Journal of Computer-Aided Molecular Design

, Volume 26, Issue 11, pp 1229–1246 | Cite as

Derivatives in discrete mathematics: a novel graph-theoretical invariant for generating new 2/3D molecular descriptors. I. Theory and QSPR application

  • Yovani Marrero-Ponce
  • Oscar Martínez Santiago
  • Yoan Martínez López
  • Stephen J. Barigye
  • Francisco Torrens


In this report, we present a new mathematical approach for describing chemical structures of organic molecules at atomic-molecular level, proposing for the first time the use of the concept of the derivative (\( \partial \)) of a molecular graph (MG) with respect to a given event (E), to obtain a new family of molecular descriptors (MDs). With this purpose, a new matrix representation of the MG, which generalizes graph’s theory’s traditional incidence matrix, is introduced. This matrix, denominated the generalized incidence matrix, Q, arises from the Boolean representation of molecular sub-graphs that participate in the formation of the graph molecular skeleton MG and could be complete (representing all possible connected sub-graphs) or constitute sub-graphs of determined orders or types as well as a combination of these. The Q matrix is a non-quadratic and unsymmetrical in nature, its columns (n) and rows (m) are conditions (letters) and collection of conditions (words) with which the event occurs. This non-quadratic and unsymmetrical matrix is transformed, by algebraic manipulation, to a quadratic and symmetric matrix known as relations frequency matrix, F, which characterizes the participation intensity of the conditions (letters) in the events (words). With F, we calculate the derivative over a pair of atomic nuclei. The local index for the atomic nuclei i, Δ i , can therefore be obtained as a linear combination of all the pair derivatives of the atomic nuclei i with all the rest of the j′s atomic nuclei. Here, we also define new strategies that generalize the present form of obtaining global or local (group or atom-type) invariants from atomic contributions (local vertex invariants, LOVIs). In respect to this, metric (norms), means and statistical invariants are introduced. These invariants are applied to a vector whose components are the values Δ i for the atomic nuclei of the molecule or its fragments. Moreover, with the purpose of differentiating among different atoms, an atomic weighting scheme (atom-type labels) is used in the formation of the matrix Q or in LOVIs state. The obtained indices were utilized to describe the partition coefficient (Log P) and the reactivity index (Log K) of the 34 derivatives of 2-furylethylenes. In all the cases, our MDs showed better statistical results than those previously obtained using some of the most used families of MDs in chemometric practice. Therefore, it has been demonstrated to that the proposed MDs are useful in molecular design and permit obtaining easier and robust mathematical models than the majority of those reported in the literature. All this range of mentioned possibilities open “the doors” to the creation of a new family of MDs, using the graph derivative, and avail a new tool for QSAR/QSPR and molecular diversity/similarity studies.


Discrete mathematics Chemical graph theory Molecular graph Sub-graph Incidence matrix Frequency matrix Derivative of molecular graph Invariant QSPR study Physicochemical property Derivatives of 2-furylethylene 



Marrero-Ponce, Y. thanks the program ‘Estades Temporals per a Investigadors Convidats’ for a fellowship to work at Valencia University in 2011. The authors acknowledge also the partial financial support from Spanish “Comisión Interministerial de Ciencia y Tecnología” (CICYT) (Project Reference: SAF2009-10399). Finally, but not least, this work was supported in part by VLIR (Vlaamse InterUniversitaire Raad, Flemish Interuniversity Council, Belgium) under the IUC Program VLIR-UCLV.


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Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  • Yovani Marrero-Ponce
    • 1
    • 2
  • Oscar Martínez Santiago
    • 1
    • 3
  • Yoan Martínez López
    • 1
    • 4
  • Stephen J. Barigye
    • 1
  • Francisco Torrens
    • 2
  1. 1.Unit of Computer-Aided Molecular “Biosilico” Discovery and Bioinformatic Research (CAMD-BIR Unit), Faculty of Chemistry-PharmacyCentral University of Las VillasSanta ClaraCuba
  2. 2.Institut Universitari de Ciència MolecularUniversitat de ValènciaValènciaSpain
  3. 3.Department of Chemical Science, Faculty of Chemistry-PharmacyCentral University of Las VillasSanta ClaraCuba
  4. 4.Department of Computer Sciences, Faculty of InformaticsCamaguey UniversityCamagüey CityCuba

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