Let $G=(V,E)$ be a simple, undirected graph. Given a family $\{(M_v,\tau_v)\colon v\in V\}$ of tracial von Neumann algebras, their graph product over $G$ is a tracial von Neumann algebra $(M_G,\tau_G)$ admitting trace-preserving embeddings of each $M_v$ so that: if $(v,w)\in E$ then $M_v$ and $M_w$ commute; and otherwise $M_v$ and $M_w$ are freely independent with respect to $\tau_G$. This is the analogue of a construction in group theory due to Green, and which has (re)appeared in operator algebras several times in the work of Młotkowski, Caspers–Fima, and Speicher–Wysoczański. From a probabilistic viewpoint, graph products of von Neumann algebras model the so-called notion of graph independence, which is a natural interpolation of classical and free independence. Given that strong 1-boundedness is well understood at the two extremes, it is natural to consider when it appears in graph products. This turns out to be a non-trivial question even when all of the vertex algebras $M_v$ are assumed to be finite dimensional, but a sufficient condition in this case is that the $*$-algebra generated by the vertex algebras has vanishing first $\ell^2$-Betti number. In this talk, I will discuss an ingredient in the proof of this sufficiency that is of independent interest to random matrix theory. Namely, that random permutation matrices can be used to construct random matrix ensembles that are asymptotically graph independent over the diagonal subalgebra. Notably, this construction relies on the tensor-product matrix models of Charlesworth–Collins and the traffic moment techniques of Au–Cébron–Dahlqvist–Gabriel–Male. This is based on joint work with Ian Charlesworth, Rolando de Santiago, Ben Hayes, David Jekel, and Srivatsav Kunnawalkam Elayavalli.