Topological Characterization and Statistical Entropy Modeling of Hexagonal Graphene Nanostructures

The mathematical characterization and thermodynamic modeling of graphene nanostructures present significant analytical challenges in mathematical chemistry, particularly in mapping discrete structural symmetries to macroscopic physical properties. This paper focuses on the quantitative structural analysis of hexagonal graphene configurations to establish a formal statistical framework correlating their molecular graph architecture with physical entropy. To achieve this objective, a comprehensive and diverse set of topological indices—specifically focusing on degree-based, distance-based, and information-theoretic descriptors—is employed to rigorously evaluate the structural complexity, connectivity, internal symmetry variations, and boundary features of unique graphene configurations. By treating these nanostructures as discrete molecular graphs, we compute stable mathematical invariants that capture subtle topological variations across the entire structural landscape. Furthermore, advanced statistical analysis, multi-variable regression modeling, and rigorous analysis of variance are applied to investigate the direct analytical dependencies between these calculated topological descriptors and the physical entropy of the corresponding carbon networks. The analytical results reveal remarkably strong statistical correlations, demonstrating that discrete invariants derived purely from graph topology and network symmetry possess exceptionally high predictive power for complex thermodynamic behavior. Ultimately, this study highlights the profound utility of quantitative mathematical analysis as a robust tool in mathematical chemistry, providing an efficient predictive method to interpret, bound, and estimate physical entropy within extended polycyclic aromatic systems without relying on computationally expensive quantum mechanical calculations.