This note attempts to understand graph limits as defined by Lovasz and Szegedy in terms of harmonic analysis on semigroups. This is done by representing probability distributions of random exchangeable graphs as mixtures of characters on the semigroup of unlabeled graphs with node-disjoint union, thereby providing an alternative derivation of de Finetti's theorem for random exchangeable graphs.
For a finite group $G$ and a non-linear irreducible complex character $\chi $ of $G$ write $\upsilon (\chi )=\{g\in G\mid \chi (g)=0\}$. In this paper, we study the finite non-solvable groups $G$ such that $\upsilon (\chi )$ consists of at most two conjugacy classes for all but one of the non-linear irreducible characters $\chi $ of $G$. In particular, we characterize a class of finite solvable groups which are closely related to the above-mentioned question and are called solvable $\varphi $-groups. As a corollary, we answer Research Problem $2$ in [Y. Berkovich and L. Kazarin: Finite groups in which the zeros of every non-linear irreducible character are conjugate modulo its kernel. Houston J. Math.\ 24 (1998), 619--630.] posed by Y. Berkovich and L. Kazarin.