Computing First General Zagreb Index of Operations on Graphs

The numerical coding of the molecular structures on the bases of topological indices plays an important role in the subject of Cheminformatics which is a combination of Computer, Chemistry, and Information Science. In 1972, it was shown that the total pi-electron energy of a molecular graph depends upon its structure and it can be obtained by the sum of the square of degrees of the vertices of a molecular graph. Later on, this sum was named as the first Zagreb index. In 2005, for gamma is an element of R - {0, 1}, the first general Zagreb index of a graph G is defined as M-gamma(G) = Sigma(v is an element of V(G))[d(G)(v)](gamma), where d(G)(v) is degree of the vertex v in G. In this paper, for each gamma is an element of R - {0, 1}, we study the first general Zagreb index of the cartesian product of two graphs such that one of the graphs is D-sum graph and the other is any connected graph, where D-sum graph is obtained by using certain D operations on a connected graph. The obtained results are general extensions of the results of Deng et al. [Applied Mathematics and Computation 275(2016): 422-431] and Akhter et al. [AKCE Int. J. Graphs Combin. 14(2017): 70-79] who proved only for gamma = 2 and gamma = 3, respectively.