Zagreb Connection Indices of Subdivision and Semi-Total Point Operations on Graphs

Representation or coding of the molecular graphs with the help of numerical numbers plays a vital role in the studies of physicochemical and structural properties of the chemical compounds that are involved in the molecular graphs. For the first time, the modified first Zagreb connection index appeared in the paper by Gutman and Trinajstic (1972) to compute total electron energy of the alternant hydrocarbons, but after that, for a long time, it has not been studied. Recently, Ali and Trinajstic (2018) restudied the first Zagreb connection index (ZC(1)), the second Zagreb connection index (ZC(2)), and the modified first Zagreb connection index (ZC(1)*) to find entropy and acentric factor of the octane isomers. They also reported that the values provided by the International Academy of Mathematical Chemistry show better chemical capability of the Zagreb connection indices than the ordinary Zagreb indices. Assume that S-1 and S-2 denote the operations of subdivision and semitotal point, respectively.,en, the S-sum graphs Q(1) + (S)Q(2) are obtained by the cartesian product of S(Q(1)) and Q(2), where S is an element of {S-1, S-2}, Q(1) and Q(2) are any connected graphs, and S(Q(1)) is a graph obtained after applying the operation S on Q(1). In this paper, we compute the Zagreb connection indices (ZC(1), ZC(2), and ZC(1)*) of the S-sum graphs in terms of various topological indices of their factor graphs. At the end, as an application of the computed results, the Zagreb connection indices of the S-sum graphs obtained by the particular classes of alkanes are also included.