Graph molecular connectivity

Methods Based on Electrotopological State (E-state) Descriptors 

The E-state indices [72, 73] were developed to cover both topological and valence states of atoms. These indices were successfully used to build correlations between the structure and activity for different physicochemical and biological properties [72]. New applications of this methodology are also extensively reviewed in Ghapter 4. Several articles by different authors demonstrated the applicability of these indices for lipophilicity predictions [74—83]. 

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Correlation methods discussed include basic mathematical and numerical techniques, and approaches based on reference substances, empirical equations, nomographs, group contributions, linear solvation energy relationships, molecular connectivity indexes, and graph theory. Chemical data correlation foundations in classical, molecular, and statistical thermodynamics are introduced. 

Molecular Connectivity Indexes and Graph Theory. Perhaps the chief obstacle to developing a general theory for quantification of physical properties is not so much in the understanding of the underlying physical laws, but rather the inabiUty to solve the requisite equations. The plethora of assumptions and simplifications in the statistical mechanics and group contribution sections of this article provide examples of this. Computational procedures are simplified when the number of parameters used to describe the saUent features of a problem is reduced. Because many properties of molecules correlate well with stmctures, parameters have been developed which grossly quantify molecular stmctural characteristics. These parameters, or coimectivity indexes, are usually based on the numbers and orientations of atoms and bonds in the molecule. 

The E-state is based solely on atom connectivity information obtained from the molecular graph, without any input from the molecular geometry or sophisticated quantum calculations. We start this chapter with a brief presentation of the relevant notions of graph theory and continue with the definitions of a couple of important graph matrices. Then the molecular connectivity indices are mentioned... 

Hall LH (1990) In Rouvray DH (ed) Computational aspects of molecular connectivity and its role in structure-property modeling computational chemical graph theory. Nova Science Publishers, New York, NY, chap 8, p 202... 

To overcome this weakness, we are developing a quantitative structure-activity strategy that is conceptually applicable to all chemicals. To be applicable, at least three criteria are necessary. First, we must be able to calculate the descriptors or Independent variables directly from the chemical structure and, presumably, at a reasonable cost. Second, the ability to calculate the variables should be possible for any chemical. Finally, and most importantly, the variables must be related to a parameter of Interest so that the variables can be used to predict or classify the activity or behavior of the chemical (j ) One important area of research is the development of new variables or descriptors that quantitatively describe the structure of a chemical. The development of these indices has progressed into the mathematical areas of graph theory and topology and a large number of potentially valuable molecular descriptors have been described (7-9). Our objective is not concerned with the development of new descriptors, but alternatively to explore the potential applications of a group of descriptors known as molecular connectivity indices (10). 

Delta Value Schemes and Molecular Connectivity Indices A delta (6 or 6V) value is an atomic descriptor for nonhydrogen atoms in a molecular graph. The superscript-free delta value, <5, is defined as the number of adjacent nonhydrogen atoms of atom i ... 

Sabljic, A., and D. Horvatic, GRAPH III A Computer Program for Calculating Molecular Connectivity Indices on Microcomputers. J. Chem. Inf. Comput. Sci., 1993 33, 292-295. 

Several graph-theoretical properties of unbranched benzenoid hydrocarbons have been correlated with the Randic molecular connectivity indices of the equivalent Gutman trees [3], %(T). As illustrations we consider the following families of hydrocarbons (In all cases the correlation coefficient = 1). 

Go el and Madan [117] investigated the 8AR of the antiulcer activity of these compounds with Wiener s topological index (Wiener number of chemical graph, W(G)) [118] and the first-order molecular connectivity index ( x) [119] using a typical classification procedure. In the case of Wiener s... 

Molecular connectivity indices were first proposed by Randic in 1975 (16) as a means of estimating physical properties of alkanes. This formalism was quickly extended to other types of molecules (17) and, since then, a wide range of indices has been proposed, as reviewed by Hall and Kier (18) and Randic (19). The indices are derived from a graph theoret-... 

A vector of molecular topological descriptors can be calculated for the whole iterated line graph sequence. For example, a line graph Randic connectivity index was calculated as ... 

The concept of molecular complexity was introduced into chemistry only quite recently and is mainly based on the information content of molecules. Several different measures of complexity can be obtained according to the diversity of the considered structural elements such as atom types, bonds, connections, cycles, etc. The first attempts to quantify molecular complexity were based on the elemental composition of molecules later other molecular characteristics were considered such as the symmetry of molecular graphs, molecular branching, molecular cyclicity and centricity [Bonchev and Seitz, 1996]. 

Among the -+ graph-invariants, several substituent descriptors have been defined such as, for example, - Kier steric descriptor, - steric vertex topological index and - fragment molecular connectivity indices. 

Estrada, E., Guevara, N., Gutman, I. and Rodriguez, L. (1998b). Molecular Connectivity Indices of Iterated Line Graphs. A New Source of Descriptors for QSPR and QSAR Studies. SAR QSAR Environ.Res 9, 229-240. 

Hall, L.H. (1990). Computational Aspects of Molecular Connectivity and its Role in Structure-Property Modeling. In Computational Chemical Graph Theory (Rouvray, D.H., ed.). Nova Press, New York (NY), pp. 202-233. 

Bonchey D. (2001a) Overall connectivity - a next generation molecular connectivity. J. Mol. Graph. Model, 20, 65-75. 

Gupta, S., Singh, M. and Madan, A.K. (2001a) Applications of graph theory relationships of molecular connectivity index and atomic molecular connectivity index with anti-HSV activity. /. Mol. Struct. (Theochem), 571, 147-152. 

Kier, L.B. and Hall, L.H. (2001b) Molecular connectivity intermolecular accessibDity and encounter simulation. /. Mol. Graph. Modd., 20, 76-83. 

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