Polynomial and spectra factorization of graphs obtained by iteration the operad of generalized graph composition

TL;DR

This paper extends graph spectrum and polynomial factorizations using iterated operads and Schr"oder trees, generalizing Cardoso's methods to broader matrices and polynomials.

Contribution

It introduces a generalized framework for graph spectrum and polynomial factorizations via iterated operads and Schr"oder trees, expanding prior techniques.

Findings

Generalized factorizations for the universal adjacency spectrum

New polynomial factorizations using Schr"oder trees and colorings

Extension of Cardoso's spectrum factorization techniques

Abstract

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.