Extremal values on the eccentric distance sum of trees

Let G = (V-G, E-G) be a simple connected graph. The eccentric distance sum of G is defined as xi(d)(G) = Sigma(v is an element of VG) epsilon(G)(v)D-G(v), where epsilon(G)(v) is the eccentricity of the vertex v and D-G(v) = Sigma(u is an element of VG) d(G)(u, v) is the sum of all distances from the vertex v. In this paper the tree among n-vertex trees with domination number gamma having the minimal eccentric distance sum is determined and the tree among n-vertex trees with domination number gamma satisfying n = k gamma having the maximal eccentric distance sum is identified, respectively, for k = 2, 3, n/3, n/2. Sharp upper and lower bounds on the eccentric distance sums among the n-vertex trees with k leaves are determined. Finally, the trees among the n-vertex trees with a given bipartition having the minimal, second minimal and third minimal eccentric distance sums are determined, respectively. (C) 2013 Elsevier B.V. All rights reserved.