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 \(IP(G)=\sum _{v\in V(G)} d_G(v)^{1/2}\), the Misbalance Prodeg index \(MP(G)=\sum _{v\in V(G)} d_G(v)^{3/2}\), and the Yemen Prodeg index \(YP(G)=\sum _{v\in V(G)} d_G(v)^{5/2}\), together with their associated entropies \(E_{IP}(G)\), \(E_{MP}(G)\), and \(E_{YP}(G)\). More generally, for \(a>0\) we consider the degree-weighted distribution \(p_v\propto d_G(v)^a\) and the Shannon entropy \(E_a(G)\), recovering the Prodeg cases at \(a\in \{\tfrac{1}{2},\tfrac{3}{2},\tfrac{5}{2}\}\). 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 \(n\ge 2\) vertices, \(E_{IP}(G),E_{MP}(G),E_{YP}(G)\le \log n\), with equality if and only if G is regular, while highly imbalanced families (e.g., stars) exhibit strong concentration and vanishing entropy for \(a>1\) as \(n\rightarrow \infty\). 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: \(S_a(G)=\sum _v d_G(v)^a\) is multiplicative under the tensor (Kronecker) product, implying additivity of \(E_a\) and the Nordhaus–Gaddum-type bound \(E_a(G)+E_a(\overline{G})\le 2\log n\) whenever both entropies are defined. Using majorization, we also prove a monotone exponent hierarchy \(E_{IP}(G)\ge E_{MP}(G)\ge E_{YP}(G)\), with equality throughout precisely for regular graphs. To demonstrate chemical relevance, we analyze \(N=3219\) 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 \(\log _{10}(\textrm{Ipc})\)), 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 (\(R^2=0.970\)) and Molecular Weight (\(R^2=0.945\)), whereas MolLogP remains challenging (\(R^2=0.268\)). 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.
