented. We construct the Bipartite Noisy Hypercube and use it to prove the sharpness of the gap between spectral bipartite expansion and bipartite edge expansion in the dual version of the dual higher-order Cheeger's inequality.