On the extremal values of the eccentric distance sum of trees with a given domination number

Abstract Let G be a simple connected graph. The eccentric distance sum (EDS) of G is defined as ξ d ( G ) = ∑ v ∈ V e G ( v ) D G ( v ) , where e G ( v ) is the eccentricity of the vertex v and D G ( v ) = ∑ u ∈ V d G ( u , v ) is the sum of all distances from the vertex v . In this paper, the extremal tree among n -vertex trees with domination number γ satisfying 4 ≤ γ ⌈ n 3 ⌉ having the maximal EDS is characterized. This proves Conjecture 4.2 posed in Miao et al. (2015).