The Hosoya Indices And Merrifield-Simmons Indices Of Some Kinds Of Graphs

If the vertices of a graph denote the atoms of a chemical molecule and the edges denote the bonds between atoms of the molecule, then this graph is called a molecular graph. Molecular graphs can express the topological characteristics of molecules. The molecular topological index which we said is a mapping from the set of molecular graphs to the set of real numbers. For example the Wiener-index, Hosoya-index, Merrifield-Simmons-index, Vizing-index and so on. Molecular topological indices have close relationship with the physical and chemical characteristics of the molecule. The Hosoya-index (denoted by z(G)) and the Merrifield-Simmons-index (denoted by a(G)) are two important indices of molecular topology which are defined as the total number of matchings and the total number of independent sets. There are some relationship between these indices and the boiling point and melting point of molecules.This paper characterizes the extreme graphs of three kinds of graphs.(1)We characterize the extreme graphs having k-cycles which have maximum Merrifield-Simmons-index and minimum Hosoya-index, and give the maximum Merrifield-Simmons-index. (2) Then we characterize the extreme graphs of conjugate graphs having minimum Hosoya-index, containing the first, second, third minimum extreme graphs of conjugate trees and the first, second minimum extreme graphs of conjugate graphs which are not trees. (3) Finally, we characterize the extreme graphs of unicyclics with given diameter.The structure of this paper is:In chapter 1, we introduce the background and the researches of graph, Hosoya-index and Merrifield-Simmons-index. We also give some useful concepts and terminology. From chapter 2 to chapter 4, we characterize the extreme graphs of three kinds of graphs above-mentioned. In chapter 5, we summarize the main research results and discuss the problems for further research.