Characterizing trees with large Laplacian energy

We investigate the problem of ordering trees according to their Laplacian energy. More precisely, given a positive integer n, we find a class of cardinality approximately whose elements are the n-vertex trees with largest Laplacian energy. The main tool for establishing this result is a new upper bound on the sum Sk(T) of the k largest Laplacian eigenvalues of an n-vertex tree T with diameter at least four, where k∈{1,…,n}.