Zagreb indices of hyper carbon nanotube graphs.

Carbon nanotubes are cylindrical structures composed entirely of carbon atoms, renowned for their exceptional mechanical strength, electrical conductivity, and thermal stability. These unique properties have made them a cornerstone of modern nanoscience. Hyper carbon nanotube structures take this innovation further by creating hierarchical architectures, where atoms and bonds in a base scaffold are systematically replaced with larger nanotube-derived building blocks. This approach results in intricate, multi-scale carbon frameworks with enhanced functionality and wide-ranging potential applications in nanoelectronics, energy-storage devices, sensing, composite reinforcement, and targeted nanomaterials designing. Topological indices are graph invariants that encode structural information of molecular graphs and are widely used in quantitative structure property and quantitative structure activity studies to correlate chemical structure with physico-chemical behavior. In this paper, we propose and formalize a graph-theoretic construction of hyper armchair carbon nanotubes and compute closed-form expressions for Zagreb-type degree-based topological indices in both additive and multiplicative settings for this constructed family of graphs. Moreover, we present numerical comparisons along with graphical illustrations for representative parameter choices. To the best of our knowledge, this is the first paper on topological indices for this family of graphs.