Extremal values on the eccentric distance sum of trees

Let G=(VG,EG)G=(VG,EG) be a simple connected graph. The eccentric distance sum of GG is defined as ξd(G)=∑v∈VGεG(v)DG(v)ξd(G)=∑v∈VGεG(v)DG(v), where εG(v)εG(v) is the eccentricity of the vertex vv and DG(v)=∑u∈VGdG(u,v)DG(v)=∑u∈VGdG(u,v) is the sum of all distances from the vertex vv. In this paper the tree among nn-vertex trees with domination number γγ having the minimal eccentric distance sum is determined and the tree among nn-vertex trees with domination number γγ satisfying n=kγn=kγ having the maximal eccentric distance sum is identified, respectively, for k=2,3,n3,n2. Sharp upper and lower bounds on the eccentric distance sums among the nn-vertex trees with kk leaves are determined. Finally, the trees among the nn-vertex trees with a given bipartition having the minimal, second minimal and third minimal eccentric distance sums are determined, respectively.