Extending graph (discrete) derivative descriptors to N-tuple atom-relations

In the present manuscript, an extension of the previously defined Graph Derivative Indices (GDIs) is discussed. To achieve this objective, the concept of a hypermatrix, conceived from the calculation of the frequencies of triple and quadruple atom relations in a set of connected sub-graphs, is introduced. This set of subgraphs is generated following a predefined criterion, known as the event (S), being in this particular case the connectivity among atoms. The triple and quadruple relations frequency matrices serve as a basis for the computation of triple and quadruple discrete derivative indices, respectively. The GDIs are implemented in a computational program denominated DIVATI (acronym for DIscrete DeriVAtive Type Indices), a module of TOMOCOMD-CARDD program. Shannon's entropy-based variability analysis demonstrates that the GDIs show major variability than others indices used in QSAR/QSPR researches. In addition, it can be appreciated when the indices are extended over n-elements from the graph, its quality increases, principally when they are used in a combined way. QSPR modeling of the physicochemical properties Log P and Log K of the 2-furylethylenes derivatives reveals that the GDIs obtained using the triple and quadruple matrix approaches yield superior performance to the duplex matrix approach. Moreover, the statistical parameters for models obtained with the GDI method are superior to those reported in the literature by using other methods. It can therefore be suggested that the GDI method, seem to be a promissory tool to reckon on in QSAR/QSPR studies, virtualscreening of compound datasets and similarity/dissimilarity evaluations.