Cut distance identifying graphon parameters over weak* limits

TL;DR

This paper investigates graphon parameters that can identify cut distance accumulation points over weak* limits, linking them to graph norms, spectral properties, and regularity lemmas, advancing understanding in graph limit theory.

Contribution

It introduces the concept of cut distance identifying parameters, explores their properties, and connects them to graph norms, spectral methods, and the Frieze-Kannan regularity lemma.

Findings

Connected graph is weakly norming iff it is step Sidorenko.

Norming graphs are step forcing.

Spectral methods can identify cut distance limits.

Abstract

The theory of graphons comes with the so-called cut norm and the derived cut distance. The cut norm is finer than the weak* topology (when considering the predual of $L^{1}$-functions). Dole\v{z}al and Hladk\'y [J. Combin. Theory Ser. B 137 (2019), 232-263] showed, that given a sequence of graphons, a cut distance accumulation graphon can be pinpointed in the set of weak* accumulation points as a minimizer of the entropy. Motivated by this, we study graphon parameters with the property that their minimizers or maximizers identify cut distance accumulation points over the set of weak* accumulation points. We call such parameters cut distance identifying. Of particular importance are cut distance identifying parameters coming from homomorphism densities, $t(H,\cdot)$. This concept is closely related to the emerging field of graph norms, and the notions of the step Sidorenko property and the step forcing property introduced by Kr\'a\v{l}, Martins, Pach and Wrochna [J. Combin. Theory Ser. A 162 (2019), 34-54]. We prove that a connected graph is weakly norming if and only if it is step Sidorenko, and that if a graph is norming then it is step forcing. Further, we study convexity properties of cut distance identifying graphon parameters, and find a way to identify cut distance limits using spectra of graphons. We also show that continuous cut distance identifying graphon parameters have the {\guillemotleft}pumping property{\guillemotright}, and thus can be used in the proof of the Frieze-Kannan regularity lemma.