This note attempts to understand graph limits as defined by Lovasz and Szegedy (2006)} in terms of harmonic analysis on semigroups. is done representing probability distributions random exchangeable graphs mixtures characters the semigroup unlabeled with node-disjoint union, thereby providing an alternative derivation de Finetti's theorem for graphs.

We consider symmetric powers of a graph. In particular, we show that the spectra of the symmetric square of strongly regular graphs with the same parameters are equal. We also provide some bounds on the spectra of the symmetric squares of more general graphs. The connection with generic exchange Hamiltonians in quantum mechanics is discussed in an appendix.

We answer a recent question posed by Li et al. [‘Imprimitive symmetric graphs with cyclic blocks’, European J. Combin. 31 (2010), 362–367] regarding a family of imprimitive symmetric graphs. 2000 Mathematics subject classification: primary 05C25; secondary 05E99.