  Recent developments have found unexpected connections between non-commutative
probability theory and algebraic topology. In particular, Boolean cumulants
functionals seem to be important for describing morphisms of homotopy operadic
algebras.
  We provide new elementary examples which clearly resemble a connection
between algebraic topology and non-commutative probability, based on spectral
graph theory. These observations are important for bringing new ideas from
non-commutative probability into TDA and stochastic topology, and in the
opposite direction.
