The atoms of graph product von Neumann algebras

We completely classify the atomic summands in a graph product $(M,\phi )={\ast }_{v\in \mathcal{G}}(M_v,{\phi }_v)$ of von Neumann algebras with faithful normal states. Each type I factor summand $(N,\psi )$ is a tensor product of type I factor summands $(N_v,{\psi }_v)$ in the individual algebras. The existence of such a summand and its weight in the direct sum can be determined from the $(N_v,{\psi }_v)$'s using explicit polynomials associated to the graph.