Schur reduction of trees and extremal entries of the Fiedler vector

TL;DR

This paper analyzes the eigenvectors of Laplacian matrices of trees, using Schur reduction to derive formulas and bounds for eigenvector entries, particularly focusing on the Fiedler vector's extremal entries.

Contribution

It introduces a Schur complement-based reduction method for Laplacian matrices of trees, providing explicit formulas and bounds for eigenvector entry ratios, especially for the Fiedler vector.

Findings

Derived formulas for eigenvector entry ratios using Schur reduction.

Established bounds on eigenvector entry ratios along paths.

Identified conditions for extremal entries of the Fiedler vector at path endpoints.

Abstract

We study the eigenvectors of Laplacian matrices of trees. The Laplacian matrix is reduced to a tridiagonal matrix using the Schur complement. This preserves the eigenvectors and allows us to provide fomulas for the ratio of eigenvector entries. We also obtain bounds on the ratio of eigenvector entries along a path in terms of the eigenvalue and Perron values. The results are then applied to the Fiedler vector. Here we locate the extremal entries of the Fiedler vector and study classes of graphs such that the extremal entries can be found at the end points of the longest path.