Equitable partitions and spectra of symmetric trees: Revisiting Heilbronner’s composition principle

The notion of equitable partitions, first defined by Horst Sachs, embodies a notable procedure in spectral graph theory, which is far from being conveniently explored in the literature.With equitable partitions, we can deduce significant spectral properties of a graph.For trees with a high level of symmetry, we can combine this technique with the "composition principle" (developed by Edgar Heilbronner more than seven decades ago) and fully determine the entire spectrum.This is a partially survey note where we provide several descriptive examples of this combination.We show that some recent results on the factorization of the characteristic polynomials of symmetric trees can be derived by merging both methods.