Commutants of weighted shift directed graph operator algebras

We consider non-selfadjoint operator algebras L(G, λ) generated by weighted creation operators on the Fock-Hilbert spaces of countable directed graphs G. These algebras may be viewed as noncommutative generalizations of weighted Bergman space algebras, or as weighted versions of the free semigroupoid algebras of directed graphs. A complete description of the commutant is obtained together with broad conditions that ensure the double commutant property. It is also shown that the double commutant property may fail for L(G, λ) in the case of the single vertex graph with two edges and a suitable choice of left weight function λ.