Ordering trees with n vertices and matching number q by their largest Laplacian eigenvalues

Denote by Tn,qTn,q the set of trees with n vertices and matching number q  . Guo [On the Laplacian spectral radius of a tree, Linear Algebra Appl. 368 (2003) 379–385] gave the tree in Tn,qTn,q with the greatest value of the largest Laplacian eigenvalue. In this paper, we give another proof of this result. Using our method, we can go further beyond Guo by giving the tree in Tn,qTn,q with the second largest value of the largest Laplacian eigenvalue.