NM-polynomial and neighborhood degree-based indices in graph theory: a study on non-kekulean benzenoid graphs

In this study, we explore the neighborhood degree sum-based topological indices of Non-Kekulean Benzenoid graphs Kn using graph theory and computational tools. The novelty of this work lies in the application of the neighborhood M-polynomial (NM-polynomial) to derive various topological indices, which provide deep insights into the structural properties of Non-Kekulean Benzenoid systems. We compute several indices, including the third version of the Zagreb index, neighborhood second Zagreb index, neighborhood forgotten topological index, and others, using edge partitioning and combinatorial methods. The results are graphically represented and compared using MATLAB and Maple, revealing significant relationships between the molecular topology and the computed indices. Our findings demonstrate that the ND3 index is the most dominant, while the index increases more slowly compared to other indices. This study not only advances the understanding of Non-Kekulean Benzenoid graphs but also highlights the effectiveness of combining mathematical methodologies with computational tools for molecular structure analysis. The results contribute to the fields of graph theory and computational chemistry, offering a foundation for future research on diverse molecular structures.