Given a finite, directed, connected graph Γ\documentclass[12pt]{minimal}
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\begin{document}$$\Gamma $$\end{document} equipped with a weighting μ\documentclass[12pt]{minimal}
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\begin{document}$$\mu $$\end{document} on its edges, we provide a construction of a von Neumann algebra equipped with a faithful, normal, positive linear functional (M(Γ,μ),φ)\documentclass[12pt]{minimal}
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\begin{document}$$(\mathcal {M}(\Gamma ,\mu ),\varphi )$$\end{document}. When the weighting μ\documentclass[12pt]{minimal}
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\begin{document}$$\mu $$\end{document} is instead on the vertices of Γ\documentclass[12pt]{minimal}
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\begin{document}$$\Gamma $$\end{document}, the first author showed the isomorphism class of (M(Γ,μ),φ)\documentclass[12pt]{minimal}
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\begin{document}$$(\mathcal {M}(\Gamma ,\mu ),\varphi )$$\end{document} depends only on the data (Γ,μ)\documentclass[12pt]{minimal}
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\begin{document}$$(\Gamma ,\mu )$$\end{document} and is an interpolated free group factor equipped with a scaling of its unique trace (possibly direct sum copies of C\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {C}$$\end{document}). Moreover, the free dimension of the interpolated free group factor is easily computed from μ\documentclass[12pt]{minimal}
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\begin{document}$$\mu $$\end{document}. In this paper, we show for a weighting μ\documentclass[12pt]{minimal}
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\begin{document}$$\mu $$\end{document} on the edges of Γ\documentclass[12pt]{minimal}
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\begin{document}$$\Gamma $$\end{document} that the isomorphism class of (M(Γ,μ),φ)\documentclass[12pt]{minimal}
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\begin{document}$$(\mathcal {M}(\Gamma ,\mu ),\varphi )$$\end{document} depends only on the data (Γ,μ)\documentclass[12pt]{minimal}
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\begin{document}$$(\Gamma ,\mu )$$\end{document}, 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 C\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {C}$$\end{document}). The latter occurs when the subgroup of R+\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {R}}^+$$\end{document} generated by μ(e1)⋯μ(en)\documentclass[12pt]{minimal}
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\begin{document}$$\mu (e_1)\cdots \mu (e_n)$$\end{document} for loops e1⋯en\documentclass[12pt]{minimal}
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\begin{document}$$e_1\cdots e_n$$\end{document} in Γ\documentclass[12pt]{minimal}
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\begin{document}$$\Gamma $$\end{document} 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.
