Remarks on Graphons

Abstract

L. Lovász 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 $s\in [0,1{]}^2$, $(f\circ g)(s)={\vee }_{x\in [0,1{]}^2}(f(x)\wedge g(s))$.

8 pages