The Research On The Wiener Index Of Trees With K-pendent

The Wiener index of a graph is just the sum of distances between all unordered pairs of vertices of the graph. This concept is put forward by the chemist Wiener in 1947 for the first time. The Wiener index is widely used in the fields of chemistry and communication network and has been extensively studied since the middle of 1970s.Many new results are obtained. It may be especially interesting to determine the tree with extremal Wiener index in all k-pendent trees.Recently Entringer obtained the following result: If T is a tree of order n with k pendent vertices,2≤k≤n,then W(S(n,k))≤W(T)≤W(D(n,(?),(?))).The lower bound is realized if and only if T(?)S(n,k) and the upper if and only if T(?)D(n,(?),(?)).We are naturally curious to know the tree(s) with the second maximum Wiener index in all k-pendent trees. The paper contains three chapters. In chapter 1, we introduce some notations as well as some known results of Wiener index of trees. In chapter 2, in order to determine the tree(s) with the second maximum Wiener index in all k-pendent trees, first we analyse the relations between the Wiener index and the transformation of trees, and reduce the set of the trees to the set of caterpillar. Then we find the recursion of Wiener index of caterpillar, and analyse the change law of the Wiener index of the trees under edge-moving transformation. Thereby we obtain the tree(s) with k-pendent, which reach(s) the second maximum Wiener index. In chapter 3, we obtain some properties of trees with k-pendent whose Wiener index reach the third maximum and some other ordering relations.