On topological characterization of hierarchical hypercube interconnection networks using M-polynomials

Abstract

The application of the M-polynomial to chemical networks, particularly to chemical compounds, is a relatively recent development in chemical graph theory. Despite its novelty, this approach has proven to be a powerful and effective tool for deriving degree-based topological indices. These indices play a crucial role in modeling and predicting various physicochemical properties as well as biological activities of chemical substances and nanostructured systems. The M-polynomial offers a unified and systematic framework by establishing mathematical relationships between molecular structure and chemical behavior. In this work, we focus on computing the general form of M-polynomials for Hierarchical Hypercube Networks (HHNs). The selected HHNs exhibit a high degree of structural symmetry, which significantly simplifies the analytical computations and enables the exact determination of the corresponding M-polynomials. By exploiting these symmetries, we derive closed-form expressions that describe the underlying degree distributions of the considered nanostructures. Furthermore, several important degree-based topological indices are obtained from the derived M-polynomials. These indices provide valuable insights into the structural complexity and potential chemical dynamics of the studied Hierarchical Hypercube Networks. Finally, the M-polynomials are presented graphically to highlight and visualize the structural characteristics of the HHNs.