NM-polynomial and neighborhood degree-based indices in graph theory: a study on non-kekulean benzenoid graphs
Bibliographic record
Abstract
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.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.002 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".