An Explicit Formula for the Eigenvectors of Acyclic Matrices and Weighted Trees

TL;DR

This paper derives an explicit formula for eigenvectors of acyclic symmetric matrices by leveraging matching polynomial results, applicable to weighted trees modeled as forests.

Contribution

It introduces a novel explicit formula for eigenvectors of acyclic matrices, connecting spectral properties with matching polynomials in weighted trees.

Findings

Provides an explicit eigenvector formula for acyclic matrices.

Establishes a link between eigenvectors and matching polynomials.

Applicable to weighted forests and trees.

Abstract

Let $A$ be an acyclic symmetric matrix of order $n$. There is a weighted forest $F$ whose adjacency matrix is $A$. In this paper, using some results on matching polynomials, we provide an explicit formula for eigenvectors of $A$.