TREES WITH MAXIMUM SUM OF THE TWO LARGEST LAPLACIAN EIGENVALUES

Let T be a tree of order n and S-2(T) be the sum of the two largest Laplacian eigenvalues of T. Fritscher et al. proved that for any tree T of order n, S-2(T) <= n + 2 - 2/n. Guan et al. determined the tree with maximum S-2(T) among all trees of order n. In this paper, we characterize the trees with S-2(T) >= n + 1 among all trees of order n except some trees. Moreover, among all trees of order n, we also determine the first [ n-2/2 j trees according to their S-2(T). This extends the result of Guan et al.