Degree-based Inverse Prodeg indices linking graph theory and cheminformatics with sharp results and QSPR modeling

Degree-based topological indices are widely used in mathematical chemistry because they provide simple numerical descriptions of molecular graphs and can support structure-property analysis. In this work, we introduce the Inverse Prodeg index [Formula: see text] and its coindex [Formula: see text] as new degree-derived graph invariants. Unlike product-, sum-, and mixed-degree descriptors such as the Randić, sum-connectivity, harmonic, atom-bond connectivity, geometric-arithmetic, Sombor, Nirmala, inverse Nirmala, and misbalance prodeg indices, [Formula: see text] reduces to the vertex-wise concave sum [Formula: see text], whereas [Formula: see text] transfers the same inverse square-root degree weighting to nonedges using the original graph degrees. We establish their main mathematical properties, including bounds involving graph order, size, and degree extrema, equality cases, Nordhaus-Gaddum-type inequalities, exact expressions for standard graph families, and estimates under several graph operations. These results show that the proposed descriptors are analytically tractable and computationally efficient, with linear-time computability in the number of edges. To examine their chemical relevance, we evaluate a small Prodeg-based descriptor family on a dataset of 90 aromatic-carboxylate compounds using training and external test sets generated by the Kennard-Stone algorithm. The models indicate that these descriptors capture useful structure-property information, especially for size- and thermodynamics-related endpoints. Linear regression and partial least-squares regression achieved the strongest average external-test performance among the considered Prodeg-only models, with mean external-test [Formula: see text] across the nine core endpoints, while Ridge regression was close with mean external-test [Formula: see text]. Nonlinear methods did not improve the average prediction accuracy. Additional validation through bootstrap analysis, Y-randomization, residual diagnostics, and applicability-domain assessment supports a non-spurious but dataset-dependent predictive signal. Overall, the Inverse Prodeg index and its coindex provide mathematically well-founded and practically useful graph descriptors, although broader validation and combination with chemically richer descriptors are needed before claiming general predictive superiority.