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# Total and Local Quadratic Indices of the Molecular Pseudograph's Atom Adjacency Matrix: Applications to the Prediction of Physical Properties of Organic Compounds

Received: 17 June 2003 / Revised: 29 July 2003 / Accepted: 3 August 2003 / Published: 15 August 2003

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# Abstract

A novel topological approach for obtaining a family of new molecular descriptors is proposed. In this connection, a vector space E (*molecular vector space*), whose elements are organic molecules, is defined as a “direct sum“ of different ℜi spaces. In this way we can represent molecules having a total of i atoms as elements (

*vectors*) of the vector spaces ℜi (i=1, 2, 3,..., n; where n is number of atoms in the molecule). In these spaces the components of the vectors are atomic properties that characterize each kind of atom in particular. The total quadratic indices are based on the calculation of mathematical quadratic forms. These forms are functions of the k-th power of the molecular pseudograph's atom adjacency matrix (M). For simplicity, canonical bases are selected as the quadratic forms' bases. These indices were generalized to “higher analogues“ as number sequences. In addition, this paper also introduces a local approach (

*local invariant*) for molecular quadratic indices. This approach is based mainly on the use of a local matrix [M

^{k}(G, F

_{R})]. This local matrix is obtained from the k-th power (M

^{k}(G)) of the atom adjacency matrix M. M

^{k}(G, F

_{R}) includes the elements of the fragment of interest and those that are connected with it, through paths of length k. Finally, total (and local) quadratic indices have been used in QSPR studies of four series of organic compounds. The quantitative models found are significant from a statistical point of view and permit a clear interpretation of the studied properties in terms of the structural features of molecules. External prediction series and cross-validation procedures (leave-

*one*-out and leave-

*group*-out) assessed model predictability. The reported method has shown similar results, compared with other topological approaches. The results obtained were the following: a) Seven physical properties of 74 normal and branched alkanes (boiling points, molar volumes, molar refractions, heats of vaporization, critical temperatures, critical pressures and surface tensions) were well modeled (R>0.98, q2>0.95) by the total quadratic indices. The overall MAE of 5-fold cross-validation were of 2.11 oC, 0.53 cm

^{3}, 0.032 cm

^{3}, 0.32 KJ/mol, 5.34 oC, 0.64 atm, 0.23 dyn/cm for each property, respectively; b) boiling points of 58 alkyl alcohols also were well described by the present approach; in this sense, two QSPR models were obtained; the first one was developed using the complete set of 58 alcohols [R=0.9938, q

^{2}=0.986, s=4.006

^{o}C, overall MAE of 5-fold cross-validation=3.824

^{o}C] and the second one was developed using 29 compounds as a training set [R=0.9979, q

^{2}=0.992, s=2.97

^{o}C, overall MAE of 5-fold cross-validation=2.580

^{o}C] and 29 compounds as a test set [R=0.9938, s=3.17

^{o}C]; c) good relationships were obtained for the boiling points property (using 80 and 26 cycloalkanes in the training and test sets, respectively) using 2 and 5 total quadratic indices: [Training set: R=0.9823 (q

^{2}=0.961 and overall MAE of 5-fold crossvalidation= 6.429

^{o}C) and R=0.9927 (q

^{2}=0.977 and overall MAE of 5-fold crossvalidation= 4.801

^{o}C); Test set: R=0.9726 and R=0.9927] and d) the linear model developed to describe the boiling points of 70 organic compounds containing aromatic rings has shown good statistical features, with a squared correlation coefficient (R

^{2}) of 0.981 (s=7.61

^{o}C). Internal validation procedures (q

^{2}=0.9763 and overall MAE of 5-fold cross-validation=7.34

^{o}C) allowed the predictability and robustness of the model found to be assessed. The predictive performance of the obtained QSPR model also was tested on an extra set of 20 aromatic organic compounds (R=0.9930 and s=7.8280

^{o}C). The results obtained are valid to establish that these new indices fulfill some of the ideal requirements proposed by Randić for a new molecular descriptor.

*Keywords:*Molecular Vector Space; Total and Local Quadratic Index; QSPR; Physical Property; Organic Compound

*This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.*

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**MDPI and ACS Style**

Ponce, Y.M. Total and Local Quadratic Indices of the Molecular Pseudograph's Atom Adjacency Matrix: Applications to the Prediction of Physical Properties of Organic Compounds. *Molecules* **2003**, *8*, 687-726.