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For a molecular graph <inline-formula> <tex-math notation="LaTeX">$\Gamma $ </tex-math></inline-formula>, the general sum-connectivity index is defined as <inline-formula> <tex-math notation="LaTeX">$\chi _{\beta }(\Gamma) = \sum _{vw \in E(\Gamma)}[d_{\Gamma }(v)+d_{\Gamma }(w)]^{\beta }$ </tex-math></inline-formula>, where <inline-formula> <tex-math notation="LaTeX">$\beta \in \mathbb{R}$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$d_{\Gamma }(v)$ </tex-math></inline-formula> denotes the degree of the vertex <inline-formula> <tex-math notation="LaTeX">$v$ </tex-math></inline-formula> in the molecular graph <inline-formula> <tex-math notation="LaTeX">$\Gamma $ </tex-math></inline-formula>. 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 (<inline-formula> <tex-math notation="LaTeX">$F$ </tex-math></inline-formula>-sum graphs) introduced by Eliasi and Taeri [Discrete Appl. Math. 157: 794-803, 2009)]. In this paper, for <inline-formula> <tex-math notation="LaTeX">$\beta \in \mathbb{N}$ </tex-math></inline-formula>, we figured out and improved the sharp bounds of the general sum-connectivity index for <inline-formula> <tex-math notation="LaTeX">$F$ </tex-math></inline-formula>-sum graphs, where <inline-formula> <tex-math notation="LaTeX">$F \in \{R,Q,T\}$ </tex-math></inline-formula>. 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 <inline-formula> <tex-math notation="LaTeX">$F$ </tex-math></inline-formula>-sum graphs, when <inline-formula> <tex-math notation="LaTeX">$F=S$ </tex-math></inline-formula>.

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