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

Mohammed Alsharafi1,2, Yusuf Zeren3,4

  • 1Department of Mathematics, Faculty of Science, Sana'a University, Sana'a, Yemen. alsharafi205010@gmail.com.

Scientific Reports
|June 12, 2026
PubMed
Summary

We introduce the Inverse Prodeg index and its coindex, new graph invariants for mathematical chemistry. These novel descriptors are computationally efficient and show promise in predicting chemical properties of aromatic-carboxylate compounds.

Keywords:
CoindexDegree-based topological descriptorsExternal validationGraph operationsInverse Prodeg indexQSPR

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Area of Science:

  • Mathematical Chemistry
  • Graph Theory
  • Cheminformatics

Background:

  • Degree-based topological indices are crucial for describing molecular structures and analyzing structure-property relationships.
  • Existing descriptors like Randić and harmonic indices offer valuable insights but new invariants are needed for enhanced analysis.
  • The development of novel graph invariants can lead to more accurate predictions in chemical research.

Purpose of the Study:

  • To introduce and define the Inverse Prodeg index and its coindex as new degree-derived graph invariants.
  • To explore the mathematical properties, bounds, and computational efficiency of these new indices.
  • To evaluate the chemical relevance and predictive power of these indices for structure-property relationships in aromatic-carboxylate compounds.

Main Methods:

  • Introduction of the Inverse Prodeg index and its coindex, distinct from existing sum-, product-, and mixed-degree descriptors.
  • Establishment of mathematical properties, including bounds, equality cases, Nordhaus-Gaddum inequalities, and behavior under graph operations.
  • Application of a Prodeg-based descriptor family to a dataset of 90 aromatic-carboxylate compounds using regression models (Linear, PLS, Ridge) and validation techniques (Kennard-Stone, bootstrap, Y-randomization).

Main Results:

  • The Inverse Prodeg index and its coindex are analytically tractable and computationally efficient, computable in linear time with respect to the number of edges.
  • These new indices capture useful structure-property information, particularly for size- and thermodynamics-related endpoints in aromatic-carboxylate compounds.
  • Linear regression and partial least-squares regression demonstrated strong predictive performance, with mean external-test R² values, indicating a non-spurious predictive signal.

Conclusions:

  • The Inverse Prodeg index and its coindex are mathematically sound and practically useful graph descriptors.
  • These novel indices offer a valuable addition to the toolkit for quantitative structure-activity relationship (QSAR) studies.
  • Further validation and integration with other chemical descriptors are recommended for broader applicability and to claim general predictive superiority.