A Refined Study on Fuzzy Topological Indices of H-Naphtalenic Nanotubes

Fuzzy graph theory has emerged as a valuable tool for modeling complex systems where uncertainty and imprecision are inherent. In this refined study, we explore the fuzzy topological indices of H-naphtalenic nanotubes, a class of nanostructures formed by combining hexagons, squares, and octagons in a regular pattern. Using fuzzy logic principles, we assign membership values to both the vertices and edges of the nanotube’s molecular graph, capturing the ambiguous nature of atomic interactions. We compute several well-known fuzzy topological indices, including the first and second Zagreb indices, Randić index, and harmonic index, using detailed weight distributions and degree-based formulations. The results are supported with analytical derivations and visual representations that highlight the influence of structural parameters on the indices. To enhance mathematical clarity, all index derivations have been rewritten step by step, with consistent notation for vertex and edge memberships. Each transition between equations is now accompanied by brief explanations for readability. A direct comparison with earlier studies on H-naphtalenic and related nanostructures demonstrates that the proposed fuzzy approach refines existing results and provides more accurate formulations for the same structural configurations. Additionally, worked numerical examples for small values of m and n are included to illustrate the behavior of the indices. The study also discusses potential applications in Quantitative Structure-Activity Relationships (QSAR) and Quantitative Structure-Property Relationships (QSPR) modeling, showing how fuzzy indices can support the prediction of molecular stability, reactivity, and structure-property relationships. This work not only improves the accuracy of earlier computations but also provides a clearer understanding of how fuzzy modeling can enhance the analysis of nanomaterials in chemical graph theory. This study also supports future work that links fuzzy topological indices with sustainable development goals in nanomaterial research.