Distance-based topological indices of M\"{o}bius ladder graphs: a mathematical perspective

Topological indices---quantitative descriptors of connected graphs---play a pivotal role in mathematical chemistry and network analysis by capturing key structural properties. In this research, a set of well-known distance-based topological indices, including Wiener, Hyper-Wiener, Padmakar--Ivan, Szeged, Gutman and Degree Distance have been computed for the M"{o}bius ladder graph \(M_n\). By employing the automorphism group of the graph, \(\mathrm{Aut}(G)\), we derive closed-form expressions for each index for arbitrary \(n\). Detailed comparison for \(n\) ranging from 6 to 50, based on mathematical analysis and graphical illustrations, reveals that the Hyper-Wiener index effectively serves as a measure of central tendency and closely approximates the numerical mean of the other indices. Interestingly, the pairs (Wiener, Padmakar–Ivan) and (Szeged, Degree Distance) exhibit similar growth patterns and numerical behavior. These findings, presented in both tabular and graphical form, highlight the variations and interrelationships among the indices. To the best of our knowledge, this study offers the first systematic computation of distance-based indices for \(M_n\), revealing unique structural features. The comparative analysis not only enriches the understanding of topological indices in graph theory but also opens new avenues for applications in molecular structure modeling and biological network analysis.‎