On the sum of the squares of all distances in trees with fixed diameter

Denote the sum of the squares of all distances between all pairs of vertices in G by S(G) = Sigma(u, v is an element of VG) d(G)(2)(u,v) = 1/2 Sigma(v is an element of VG) L-G(v), where d(G)(u, v) is the distance between u and v in G and the sum goes over all the pairs of vertices, L-G(v) denotes the sum of the squares of all distances from u in G. In this paper, we obtain the trees with minimum and second-minimum S(G) among all the trees with n vertices and diameter d, respectively.