Algebraic properties of perfect structures

A perfect structure is a triple $(M,P,S)$ of matrices $M, P$ and $S$ of consistent sizes such that $MP = PS$. Perfect structures comprise similar matrices, eigenvectors, perfect colorings (equitable partitions) and graph coverings. In this paper we study general algebraic properties of perfect structures and characterize all perfect structures with identity or unity matrix $M$. Next, we consider a graph product generalizing most standard products (e.g. Cartesian, tensor, normal, lexicographic graph products). For this product we propose a construction of perfect structures and prove that it can be reversed for eigenvectors. Finally, we apply obtained results to calculate the spectra of several classes of graphs and to prove some properties of perfect colorings.