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

For a molecular graph Γ, the general sum-connectivity index is defined as χβ(Γ) =Σvw∈E(Γ)[dΓ(v) + dΓ(w)]β, where β ∈ R and dΓ(v) denotes the degree of the vertex v in the molecular graph Γ. 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 β ∈ N, we figured out and improved the sharp bounds of the general sum-connectivity index for F-sum graphs, where F ∈ {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.