The generalized composition graph is used by Cardoso and some researchers for factorization of the adjacency spectrum and Laplacian of a simple graph. Because the generalized composition graph is an example of a set-theoretic linear operad, this operation can be iterated at more than one level, where the complex language of partition refinement in the iteration is represented in terms of Schr"oder trees. This allows us to generalize the factorization of the adjacency spectrum and Laplacian of a simple graph presented by Cardoso in terms of Schr"oder trees and colorings over the edges of a graph. Cardoso's technique has been generalized by other authors for the universal adjacency matrix of a graph. This work also presents generalized factorizations in terms of Schr"oder trees and colorings on the edges of a graph for the universal adjacency spectrum, the characteristic polynomial of the universal adjacency matrix, and the generalized characteristic polynomial of a graph.