Monotonicity properties of certain Laplacian eigenvectors associated with trees

TL;DR

This paper provides an alternative proof of the monotonicity properties of Laplacian eigenvectors in trees, generalizing previous results and supporting the conjecture that such properties hold for all trees.

Contribution

It offers a new proof and extends the understanding of Laplacian eigenvector properties from paths to general trees.

Findings

Largest distance Laplacian eigenvalue of a path is simple.

Eigenvector exhibits properties similar to the Fiedler vector.

Supports the conjecture for all trees.

Abstract

Nath and Paul (Linear Algebra Appl.,460(2014),97-110) have shown that the largest distance Laplacian eigenvalue of a path is simple and the corresponding eigenvector has properties similar to the Fiedler vector. We given an alternative proof, establishing a more general result in the process. It is conjectured that a similar phenomenon holds for any tree.