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 ξ d ( G ) = ∑ v ∈ V ( G ) ε ( v ) D G ( v ) , where ε ( v ) is the eccentricity of the vertex v and D G ( 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.