On the sum of distance Laplacian eigenvalues of graphs

Let G be a connected graph with n vertices, m edges and having diameter d. The distance Laplacian matrix DL is defined as DL = Diag(Tr) - D, where Diag(Tr) is the diagonal matrix of vertex transmissions and D is the distance matrix of G. The distance Laplacian eigenvalues of G are the eigenvalues of DL and are denoted by 61, 61, ... , 6n. In this paper, we obtain (a) the upper bounds for the sum of k largest and (b) the lower bounds for the sum of k smallest non-zero, distance Laplacian eigenvalues of G in terms of order n, diameter d and Wiener index W of G. We characterize the extremal cases of these bounds. Also, we obtain the bounds for the sum of the powers of the distance Laplacian eigenvalues of G. Finally, we obtain a sharp lower bound for the sum of the beta th powers of the distance Laplacian eigenvalues, where beta =6 0, 1.