The Chemical Molecule Graph Method Of Calculation Of The Wiener Number Of A Few Classes

In this thesis, we mainly study one of the topics in graph theory with applications: calculate the Wiener Index / a number for a molecular graph in chemistry.The Wiener topological Index is also called as Wiener number, it was first introduced by a well-known chemist Horald Wiener in 1947, and it is a very important index for molecular graphs. During the past fifty years, it was widely used to measure the chemical properties of molecules in the fields of structural chemistry, organic chemistry, and pharmacy etc. In 1997, a particular international conference was held to celebrate the fiftieth anniversary for the invention of the index, and published a special issue for the proceedings of the conference in Discrete Mathematics to memorize it. The reasons why it is widely used are that Combinatorial Chemistry becomes a hot field in recent years, and it lays theoretical foundation for the searching of new substances (especially, new organic material, and new drugs). Below is the framework of the theory: first, find molecular graphs (structures) for a given value of Wiener index, and then synthesize the molecules for the corresponding structures, which is called the inverse problem. The thesis is organized as follows.In the first chapter, we introduce the needed basic concepts and terms in graph theory, organic chemistry, and the definition of Wiener Index. In the end of this chapter, we give the main result of the paper.In the second chapter, we calculate the values of Wiener Index for two classes of linear phenylene molecular graphs, and obtain the linear recursive formulas, and then we get the exact expressions by solving some difference equations of the recursions. In the end of this chapter, we generalize the results to more general molecular graphs.In the third chapter, we study one of the most important class of molecularstructures in organic chemistry-the benzenoid systems, which is also calledhexagonal systems- in which we discuss benzenoid systems with one hole. Themethod, in which we used to calculate the value of the Wiener Index, breaks the limitation of the calculation for the benzenoid systems without holes. The method for dealing with benzenoid systems with one hole can be easily generalized to benzenoid systems with many holes.In the forth chapter, we study the Wiener Index of peptoids. The Wiener Index of peptoids, when the fragment graph is a path, has been calculated, see the reference [40]. We study the Wiener Index of peptoids, when the fragment graph is more complicated: a complete graph, a wheel, a cycle and a circulant graph. We get the exact expressions for them. And we get the Wiener Index for the unicyclic graphs. Inthe end of this chapter, we study the Wiener Index for the (M, N)- graph. This mayhelp for the study of the following problem: which graphs have the Wiener Index equal to the natural numbers besides 2 and 5, or the extreme value problem of the Wiener Index.