Zeroth-order general Randić index for four new sum-graphs

For molecular graphs, the topological indices (TIs) proved to be a bridge between graph theory and mathematical chemistry to understand and predict certain properties of underlying chemical substances. In chemical graph theory (CGT), for a (molecular) graph Γ, zeroth-order general Randić index is considered to be a prominent and comprehensive topological index represented as ${}^{0}R_{\alpha }\left(\right(\Gamma \left)\right)$. Many prominent and useful indices are special cases of ${}^{0}R_{\alpha }\left(\Gamma \right)$. For instance, for $\alpha =-1$, $\frac{-1}{2}$, 2, and 3 we get inverse degree index (IDI) or modified total adjacency index, zeroth order Randić index $M_1^{\frac{-1}{2}}\left(\Gamma \right)$, first Zagreb index $M_1\left(\Gamma \right)$, and forgotten index $F\left(\Gamma \right)$, respectively. The TI ${}^{0}R_{\alpha }\left(\Gamma \right)$ has a myriad of applications in chemistry, such as estimating the structural dependence of alternant hydrocarbons on the π-electron energy. In CGT, graph products provide a framework for generating new (molecular) graphs of our choice by combining two graphs under a specific binary operation. Adequate research has been conducted on numerous TIs, including ${}^{0}R_{\alpha }\left(\right(\Gamma \left)\right)$, for F-sum graphs under the Cartesian product. In this paper, we derived the exact formulas of the ${}^{0}R_{\alpha }\left(\Gamma \right)$ for newly defined F-sum graphs based on the tensor product. We prove the accuracy and validity of our formulas by taking diverse pertinent examples. Researchers from cheminformatics can use our formulas of zeroth-order general Randić index for QSAR/QSPR studies in the design and analysis of new molecules. In addition, we provided closed-form formulas of ${}^{0}R_3\left(\Gamma \right)$ (F-index) for some general families of graphs using our results.