A Novel Weighted First Zagreb Index of Graph

A Novel Weighted First Zagreb Index of Graph

Jibonjyoti Buragohain (Dibrugarh University, India) and A. Bharali (Dibrugarh University, India)
DOI: 10.4018/978-1-5225-9380-5.ch004

Abstract

The Zagreb indices are the oldest among all degree-based topological indices. For a connected graph G, the first Zagreb index M1(G) is the sum of the term dG(u)+dG(v) corresponding to each edge uv in G, that is, M1 , where dG(u) is degree of the vertex u in G. In this chapter, the authors propose a weighted first Zagreb index and calculate its values for some standard graphs. Also, the authors study its correlations with various physico-chemical properties of octane isomers. It is found that this novel index has strong correlation with acentric factor and entropy of octane isomers as compared to other existing topological indices.
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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 978-1-5225-9380-5.ch004.m02 in a graph 978-1-5225-9380-5.ch004.m03 is the number of vertices incident on 978-1-5225-9380-5.ch004.m04 and it is denoted as 978-1-5225-9380-5.ch004.m05 or simply as 978-1-5225-9380-5.ch004.m06 if there is no scope for confusion. The notation 978-1-5225-9380-5.ch004.m07 represents any edge between two vertices 978-1-5225-9380-5.ch004.m08 and 978-1-5225-9380-5.ch004.m09 in a graph 978-1-5225-9380-5.ch004.m10. 978-1-5225-9380-5.ch004.m11 is used to denote the distance between vertices 978-1-5225-9380-5.ch004.m12 and 978-1-5225-9380-5.ch004.m13 which is nothing but the length of the shortest path between 978-1-5225-9380-5.ch004.m14 and 978-1-5225-9380-5.ch004.m15. The status of a vertex 978-1-5225-9380-5.ch004.m16 in a graph 978-1-5225-9380-5.ch004.m17 is denoted by 978-1-5225-9380-5.ch004.m18 or simply 978-1-5225-9380-5.ch004.m19 and is defined as the sum of distances of all other vertices from 978-1-5225-9380-5.ch004.m20 in 978-1-5225-9380-5.ch004.m21.

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