Multiplicative Dharwad indices for heterogeneous graphs random trees and line graphs with molecular applications
Chemical Papers, 2026 (SCI-Expanded, Scopus)
- Yayın Türü: Makale / Tam Makale
- Basım Tarihi: 2026
- Doi Numarası: 10.1007/s11696-026-05564-0
- Dergi Adı: Chemical Papers
- Derginin Tarandığı İndeksler: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Aerospace Database, Chemical Abstracts Core, Materials Science & Engineering Collection (ProQuest), Technology Collection (ProQuest)
- Anahtar Kelimeler: Degree heterogeneity, Harmonic–arithmetic index, Line graph, Molecular tree, Multiplicative Dharwad index, Multiplicative Zagreb index, QSPR, Random spider tree
- Yıldız Teknik Üniversitesi Adresli: Evet
Özet
We develop the multiplicative Dharwad index within a general degree-power framework and connect it rigorously with several active directions in degree-based graph invariants. An exact factorization separates into a classical second multiplicative Zagreb term and a nonnegative edge-heterogeneity correction expressed through hyperbolic cosine. This decomposition yields sharp regularity criteria, degree-ratio bounds, and a new lower bound involving the harmonic–arithmetic index. A power-mean representation links the full family with mean Sombor and Stolarsky–Puebla-type descriptors. We derive an exact line-graph formula, including closed forms for regular and semiregular bipartite graphs, and prove a sharp ordering of spider trees with fixed order and number of legs. For a random spider-growth model, we obtain the almost-sure asymptotic law. Sharp general bounds, the unique path minimum and star maximum among trees, graph-operation formulas, connectivity candidates, and domination-number transfer bounds are also established. The structural theory is supported by four computational studies and exhaustive small-graph checks. Exhaustive enumeration of 12,665 molecular trees of selected orders shows a very strong inverse association between the edge-normalized logarithmic Dharwad index and the harmonic–arithmetic index, while its association with degree Gini inequality is strongly positive. On 18 octane isomers, distinguishes 16 structures, whereas the line-graph descriptor distinguishes all 18 and shows an exploratory one-descriptor association with the acentric factor, with, leave-one-out, and RMSE 0.00501. Simulations on Erdős–Rényi, random geometric, regular, and Barabási–Albert networks further show that the normalized multiplicative correction tracks degree heterogeneity. The molecular regression is limited to 18 closely related isomers and internal leave-one-out validation; it is therefore presented as a benchmark rather than evidence of universal predictive superiority. The resulting study moves the multiplicative Dharwad index from a catalogue of closed forms to a unified structural, probabilistic, and molecular-descriptor theory.