The eccentric distance sum, the Harary index and the degree powers of graphs with given diameter

Let G be a simple connected graph with the vertex set V(G). The eccentric distance sum of G is defined as xi(d)(G) = Sigma(v is an element of V(G)) epsilon(v)D-G(v), where epsilon(v) is the eccentricity of the vertex v and D-G(v) is the sum of all distances from the vertex v. The Harary index of G is defined as H(G) = Sigma({u,v}subset of V(G)) 1/d(u,v), where d(u, v) is the distance between u and v in G. The degree powers of G is defined as F-p(G) = Sigma(v is an element of V(G)) d(u)(p) for the natural number p >= 1. In this paper, we determine the extremal graphs with the minimal eccentric distance sum, the maximal Harary index and the maximal degree powers among all graphs with given diameter.