Monotonicity properties of certain Laplacian eigenvectors associated   with trees
  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. 
Ravindra B. Bapat
math.CO