Graph spectrum is a topic extensively studied in algebraic graph theory but for its asymptotics for growing graphs we need analytic techniques. In this talk, we overview how the idea of quantum (non-commutative) probability is applied to the asymptotic spectral analysis of growing graphs.
I) The method of quantum decomposition: This is closely related to orthogonal polynomials in one-variable and applied to distance-regular graphs and some generalizations. A bivariate extension is an interesting topic.
II) There are several central limit theorems corresponfing to several concepts of independence. Some graph products enjoy this structure and their asymptotic spectral distributions are obtained as a corollary of quantum central limit theorems.