Bipartite noisy hypercubes have large higher-order Cheeger separation

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.