Assigned rational functions of a rooted tree

TL;DR

This paper introduces assigned rational functions for rooted trees to analyze their spectral properties, providing formulas for characteristic polynomials and a spectrum-preserving tree merging method, with applications to balanced and Bethe trees.

Contribution

It develops recursive assigned rational functions and formulas for characteristic polynomials, enhancing spectral analysis of rooted trees and enabling spectrum-preserving tree merging.

Findings

Formulas for adjacency and Laplacian characteristic polynomials as products of rational functions.

Application of formulas to balanced trees and Bethe trees.

A tree merging procedure that preserves spectra of original trees.

Abstract

We investigate the spectral properties of rooted trees with the intention of improving the currently existing results that deal with this matter. The concept of an assigned rational function is recursively defined for each vertex of a rooted tree. Afterwards, two mathematical formulas are given which show how the characteristic polynomials of the adjacency and Laplacian matrix can be represented as products of the aforementioned rational functions. In order to demonstrate their general use case scenario, the obtained formulas are subsequently implemented on balanced trees, with a special focus on the Bethe trees. In the end, some of the previously derived results are used in order to construct a tree merging procedure which preserves the spectra of all of the starting trees.