  We consider non-selfadjoint operator algebras $\mathfrak{L}(G,\lambda)$
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 $\mathfrak{L}(G,\lambda)$ in the case of the single vertex graph with
two edges and a suitable choice of left weight function $\lambda$.
