On the Reformulated Second Zagreb Index of Graph Operations

Durbar Maji,Yaé Ulrich Gaba,Ganesh Ghorai,Haidar Ali

doi:10.1155/2021/9289534

Durbar Maji, Yaé Ulrich Gaba + Show 2 more

Open Access

https://doi.org/10.1155/2021/9289534

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Abstract

Topological indices (TIs) are expressed by constant real numbers that reveal the structure of the graphs in QSAR/QSPR investigation. The reformulated second Zagreb index (RSZI) is such a novel TI having good correlations with various physical attributes, chemical reactivities, or biological activities/properties. The RSZI is defined as the sum of products of edge degrees of the adjacent edges, where the edge degree of an edge is taken to be the sum of vertex degrees of two end vertices of that edge with minus 2. In this study, the behaviour of RSZI under graph operations containing Cartesian product, join, composition, and corona product of two graphs has been established. We have also applied these results to compute RSZI for some important classes of molecular graphs and nanostructures.

Highlights

In the whole study, we only consider the molecular graph [1, 2], a graphical representation of molecular structure, in which every vertex corresponds to the atoms and the edges to the bonds between them
Topological indices (TIs) can be expressed by real numbers related to graphs. ere exist many applications as tools for modelling chemical and other properties of molecules for TIs. ey determine the correlation between the specific properties of molecules and the biological activity with their configuration in the study of quantitative structure-activity relationships (QSARs) and quantitative structure-property relationships (QSPRs) [3]
We study different binary graph operations such as join, Cartesian product, composition, and corona product of two molecular graphs and compute some exact formulae for reformulated second Zagreb index (RSZI) with respect to those operations separately

Summary

Introduction

We only consider the molecular graph [1, 2], a graphical representation of molecular structure, in which every vertex corresponds to the atoms and the edges to the bonds between them. DJ(x) denotes the degree of a vertex (x) in J and is defined as the number of edges incident to x. Ese two indices are, respectively, defined for molecular graph (J) as. In 2004, Milicevic et al [7] reformulated the Zagreb indices by replacing vertex degree with edge degree, and the edge degree of an edge is defined as d(e) d(x) + d(y) − 2. E first and second reformulated Zagreb indices [8] of a graph J are defined as EM1(J) 􏽘 d2(e),. In [10], Khalifeh et al computed the first and second Zagreb indices under some graph operations. In [12], Das et al derived multiplicative Zagreb indices of different graph operations. We refer to [17,18,19,20,21,22,23] in this regard for interested readers

Main Results

Step 3:

Conclusion