On the eccentric distance sum of trees and unicyclic graphs

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) = Sigma(u is an element of V(G))d(u, v) is the sum of all distances from the vertex v. In this paper we characterize the extremal unicyclic graphs among n-vertex unicyclic graphs with given girth having the minimal and second minimal eccentric distance sum. In addition, we characterize the extremal trees with given diameter and minimal eccentric distance sum. (C) 2010 Elsevier Inc. All rights reserved.