Prodeg type Shannon graph entropies with closed forms bounds and QSPR modeling

Degree-based graph entropies quantify structural heterogeneity by transforming vertex-degree information into a probability distribution and applying Shannon entropy. We develop a unified framework for three Prodeg-type degree-power invariants, namely the Inverse Prodeg index [Formula: see text], the Misbalance Prodeg index [Formula: see text], and the Yemen Prodeg index [Formula: see text], together with their associated entropies [Formula: see text], [Formula: see text], and [Formula: see text]. More generally, for [Formula: see text] we consider the degree-weighted distribution [Formula: see text] and the Shannon entropy [Formula: see text], recovering the Prodeg cases at [Formula: see text]. We derive closed-form expressions for representative graph families (complete graphs, cycles, paths, stars, and complete bipartite graphs) and establish sharp extremal behavior: for connected graphs on [Formula: see text] vertices, [Formula: see text], with equality if and only if G is regular, while highly imbalanced families (e.g., stars) exhibit strong concentration and vanishing entropy for [Formula: see text] as [Formula: see text]. We further provide explicit two-sided bounds in terms of degree extremes and concentration control via the heaviest weight. A main structural result is a tensor-product principle: [Formula: see text] is multiplicative under the tensor (Kronecker) product, implying additivity of [Formula: see text] and the Nordhaus-Gaddum-type bound [Formula: see text] whenever both entropies are defined. Using majorization, we also prove a monotone exponent hierarchy [Formula: see text], with equality throughout precisely for regular graphs. To demonstrate chemical relevance, we analyze [Formula: see text] antibacterial compounds curated from the ChEMBL database and show that both classical degree-entropies and Prodeg entropies strongly track established molecular information/complexity measures (BertzCT and [Formula: see text]), while AvgIpc exhibits weaker and nonlinear associations. Finally, we benchmark entropy-only QSPR models for nine physicochemical endpoints using 5-fold cross-validation. Tree ensembles deliver the strongest performance, with particularly high accuracy for size-related properties such as MolMR ([Formula: see text]) and Molecular Weight ([Formula: see text]), whereas MolLogP remains challenging ([Formula: see text]). Across endpoints, the Prodeg block is competitive despite using fewer descriptors, and combining classical and Prodeg entropies yields consistent (typically modest) RMSE gains, supporting Prodeg entropies as compact and interpretable descriptors for chemical graph analysis and QSPR modeling.