On the sum of the Laplacian eigenvalues of a tree

Abstract

Given an n-vertex graph G = ( V , E ) , the Laplacian spectrum of G is the set of eigenvalues of the Laplacian matrix L = D - A , where D and A denote the diagonal matrix of vertex-degrees and the adjacency matrix of G, respectively. In this paper, we study the Laplacian spectrum of trees. More precisely, we find a new upper bound on the sum of the k largest Laplacian eigenvalues of every n-vertex tree, where k ∈ { 1 , … , n } . This result is used to establish that the n-vertex star has the highest Laplacian energy over all n-vertex trees, which answers affirmatively to a question raised by Radenković and Gutman [10].