Tree Decomposition By Eigenvectors

TL;DR

This paper introduces a novel technique linking tree eigenvectors to skeleton forests, providing insights into eigenvalue multiplicities and characterizations of trees with specific eigenspace bases, advancing spectral graph theory.

Contribution

It presents a new composition-decomposition method relating tree eigenvectors to skeleton forests and characterizes trees with eigenspaces having entries from {0, 1, -1}.

Findings

Matching properties of skeletons determine eigenvalue multiplicities.

Characterization of trees with eigenspaces with entries in {0, 1, -1}.

Generalization of Nylen's result on eigenvector partitioning.

Abstract

In this work a composition-decomposition technique is presented that correlates tree eigenvectors with certain eigenvectors of an associated so-called skeleton forest. In particular, the matching properties of a skeleton determine the multiplicity of the corresponding tree eigenvalue. As an application a characterization of trees that admit eigenspace bases with entries only from the set {0, 1,-1} is presented. Moreover, a result due to Nylen concerned with partitioning eigenvectors of tree pattern matrices is generalized.