Research On The Eccentric Distance Sum Of Trees

This paper which stands on the basis of previous results, do further research on sharp bounds of the eccentric distance sum of some kinds of trees.●In Section1, we introduce the background and significance of the research, including the development of a representative at home and abroad regarding this aspect. Based on this research background and profound discussion on the status quo, it fully shows the main work’s necessity and innovation.●In Section2, we introduce some definitions, symbols and lemmas which are mentioned in this paper.●In Section3, we study the minimal and second minimal eccentric distance sums of the n-vertex trees with perfect matchings and m-matchings.●In Section4, we characterize the extremal tree with the second minimal eccen-tric distance sum among the n-vertex trees of a given diameter. Consequently, we determine the trees with the third and fourth minimal eccentric distance sums among the n-vertex trees, which is a continuance study as the results in [G.H. Yu, L.H. Feng, A. Ilic, On the eccentric distance sum of trees and unicyclic graphs, J. Math. Anal. Appl.375(2011)99-107].●In Section5, we determine the minimal eccentric distance sum of n-vertex trees with domination number γ. Also, among n-vertex trees with domination number γ satisfying n=kγ having the maximal eccentric distance sum is identified, respectively, for k=2,3,n/3,n/2.●In Section6, we characterize the trees with the minimal and maximal eccentric distance sums among the n-vertex trees with k leaves.