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2012 Monotone Graph Limits and Quasimonotone Graphs
Bála Bollobás, Svante Janson, Oliver Riordan
Internet Math. 8(3): 187-231 (2012).


The recent theory of graph limits gives a powerful framework for understanding the properties of suitable (convergent) sequences $(G_n)$ of graphs in terms of a limiting object that may be represented by a symmetric function $W$ on $[0, 1]^2$ , i.e., a kernel or graphon. In this context it is natural to wish to relate specific properties of the sequence to specific properties of the kernel. Here we show that the kernel is monotone (i.e., increasing in both variables) if and only if the sequence satisfies a “quasimonotonicity” property defined by a certain functional tending to zero. As a tool we prove an inequality relating the cut and $L^1$ norms of kernels of the form $W_1 −W_2$ with $W_1$ and $W_2$ monotone that may be of interest in its own right; no such inequality holds for general kernels.


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Bála Bollobás. Svante Janson. Oliver Riordan. "Monotone Graph Limits and Quasimonotone Graphs." Internet Math. 8 (3) 187 - 231, 2012.


Published: 2012
First available in Project Euclid: 21 August 2012

zbMATH: 1258.05121
MathSciNet: MR2967065

Rights: Copyright © 2012 A K Peters, Ltd.


Vol.8 • No. 3 • 2012
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