Characterization Extremal Trees Of The Average Distance And Eccentric Distance Sum

Let G =(VG,EG)be a simple connected graph.The average distance ?(G)is defined to the average of all distance in G,i.e.,?(G)=1/n(n-)(?)dG(u,v)where |VG|? and dG(u,v)denotes the length of a shortest path joining the vertices u and v in G.The ec-centric distance sum is defined as ?d(G)=(?)?G(v)DG(v),where ?G(v)=max{d(v,u):u?VG} is the eccentricity of the vertex v and DG(v)=(?)dG(u,v)is sum of all distance from the vertex v.The paper mainly studied two questions.First,Dankelmann gave the tight upper bounds on the average distance for all graphs with order n and domination number ?,and determined the extremal graphs.In this paper,we give the tight upper and lower bounds on the average distance for all trees with order n and total domination number?t satisfying 2??t?[n/2],and characterize the extremal graphs;in addition,if the tree with order n = 2(mod 4)and total domination number ?t=-n/2 + 1,then Pn has the maximal average distance.Second,Geng,Li and Zhang characterized the trees with minimal eccentric distance sum among n-vertex trees with fixed domination number ?;Miao,Pang and others characterized the trees with the maximal eccentric distance sum among n-vertex trees with domination number ? satisfying ??[n/3].In this paper,we characterize the trees with the maximal eccentric distance sum among n-vertex trees with domination number ? satisfyina[n/3]<??n/2.