Topological indices of a class of graphs of some finite groups and applications to molecular structures

Abstract

In mathematical chemistry, a topological index is a molecular descriptor that is calculated based on the molecular graph of chemical compound, where the molecular graph can be represented as a graph in graph theory. The motivation of the study comes from chemistry, where it is used to predict the boiling point of some alkanes. Until today, there are many types of topological indices that have been introduced, such as the Wiener index, the Zagreb index, the Szeged index, the Harary index, the Hosoya index, the Kirchoff index, and the degree-distance index. However, in this thesis, only the Wiener index, the Zagreb index, the Szeged index, and the Harary index are considered since they are closely related to each other. The Wiener index of a graph is defined as the summation of the distance for all vertices in the graph. The Zagreb index is divided into two which are the first Zagreb index and the second Zagreb index. The first Zagreb index is the summation of all square degree of all vertices in the graph. Meanwhile, the second Zagreb index is a summation of the product for the degree of two vertices which are adjacent to each other. The Szeged index involves the product of the number of vertices in a graph which is closer to one particular vertex. Meanwhile, the Harary index is defined as half of the total of entries of reciprocal distance matrix. Over the past decades, the topological indices have been applied to graph theory, particularly a simple connected graph. Recently, the topological indices of graphs have become an area of interest to many mathematicians. Hence, this thesis focuses on finding the topological indices of some graphs related to some groups. In the first part of this research, these indices are determined for three types of graphs associated with three types of groups by using their definitions and some previous results. The graphs considered are the non-commuting graph, the conjugacy class graph, and the coprime graph associated with the dihedral groups, the generalized quaternion groups, and the quasidihedral groups. Then, the topological indices are computed and their general formulas are determined. The computation of the topological indices is further implemented to a larger group which is the direct product of two groups. This research focuses on the direct product of an abelian group with dihedral groups, and the direct product of two dihedral groups of not necessarily the same order. The second part of this research involves a set of point groups which is a collection of symmetry elements possessed by a shape or form in which all pass through one point in space. The point groups of order eight for some molecular structures are analyzed and the isomorphism of these three types of groups with the point groups are investigated. It is found that not all of the point groups of order eight of the molecular structures are isomorphic to the dihedral group of order eight. Lastly, the topological indices of the point groups of the molecular structures are computed.