Graph Energy Variants and Topological Indices in Platinum Anticancer Drug Design: Mathematical Insights and Computational Analysis with DFT and QTAIM

In this paper, we present a new approach that explores the application of graph energy variants in chemistry, specifically in the development of platinum anticancer drugs. While previous energy variants have been proposed without considering their direct relevance to chemistry, our study focuses on two key aspects. First, we investigate the correlation between seven degree-based and four distance-based topological indices and their corresponding energies in platinum anticancer drugs. Furthermore, we mathematically analyze the properties of these energies, establishing upper and lower bounds that can be generalized to other structures. Second, we examine the possibility of utilizing the energies of these topological indices as structural descriptors. Our research showcases promising results, suggesting potential improvements in the future manufacturing of anticancer drugs. In addition, we employ density functional theory $\left(\mathrm{D}\mathrm{F}\mathrm{T}\right)$ calculations to optimize the molecular structures of platinum anticancer drugs and identify local reactive sites using Fukui functions. Quantum theory of atoms in molecules $\left(\mathrm{Q}\mathrm{T}\mathrm{A}\mathrm{I}\mathrm{M}\right)$ was carried out at the bond critical point $\left(\mathrm{B}\mathrm{C}\mathrm{P}\right)$ , to reveal the nature of the intermolecular interactions in the investigated ten Pt anticancer drugs, especially, the nature of bonds between Pt atoms and their bond atoms. Overall, this study presents an innovative approach that bridges graph energies, topological indices, and DFT with the properties (physical and chemical) of platinum anticancer drugs, offering insights into their molecular properties and potential for enhanced drug design.