Relations frequency hypermatrices in mutual, conditional and joint entropy-based information indices.

Graph-theoretic matrix representations constitute the most popular and significant source of topological molecular descriptors (MDs). Recently, we have introduced a novel matrix representation, named the duplex relations frequency matrix, F, derived from the generalization of an incidence matrix whose row entries are connected subgraphs of a given molecular graph G. Using this matrix, a series of information indices (IFIs) were proposed. In this report, an extension of F is presented, introducing for the first time the concept of a hypermatrix in graph-theoretic chemistry. The hypermatrix representation explores the n-tuple participation frequencies of vertices in a set of connected subgraphs of G. In this study we, however, focus on triple and quadruple participation frequencies, generating triple and quadruple relations frequency matrices, respectively. The introduction of hypermatrices allows us to redefine the recently proposed MDs, that is, the mutual, conditional, and joint entropy-based IFIs, in a generalized way. These IFIs are implemented in GT-STAF (acronym for Graph Theoretical Thermodynamic STAte Functions), a new module of the TOMOCOMD-CARDD program. Information theoretic-based variability analysis of the proposed IFIs suggests that the use of hypermatrices enhances the entropy and, hence, the variability of the previously proposed IFIs, especially the conditional and mutual entropy based IFIs. The predictive capacity of the proposed IFIs was evaluated by the analysis of the regression models, obtained for physico-chemical properties the partition coefficient (Log P) and the specific rate constant (Log K) of 34 derivatives of 2-furylethylene. The statistical parameters, for the best models obtained for these properties, were compared to those reported in the literature depicting better performance. This result suggests that the use of the hypermatrix-based approach, in the redefinition of the previously proposed IFIs, avails yet other valuable tools beneficial in QSPR studies and diversity analysis.