Abstract: In both classical and free probability theory, the central limit distribution can be modeled on a symmetric or free Fock space. The $q$-deformed Gaussians are the corresponding variables on a $q$-deformed Fock space (being the free semicirculars when $q = 0$ and the classical Gaussians when $q=1$), which raises the question of whether they arise from a central limit-type theorem. To find such a situation, Młotkowski introduced $\varepsilon$-independence as an interpolation between free and classical independence, where distributions (or von Neumann algebras) are assigned to the vertices of a graph with adjacency matrix $\varepsilon$, and are placed in a larger algebra in such a way that they are independent when they correspond to adjacent vertices and free otherwise. The corresponding product operation on von Neumann algebras corresponds to the idea of a graph product of groups, studied by Green. In this talk I will be interested in the following question: when do type I summands appear in the graph product of von Neumann algebras? The answer is pleasantly combinatorial, and can be described based on a family of polynomials built using the cliques in the graph (first arising in work of Cartier--Foata in 1969), and the behaviour of type I summands in the input algebras. This is joint work with David Jekel. Note: This talk is part of the activity of the MUR Excellence Department Project MatMod@TOV (CUP E83C23000330006) The Operator Algebra Seminar schedule is here: https://sites.google.com/view/oastorvergata/home-page