Cut distance identifying graphon parameters over weak* limits
| Autor: | Martin Doležal, Jan Grebík, Jan Hladký, Israel Rocha, Václav Rozhoň |
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| Jazyk: | angličtina |
| Rok vydání: | 2018 |
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| Popis: | 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. Comment: 49 pages, 5 figures. Referees' comments incorporated. The most substantial change is a simplification of the proof of Proposition 2.15 (which does not rely on untrue (as we discovered) Exercise 4.18 from Lovasz's book anymore) |
| Databáze: | OpenAIRE |
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