# Algebraic properties of perfect structures

## TL;DR

This paper explores the algebraic properties of perfect structures, characterizes those with identity matrices, and applies these findings to graph spectra and colorings, unifying various graph products.

## Contribution

It provides a comprehensive algebraic framework for perfect structures, characterizes cases with identity matrices, and introduces a new graph product with applications to spectra and colorings.

## Key findings

- Characterization of perfect structures with identity matrices
- Construction and reversal of perfect structures for a generalized graph product
- Calculation of spectra for various classes of graphs

## Abstract

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.