The dual Cheeger-Buser inequality for graphons

TL;DR

This work extends the dual Cheeger–Buser inequality to graphons by defining the bipartiteness ratio $\beta_W$ for connected graphons and relating it to the top Laplacian eigenvalue $\lambda_W^{\max}$ via the bound $\frac{\beta_W^2}{2} \le 2 - \lambda_W^{\max} \le 2 \beta_W$. The upper bound uses a Trevisan-style argument, recasting $\beta_W$ in terms of test functions $f: I \to \{-1,0,1\}$ to control the Rayleigh quotient, while the lower bound leverages approximation by essentially bounded functions and a variational characterization of the top spectrum. The paper also establishes an equivalence between bipartiteness, the top spectral value $2$, and the degree-positivity condition, and it connects graph and graphon notions by showing the top spectrum and bipartiteness ratio behave compatibly under the graphonification of finite graphs. Consequently, the results bridge finite graphs and graph limits, enabling spectral and bipartiteness analyses in the graph limit regime with potential implications for extremal graph theory and network modeling. The work thereby broadens the spectral toolkit for graph limits and clarifies how bipartiteness manifests in the graphon setting.

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.

Theorems & Definitions (36)

• Theorem 1.1: The dual Cheeger--Buser inequality for graphons
• Definition 2.1: Top of the spectrum
• Lemma 2.2
• proof
• Definition 3.1: Bipartiteness ratio
• Definition 3.2: Strong mixing
• Lemma 3.3: Strong mixing property
• Lemma 3.4
• proof
• Remark