A Class Of Wiener Index Most Value

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, As a important topological index, it has been used with the physical characteristics and chemical characteristics of many compounds in the field of Chemistry. As been now, The Wiener index has been widely studied, For example:parting the trees by the number of the pendent vertices, Entrnger 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,(?)k/2」,「k/2(?))). The lower bound is realized if and only if T≌S(n, k) and the upper if and only if T≌D(n, (?)k/2」,「k/2(?)).In this paper, we study this graph:let G be a connected graph, uv is the cut-edge of the graph G, G1 and G2 are the two part of the G\uv. if translating the G2 into a tree T and parting the trees by the number of the pendent vertices,noted byτnk,keeping the G1 and the cut-edge uv, we have:the Wiener index of G get the minimum value if and only if T≌S(n, k), and vertex u is the center vertex of S(n, k). The paper contains four chapters. In chapter 1, we introduce some notations as well as some results of Wiener index; In chapter 2, when the k=n-1, n-2, n-3, we get some principles of the extremal vertex Wiener index inτnk and the extremal Wiener index of G; In chapter 3, we give the minimum vertex Wiener index inτnk; In chapter 4, we research some properties of maximum vertex Wiener index in tree T.