Abstract
The recent theory of graph limits gives a powerful framework for understanding the properties of suitable (convergent) sequences (_G<sub>n</sub>_) of graphs in terms of a limiting object that may be represented by a symmetric function _W_ on [0, 1]<sup>2</sup>, 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_<sup>1</sup> norms of kernels of the form _W_<sub>1</sub>−_W_<sub>2</sub> with _W_<sub>1</sub> and _W_<sub>2</sub> monotone that may be of interest in its own right; no such inequality holds for general kernels.
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