Given a finite, directed, connected graph $$\Gamma $$ equipped with a weighting $$\mu $$ on its edges, we provide a construction of a von Neumann algebra equipped with a faithful, normal, positive linear functional $$(\mathcal {M}(\Gamma ,\mu ),\varphi )$$ . When the weighting $$\mu $$ is instead on the vertices of $$\Gamma $$ , the first author showed the isomorphism class of $$(\mathcal {M}(\Gamma ,\mu ),\varphi )$$ depends only on the data $$(\Gamma ,\mu )$$ and is an interpolated free group factor equipped with a scaling of its unique trace (possibly direct sum copies of $$\mathbb {C}$$ ). Moreover, the free dimension of the interpolated free group factor is easily computed from $$\mu $$ . In this paper, we show for a weighting $$\mu $$ on the edges of $$\Gamma $$ that the isomorphism class of $$(\mathcal {M}(\Gamma ,\mu ),\varphi )$$ depends only on the data $$(\Gamma ,\mu )$$ , and is either as in the vertex weighting case or is a free Araki–Woods factor equipped with a scaling of its free quasi-free state (possibly direct sum copies of $$\mathbb {C}$$ ). The latter occurs when the subgroup of $${\mathbb {R}}^+$$ generated by $$\mu (e_1)\cdots \mu (e_n)$$ for loops $$e_1\cdots e_n$$ in $$\Gamma $$ is non-trivial, and in this case the point spectrum of the free quasi-free state will be precisely this subgroup. As an application, we give the isomorphism type of some infinite index subfactors considered previously by Jones and Penneys.