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

Abstract

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 Rα0((Γ)). Many prominent and useful indices are special cases of Rα0(Γ). For instance, for α=−1, −12, 2, and 3 we get inverse degree index (IDI) or modified total adjacency index, zeroth order Randić index M1−12(Γ), first Zagreb index M1(Γ), and forgotten index F(Γ), respectively. The TI Rα0(Γ) 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 Rα0((Γ)), for F-sum graphs under the Cartesian product. In this paper, we derived the exact formulas of the Rα0(Γ) 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 R30(Γ) (F-index) for some general families of graphs using our results.

Keywords

• First Zagreb index
• Molecular graph
• Cartesian product
• Graph operations
• Tensor product