On the topological indices of some graph operations on graphs

Abstract

A topological representation of a molecule is called molecular graph. Molecular graph of a corresponding molecule is a simple, undirected graph G = (V;E), which newlinerepresents the topological structure given the chemical compound. A topological index newlineis a numerical value that can be obtained logically from the given graph. The collection of vertices represents atoms in the molecule and set of edges represents covalent bonds among the atoms. There are tremendous applications in the development of Quanti- newlinetative Structure- Activity Relationship (QSAR) and Quantitative Structure- Property newlineRelationship (QSPR). A molecular graph is also known as hydrogen suppressed molecular graph or hydrogen-depleted molecular graph, where hydrogen vertices are deleted. newlineIn 1947, Wiener , a chemist introduced two parameters namely polarity number and newlinepath number and they are denoted by P and W respectively. He also assumed the newlinelinear formula to and boiling point of alkenes in terms of polarity and path number. The linear formula is tB = aw + bp + c, where a,b and c are constants for given isomeric group and p and w are structural variables. The path number was renamed with newlineWiener number or Wiener index . The Wiener index is measure based on the distance newlinebetween pair of vertices in the graph. Many more distance based indices were defined newlinelater. There are another types known as degree based topological indices, which are newlinemeasured by calculating the degree of the vertices of the given graph. Degree based newlineindices also have as many applications as distance based indices have. In this thesis we have studied six important topological indices (Zagreb indices and its coindices, Hyper Zagreb index, Reverse Zagreb indices, Forgotten index and its coindex and Harmonic index) of graph operations (Four operations, F) on graphs and some composite graphs (Jump graph, Mycielki graph, Total graph)as follows. (i) Zagreb indices and its coindices of some graph operations.(ii) Hyper-Zagreb indices of four operation on graphs.