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 \(IP(G)\) and its coindex \(\overline{IP}(G)\) 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, \(IP(G)\) reduces to the vertex-wise concave sum \(\sum _{u\in V(G)}\sqrt{\delta _G(u)}\) , whereas \(\overline{IP}(G)\) 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 \(R^2=0.832\) across the nine core endpoints, while Ridge regression was close with mean external-test \(R^2=0.829\) . 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.
