Remarks on Graphons

Abstract

L. Lov\'asz and B. Szegedy proved in 2006 that the limits of convergent graph sequences can be described by measurable symmetric functions $W: [0, 1]\times [0, 1]\to [0, 1]$ called graphons. In our present paper we investigate the structure of the set of all graphons within the semigroup $(\mathfrak{F}([0, 1]^2); \circ)$ of all fuzzy subsets of the unit square $[0,1]^2=[0, 1]\times [0, 1]$, where the operation $\circ$ is defined by: for every $f, g\in \mathfrak{F}([0,1]^2)$ and every $s\in [0,1]^2$, $(f\circ g)(s)=\vee_{x\in [0,1]^2}(f(x)\wedge g(s))$.