On Generalization of Molecular Descriptors Based on Certain Techniques for 2D Structures

Abstract

Graph theory has vast applications in every aspect of science particularly in chemistry. It is used to find the precise solutions of the problems and also to discuss the molecular compounds in the form of graphs. The graph of a chemical compound can be represented by a matrix, by a polynomial or by molecular descriptor. The graph polynomials, molecular descriptors and the sequences related to molecular descriptors for 2D-molecular graphs have numeral applications in chemical graph theory, mathematical chemistry, QSAR, QSPR researches and in quantum chemistry as well. In this dissertation counting polynomials have been constructed for different molecular graphs and also established the relations to analyze their topological indices. We discussed the Omega polynomial, Theta, PI and Sadhana polynomials for different 2D-molecular structures. The Omega polynomial of a graph is denoted by , and defined as ∑

, where is the length of the strip, and „ ‟ is the total number of strips having length . The Sadhana polynomial is computed as, ∑ | | where | | is the size of the graph . The Theta and PI polynomial of the graph have the relations, ∑ and =∑ | | respectively. We computed the molecular descriptors i.e Omega ( , Theta ( , PI ( and Sadhana indices ( , based on counting polynomials for Benzenoid Coronene graph, Concealed non-Kekulean Benzenoid graph and Ring type Benzenoid graph, Pyrene Network and P-Type-Surface of Benzenoid network. By working on the Benzenoid Coronene graph, we generated the sequences for Omega, Theta, PI and Sadhana indices. Moreover, we introduced a novel a technique to construct Theta, PI, Sadhana polynomials and their related indices for 2D-planar graphs. We also extended the idea of P. E. John et.al and evolved Sadhana Index in-terms of Omega polynomial and Omega index and also, the PI index in-terms of Theta polynomial and Theta index for the first time. The Degree based molecular descriptors are based on the degree of vertices of a graph such as, the Randic index, the Geometric-arithmetic index , the Atom-Bond connectivity index and derived versions of these indices named, the fifth version of Geometric-arithmetic index and the fourth version of Atom-Bond connectivity index have been studied frequently for some molecular graphs are also the part of this work. The relations between and indices have been studied for complete graph in v the last section of chapter six. This work will provide the interesting information about the chemical graphs and will be a valuable contribution in the existing knowledge