A Novel Weighted First Zagreb Index of Graph

Introduction

The introduction of topological index or molecular structure descriptor in literature is a great success as it can correlate various physical properties, biological reactivity or chemical activity of molecules without undergoing actual experimentation. A topological index is a real number that can be associated to a molecule based on its molecular graph. A molecular graph is a simple graph corresponding to a molecule in which vertices represents atoms and edges represents various chemical bonds between them. For historical background of the topological index interested reader can go through (Trinajstić, 2011). The success of this simple mathematical quantity is because of the fact that the properties of a chemical compound is highly related to its molecular structure and we calculate the indices directly from the molecular graph of a molecule using various graph theoretic notions (e.g. degree, distance, etc). Some of the topological indices may be found in (Das & Tinajstić, 2010; Gutman, 2013; Estrada, 2000; Gutman, Milovanović & et al., 2018) and the references therein.

Throughout the chapter only undirected, finite and simple connected graphs are considered. The degree of a vertex in a graph is the number of vertices incident on and it is denoted as or simply as if there is no scope for confusion. The notation represents any edge between two vertices and in a graph . is used to denote the distance between vertices and which is nothing but the length of the shortest path between and . The status of a vertex in a graph is denoted by or simply and is defined as the sum of distances of all other vertices from in .