Extremal values on the eccentric distance sum of trees ?
Let G=(VG,EG) G = ( V G , E G ) be a simple connected graph. The eccentric distance sum of G G is defined as ?d(G)=?v?VG?G(v)DG(v) ? d ( G ) = ? v ? V G ? G ( v ) D G ( v ) , where ?G(v) ? G ( v ) is the eccentricity of the vertex v v and DG(v)=?u?VGdG(u,v) D G ( v ) = ? u ? V G d G ( u , v ) is the sum of all distances from the vertex v v . In this paper the tree among n n -vertex trees with domination number ? ? having the minimal eccentric distance sum is determined and the tree among n n -vertex trees with domination number ? ? satisfying n=k? n = k ? having the maximal eccentric distance sum is identified, respectively, for k = 2 , 3 , n 3 , n 2 . Sharp upper and lower bounds on the eccentric distance sums among the n n -vertex trees with k k leaves are determined. Finally, the trees among the n n -vertex trees with a given bipartition having the minimal, second minimal and third minimal eccentric distance sums are determined, respectively.
eccentric distance sum domination number leaves bipartition