The structure and topology of chemical compounds can be determined using chemical graph theory. Using topological indices, we may uncover much about connectivity, complexity, and other important aspects of molecules. Numerous research investigations have been conducted on the K-Banhatti indices and entropy measurements in various fields, including the study of natural polymers, nanotubes, and catalysts. At the same time, the Shannon entropy of a graph is widely used in network science. It is employed in evaluating several networks, including social networks, neural networks, and transportation systems. The Shannon entropy enables the analysis of a network’s topology and structure, facilitating the identification of significant nodes or structures that substantially impact network operation and stability. In the past decade, there has been a considerable focus on investigating a range of nanostructures, such as nanosheets and nanoparticles, in both experimental and theoretical domains. As a very effective catalyst and inert substrate, the $MgO$ nanostructure has received a lot of interest. The primary objective of this research is to study different indices and employ them to look at entropy measures of magnesium oxide(111) nanosheets over a wide range of p values, including $p=1,2,3,\dots ,j$. Additionally, we conducted a linear regression analysis to establish the correlation between indices and entropies.

Topological indices are widely used for identifying structure-property relationships due to their computational simplicity. Graph energies are an active research area nowadays, with over a thousand publications and an average of two papers published weekly. This study has two parts. The first part aims to explore the relationship between various graph energies of random tree graphs and well-known graph structural features, and the second part utilizes energies to predict the physicochemical properties of real-world molecular graphs. For both studies, we used XGBoost and SHAP (SHapley Additive Explanations) to build a decision-making model. We employed the Randomised Search CV to enhance XGBoost's performance further. This algorithm randomly selects a set of hyperparameters and evaluates the model's performance using cross-validation, resulting in improved accuracy. According to our research, XGBoost and SHAP (SHapley Additive Explanations) can help examine the relationships between topological indices and the structural features and physicochemical properties of drug molecules.