Research Article Zagreb Connection Indices of Molecular Graphs Based on Operations Jinde Cao , 1 Usman Ali, 2 Muhammad Javaid , 2 and Chuangxia Huang 3 1 School of Mathematics, Southeast University, Nanjing 210096, China 2 Department of Mathematics, School of Science, University of Management and Technology, Lahore, Pakistan 3 School of Mathematics, Changsha University of Science and Technology, Changsha 410114, China Correspondence should be addressed to Jinde Cao; jdcao@seu.edu.cn Received 15 November 2019; Accepted 5 February 2020; Published 30 March 2020 Academic Editor: Qingling Wang Copyright © 2020 Jinde Cao et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 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. e 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, AliandTrinajstic [Mol. Inform. 37(2018), 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 poly- nomial 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. 1. Introduction Graph theory has provided a variety of useful tools in which one of the best tools is a topological index (TI). Molecules and molecular compounds are often modeled by molecular graphs. e topological indices (TIs) predict hydrocarbon, physicochemical, and structural properties of the molecular graphs such as critical temperature, ZE-isomerism, chirality, solubility, molecular mass, and connectivity, see [1–4]. Medical behaviours of the drugs, crystallin materials, and nanomaterials which are very important for chemical and pharmaceutical industries are also studied by TIs, see [5–8]. Todeschini et al. [9] also reported that TIs are widely used in the study of quantitative structure-activity relationships (QSARs) and quantitative structure-property relationships (QSPRs).eserelationshipsplayavitalroleinthesubjectof cheminformatics, see [9–13]. TIs have been divided into different classes, but degree- based are studied more, see [1, 4, 7, 14, 15, 16]. Gutman and Trinajsti´ c [17] investigated the correlation value between the total π-electron energy and the structure of a molecule using the first Zagreb index. Gutman et al. [18] developed their work and established another TI for molecular structures called the second Zagreb index. After that, many extended works have been appeared on these invariants. For more study, we refer to [9, 10, 19, 20]. Another TI was studied by Gutman and Trinajsti´ c in the same paper [17], but there was not more attention on this index by other researchers up to 2017.AliandTrinajsti´ c[21]restudiedthisTIandrenamedit as the modified first Zagreb connection index (ZCI). ey Hindawi Complexity Volume 2020, Article ID 7385682, 15 pages https://doi.org/10.1155/2020/7385682