  The present paper introduces a modified version of cyclic-monotone
independence which originally arose in the context of random matrices, and also
introduces its natural analogy called cyclic-Boolean independence. We
investigate formulas for convolutions, limit theorems for sums of independent
random variables, and also classify infinitely divisible distributions with
respect to cyclic-Boolean convolution. Finally, we provide applications to the
eigenvalues of the adjacency matrices of iterated star products of graphs and
also iterated comb products of graphs.
