Abstract
Topological index (numeric number) is a mathematical coding of the molecular graphs that predicts the physicochemical, biological, toxicological, and structural properties of the chemical compounds that are directly associated with the molecular graphs. The Zagreb connection indices are one of the TIs of the molecular graphs depending upon the connection number (degree of vertices at distance two) appeared in 1972 to compute the total electron energy of the alternant hydrocarbons. But after that, for a long period, these are not studied by researchers. Recently, Ali and Trinajstic Mol. Inform. 372018,1−7 restudied the Zagreb connection indices and reported that the Zagreb connection indices comparatively to the classical Zagreb indices provide the better absolute value of the correlation coefficient for the thirteen physicochemical properties of the octane isomers (all these tested values have been taken from the website http://www.moleculardescriptors.eu). In this paper, we compute the general results in the form of exact formulae & upper bounds of the second Zagreb connection index and modified first Zagreb connection index for the resultant graphs which are obtained by applying operations of corona, Cartesian, and lexicographic product. At the end, some applications of the obtained results for particular chemical structures such as alkanes, cycloalkanes, linear polynomial chain, carbon nanotubes, fence, and closed fence are presented. In addition, a comparison between exact and computed values of the aforesaid Zagreb indices is also included.
Highlights
Graph theory has provided a variety of useful tools in which one of the best tools is a topological index (TI)
Todeschini et al [9] reported that TIs are widely used in the study of quantitative structure-activity relationships (QSARs) and quantitative structure-property relationships (QSPRs). ese relationships play a vital role in the subject of cheminformatics, see [9,10,11,12,13]
We compute the second Zagreb connection index (ZCI) and modified first ZCI of the resultant graphs which are obtained by applying various operations of corona product, Cartesian product, and lexicographic product in the form of exact formulae and upper bounds. e rest of the paper is settled as Section 2 represents the preliminary definitions and results, Section 3 covers the general results of molecular graphs based on operations, and Section 4 includes the applications and conclusion
Summary
Graph theory has provided a variety of useful tools in which one of the best tools is a topological index (TI). Gutman et al [18] developed their work and established another TI for molecular structures called the second Zagreb index. Graovac and Pisanski [27] computed different results of the Wiener index using product based on operations. Many results of the various TIs have been presented under different molecular graphs based on operations, see [26, 28,29,30,31,32,33,34,35,36,37,38]. We compute the second ZCI and modified first ZCI of the resultant graphs which are obtained by applying various operations of corona product, Cartesian product, and lexicographic product (composition) in the form of exact formulae and upper bounds. We compute the second ZCI and modified first ZCI of the resultant graphs which are obtained by applying various operations of corona product, Cartesian product, and lexicographic product (composition) in the form of exact formulae and upper bounds. e rest of the paper is settled as Section 2 represents the preliminary definitions and results, Section 3 covers the general results of molecular graphs based on operations, and Section 4 includes the applications and conclusion
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