Graph-theoretical indices

Number of pairs of carbon atoms two bonds apart Pcic = 2 

Number of pairs of methyl groups three bonds apart T3 =0 

A is a square nxn symmetric matrix, where n is the number of nonhydrogen atoms in the molecule. The adjacency matrix of G-I is as follows  

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Most of the 2D QSAR methods are based on graph theoretic indices, which have been extensively studied by Randic [29] and Kier and Hall [30,31]. Although these structural indices represent different aspects of molecular structures, their physicochemical meaning is unclear. Successful applications of these topological indices combined with multiple linear regression (MLR) analysis are summarized in Ref. 31. On the other hand, parameters derived from various experiments through chemometric methods have also been used in the study of peptide QSAR, where partial least square (PLS) [32] analysis has been employed [33]. 

S.C. Basak, V.R. Magnuson, G.J. Niems and R.R. Regal, Determining structural similarity of chemicals using graph-theoretic indices. Discr. Appl. Math., 19 (1988) 17-44. 

Three different variables are used for calculating molecular descriptors indicator variable, count variable, and graph-theoretical indices, as described below. Molecular... 

Basak, S. C., et al., Determining Structural Similarity of Chemicals Using Graph-Theoretical Indices. Discrete Appl. Math., 1988 19, 17-44. 

An interesting comparison between several aromaticity indices (harmonic oscillator model, nucleus-independent chemical shift, para-delocalization index, aromatic fluctuation index, multicenter indices, atoms-in-molecules theoretical indices and graph-theoretical indices) concluded that the most reliable ones are based on electron delocalization (08JCC1543). 

Agrawal et al. have studied the anti-histaminic activity of these 2,3-disubstituted thiazolidin-4-one analogues (Table 9) with graph theoretical indices and discovered the following equation for the activity in terms of the rooted Wiener index (Ww), rooted Szeged index (Szw), molecular redundancy index (MRI), and an indicator variable Ipi (for the p-alkyl of the aryl... 

Many different approaches to QSAR have been developed since Hansch s seminal work. These include both 2D (two-dimensional) and 3D (three-dimensional) QSAR methods. Most of the 2D QSAR methods employ graph theoretic indices to characterize molecular structures, which have been extensively studied by Radio, Kier and Hall. Similarly, ADAPT system employs topo-... 

Basak, S.C., Magnuson, V.R., Niemi, G.J. and Regal, R.R. (1988b). Determining Structural Similarity of Chemicals Using Graph-Theoretic Indices. Disc.Appl.Math., 19, YI-4A. 

Osmialowski, K. and Kaliszan, R. (1991). Studies of Performance of Graph Theoretical Indices in QSAR Analysis. Quant.Struct.-Act.Relat., 10,125-134. 

Basak, S.C., Niemi, G.J. and Veith, G.D. (1990c) Recent developments in the characterization of chemical structure using graph-theoretic indices, in Computational Chemical Graph Theory (ed. D.H. Rouvray), Nova Science Publishers, New York, pp. 235-277. 

Randic, M., Sabljic, A., Nikolic, S. and Trinajstic, N. (1988) A rational selection of graph-theoretical indices in the QSAR. Int.J. Quantum Chem. Quant. Biol Symp., 15, 267-285. 

Anisometry. Characterized by properties depending on inertia axes, symmetry axes, and distance distributions. Descriptors include the moments of inertia X,, the molecular eccentricity e, the asphericity fl, and the length of the principal axes of minimal ellipsoidal cavities, as well as distance matrices, contact maps, and some graph-theoretical indices. 

Determining Structural Similarity of Chemicals Using Graph-Theoretic Indices. 

A series of graph theoretical indices have been developed by Kier related to the molecular shape of a molecule. 

Mathematical representation of a structure allows one to extract from a structure a number of mathematical invariants, quantities that do not depend on assumed labeling of atoms in the structure. One refers to structural invariants as topological or topographic indices if they reflect upon the molecular connectivity or molecular geometry, respectively. The former are based on a graph as a model for a molecule. Hence, correctly these invariants should have been called graph theoretical indices. However, the label topological index prevailed. 

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