The dual Cheeger-Buser inequality for graphons

TL;DR

This paper extends the dual Cheeger-Buser inequality from finite graphs to graphons, relating spectral gaps to bipartiteness ratio in a continuous graph limit setting.

Contribution

It introduces the bipartiteness ratio for graphons and proves an analogous dual Cheeger-Buser inequality, expanding spectral graph theory to the graphon framework.

Findings

Established the bipartiteness ratio for graphons

Proved the dual Cheeger-Buser inequality for graphons

Connected spectral properties with bipartiteness in graphons

Abstract

We introduce the notion of bipartiteness ratio for graphons. We prove the dual Cheeger-Buser inequality for graphons, which relates the gap between $2$ and the top of the spectrum of the Laplacian of a graphon with its bipartiteness ratio. The dual Cheeger-Buser inequality was established by Trevisan and Bauer-Jost for graphs. Our result is an analog of that for graphons.