On the extremal values of 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 ξ d ( G ) = ∑ v ∈ V G ε G ( v ) D G ( v ) , where ε G ( 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 first characterize the extremal trees among n-vertex conjugated trees (trees with a perfect matching) having the minimal and second minimal eccentric distance sums. Then we identify the trees with the minimal and second minimal eccentric distance sums among the n-vertex trees with matching number m. Finally, we characterize the extremal tree with the second minimal eccentric distance sum among the n-vertex trees of a given diameter. Consequently, we determine the trees with the third and fourth minimal eccentric distance sums among the n-vertex trees, which is a continuance study as the results in [G.H. Yu, L.H. Feng, A. Ilić, On the eccentric distance sum of trees and unicyclic graphs, J. Math. Anal. Appl. 375 (2011) 99–107].