A Study On Fuzzy Zagreb Indices And Its Applications

Abstract

In mathematical chemistry, topological indices are molecular descriptors that are calculated on the molecular graph of a chemical compound. Topological indices are Numerical quantities of a graph that describe its topology. An atom represents a vertex and a bond between two atoms represents an edge in a molecular graph. In case of classical graphs, both the vertices and edges have membership value one, but in case of fuzzy graphs both vertices and edges are equally important along with their fuzzy membership values. If the description of objects or their relationships or both are vague in nature, then we design a Fuzzy Graph model. This thesis is devoted to study of topological indices of fuzzy graphs and in particularly, fuzzy graph operations. The thesis is structured into eight chapters. Each of the chapters is sectioned to contain introduction to respective chapter, literature survey, preliminaries with auxiliary lemmas, main result and its proof. The main theorems are supplemented with suitable examples wherever necessary and the chapter is culminated with conclusion. The introduction chapter gives an overview of the origin of the research area, its development and some of the vital path breaking results. The definitions which are used throughout the thesis are also mentioned here. The chapters 2-6 contain the core research work. In particular, chapter 2 explore the first entire Zagreb index for various classes of generalized fuzzy transformation graphs. This research chapter is also deals with the bounds for FEZI for generalized fuzzy transformation graphs in terms of elements of a fuzzy graph G. In chapter 3 provides the importance of Sombor index in mathematical chemistry due to its high predicting power in physico-chemical properties of chemical compounds. In this chapter, we have explore the properties of the Sombor index of fuzzy graphs. The study presented the relationship between the Sombor index and the first Zagreb index of fuzzy graphs.