Research On The Wiener Index And The Wiener Polarity Index Of Trees

Wiener index and Wiener polarity index are very important in chemistry and math-ematics fields. They are widely used molecular topological index. Molecular topological index can reflect the physical and chemical properties and pharmacological properties of molecules, such as boiling point, water-solubility, the volume and surface area of molecular, energy levels, the election distribution, etc. Molecular topological index mainly used in the research and analysis of the "structure-property" quantitative relationship and the "structure - activity" relationship. Topological index of molecu-lar graphs is one of the most widely used descriptors in quantitative structure activity relationships. Quantitative structure activity relationships (QSAR) is a popular com-putational biology paradigm in modern drug design, where the Wiener index W(G) of a graph G, denoted by W(G), which is defined as the sum of the distances between all pairs of vertices of a connected graph. Namely, the Wiener index of a graph G is W(G):=?{u,v}?V(G) dG(u,v). The Wiener polarity index of a graph G, denoted by WP(G), Which is the number of unordered pairs of vertices u, v of G such that dG(u,v)= 3, WP(G):= #{u,v}|dG(u,v)= 3,u,v ? V}, where dG{u,v) denotes the distance of the vertex u the vertex v in G. In this article, we mainly take our attention to two special kinds of graph classes, edge-weighted trees and chemical trees. By using the combination method, we firstly expound the minimum, the second minimum, the third minimum, the maximum, the second maximum values of the Wiener index for the edge-weighted trees of order n, and characterize the corresponding extremal trees, respectively. Then we study the chemical trees of order n and diameter d, the corre-sponding maximum value is classified discussed. Furthermore, we characterize the trees maximizing the Wiener polarity index. The full text is divided into five parts.In the first part, we briefly introduce the research background and current situation of index, and probe into the main content of the paper.In the second part, we introduce the notations, basic definitions and theoremsof graphs, Wiener index and the Wiener polarity index.In the third part, we deduce conclusion that the minimum, the second minimum, the third minimum, the maximum, the second maximum values of the Wiener index for weighted trees, and characterize the corresponding extremal trees respectively.In the fourth part, we get the maximum values of the Wiener polarity index for chemical trees of given order n and diameter d, and characterize the corresponding extremal trees.In the fifth part, we summarize the main work of this thesis, and suggest some further studies on Wiener index and Wiener polarity index in future.