Exact Formula and Improved Bounds for General Sum-Connectivity Index of Graph-Operations

For a molecular graph Gamma, the general sum-connectivity index is defined as chi(beta) (Gamma) = Sigma(vw is an element of E(Gamma))[d(Gamma) (v) + d(Gamma) (w)](beta); where beta is an element of R and d(Gamma) (v) denotes the degree of the vertex v in the molecular graph Gamma: The problem of finding best possible upper and lower bound for certain topological index is of fundamental nature in extremal graph theory. Akhtar and Imran [J. Inequal. Appl. (2016) 241] obtained the sharp bounds of general sum-connectivity index for four graph operations (F-sum graphs) introduced by Eliasi and Taeri [Discrete Appl. Math. 157: 794-803, 2009)]. In this paper, for beta is an element of N; we figured out and improved the sharp bounds of the general sum-connectivity index for F-sum graphs, where F is an element of {R, Q, T}. Several examples are presented to elaborate and compare the results of improved bounds with existing sharp bounds. In addition, we obtained exact formula of general sum-connectivity index for F-sum graphs, when F = S.