Graph entropy reveals hidden order in hyper carbon nanotubes

Carbon nanotubes are pivotal in nanotechnology, serving roles in nanoelectronics, molecular sensors, and energy-storage devices. As researchers explore more complex architectures, quantifying the structural complexity of these materials becomes essential. A recent study in Scientific Reports by Muhammad Faisal Nadeem, Ayesha Shabbir, Mohamad Nazri Husin, and Yuslenita Muda addresses this challenge using graph entropy to analyze hyper armchair carbon nanotubes (HCN). These structures replace vertices of an armchair nanotube with Y-shaped junctions and edges with nanotube segments, creating a branched network suitable for multi-terminal interconnects in nanoscale circuits.

The study employs degree-based topological indices, which condense the connectivity pattern of a molecular graph into numerical descriptors. In nanotube graphs, vertices represent atoms or junctions, and edges represent bonds. Classic indices like the Zagreb indices correlate with physicochemical properties but can obscure structural information. By applying Shannon's entropy to these indices, researchers can assess the distribution of contributions across the graph. High entropy indicates even distribution among edge types, while low entropy suggests dominance by a few edges.

The authors established vertex-degree and edge partitions for HCN graphs, ensuring these classifications align with the graph's invariants: order, size, and degree-sum identity. This rigorous approach prevents errors common in chemical graph theory. With these partitions, the team derived closed-form expressions for topological indices, allowing predictions of entropy behavior as the nanotube grows without recalculating for each size.

The study finds that as hyper armchair carbon nanotubes increase in size or complexity, entropy values of degree-based indices rise, indicating a more even distribution of edge contributions. Additionally, entropy values of different indices converge for larger HCN graphs due to the dominance of a particular degree class. This suggests that for large hyper nanotubes, the choice of index may be less critical, while for smaller structures, different indices can yield distinct entropy values.

The research, funded by Universiti Malaysia Terengganu, highlights the synergy between information theory and chemical graph theory in understanding complex nanostructures. As synthetic chemistry advances, mathematical tools like entropy measures of degree-based indices will become increasingly important for interpreting carbon architectures. For hyper carbon nanotubes, these measures provide a precise metric for structural complexity, emphasizing the significance of connection patterns over individual atoms.