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 ξd(G)=∑v∈VεG(v)DG(v), where εG(v) is the eccentricity of the vertex v and DG(v)=∑u∈VdG(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<⌈n3⌉ having the maximal EDS is characterized. This proves Conjecture 4.2 posed in Miao et al. (2015).