From graphs to free products

We investigate a construction which associates a finite von Neumann algebra $M(\mathrm{\Gamma },\mu )$ to a finite weighted graph $(\mathrm{\Gamma },\mu )$. Pleasantly, but not surprisingly, the von Neumann algebra associated to to a flower with $n$ petals' is the group von Neumann algebra of the free group on $n$ generators. In general, the algebra $M(\Gamma,\mu)$ is a free product, with amalgamation over a finite-dimensional abelian subalgebra corresponding to the vertex set, of algebras associated to subgraphs with one edge’ (or actually a pair of dual edges). This also yields `natural’ examples of (i) a Fock-type model of an operator with a free Poisson distribution; and (ii) $\text{C}\oplus \text{C}$-valued circular and semi-circular operators.