Computing Analysis of Connection-Based Indices and Coindices for Product of Molecular Networks

Usman Ali,Abdulaziz Mohammed Alanazi,Muhammad Javaid

doi:10.3390/sym12081320

Usman Ali, Abdulaziz Mohammed Alanazi + Show 1 more

Open Access

https://doi.org/10.3390/sym12081320

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Abstract

Gutman and Trinajstić (1972) defined the connection-number based Zagreb indices, where connection number is degree of a vertex at distance two, in order to find the electron energy of alternant hydrocarbons. These indices remain symmetric for the isomorphic (molecular) networks. For the prediction of physicochemical and symmetrical properties of octane isomers, these indices are restudied in 2018. In this paper, first and second Zagreb connection coindices are defined and obtained in the form of upper bounds for the resultant networks in the terms of different indices of their factor networks, where resultant networks are obtained from two networks by the product-related operations, such as cartesian, corona, and lexicographic. For the molecular networks linear polynomial chain, carbon nanotube, alkane, cycloalkane, fence, and closed fence, first and second Zagreb connection coindices are computed in the consequence of the obtained results. An analysis of Zagreb connection indices and coindices on the aforesaid molecular networks is also included with the help of their numerical values and graphical presentations that shows the symmetric behaviour of these indices and coindices with in certain intervals of order and size of the under study (molecular) networks.

Highlights

Topological indices (TIs) are functions that associate a numeric value with a finite, simple, and undirected network
These are used in chemo-informatics modelings consisting of quantitative structures activity and property relationships that create a symmetrical link between a biological property and a molecular network
At the end, an analysis of connection-based Zagreb indices and coindices on the aforesaid molecular networks is included with the help of their numerical values and graphical presentations

Summary

Introduction

Topological indices (TIs) are functions that associate a numeric value with a finite, simple, and undirected network. Gutman and Trinajstić [14] defined the first degree-based (number of vertices at distance one) TI called by the first Zagreb index to compute the total π-electron energy of the molecules in molecular networks. Ashrafi et al [16] defined the concept of coindices associated with the classical Zagreb indices for the resultant networks of different operations. We compute the coindices associated with the first and second Zagreb connection indices of the resultant networks as upper bounds in the terms of their factor networks, where resultant networks are obtained by Cartesian, corona and lexicographic products of two networks. At the end, an analysis of connection-based Zagreb indices and coindices on the aforesaid molecular networks is included with the help of their numerical values and graphical presentations.

Preliminaries

A Few Molecular Networks

Main Results

Applications, Comparisons and Conclusions