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 ξd(G)=∑v∈V(G)ε(v)DG(v), where ε(v) is the eccentricity of the vertex v and DG(v)=∑u∈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.