Molecules graph theory

In graph theory, a graph is an ensemble of points, some of them connected with lines. We call the points vertices and the lines edges. A graph is suitable to represent a molecule. Graph theory plays an important role in searching of structures and substructures. 

A connection table has been the predominant form of chemical structure representation in computer systems since the early 1980s and it is an alternative way of representing a molecular graph. Graph theory methods can equally well be applied to connection table representations of a molecule. 

Other techniques that work well on small computers are based on the molecules topology or indices from graph theory. These fields of mathematics classify and quantify systems of interconnected points, which correspond well to atoms and bonds between them. Indices can be defined to quantify whether the system is linear or has many cyclic groups or cross links. Properties can be empirically fitted to these indices. Topological and group theory indices are also combined with group additivity techniques or used as QSPR descriptors. 

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. 

According to Graph Theory, a molecule may be represented by its skeletal molecular graph which, from a mathematical point of view, is the union of a set of points, symbolising the atoms other than hydrogens, and a set of lines, symbolising bonds. Its properties can be then expressed in terms of graph-theoretical invariants, Njj, which have been defined as "the number of distinct ways in which skeleton i... 

SMILES (Simplified Molecular Input Line Entry Systems) is a line notation system based on principles of molecular graph theory for entering and representing molecules and reactions in computer (10-13). It uses a set of simple specification rules to derive a SMILES string for a given molecular structure (or more precisely, a molecular graph). A simplified set of rules is as follows ... 

Chemical graph theory is based on the observation that the connectivity of a molecule is correlated to many of its intrinsic physical properties. Chemical graph theory has been successful in providing an estimate of the intrinsic properties of various low-molecular-weight compounds when employed together with a regression analysis. ... 

Hanson, M. and D. Rouvray. 1987. The Use of Topological Indices to Estimate the Melting Points of Organic Molecules. In Graph Theory and Topology in Chemistry, King, R., Rouvray, D., Eds. Elsevier, Amsterdam, 201. 

The ideas of chemical graph theory have a special meaning for benzenoid hydrocarbons (BHs). As the structures of BHs do not allow for the cis/trans isomers, there is a rigorous correspondence between the adjacency matrices of BHs and their properties. In other words, any property JP(G) of the benzenoid molecule G is a function of only its adjacency matrix A(G)... 

In terms of graph theory, it is convenient to represent several non-empiri-cal and semi-empirical methods of quantum chemistry. Energy and charge characteristics of molecules are treated as various structural characteristics of molecular graphs. 

Simon, J. Proc. Symp. Appl. Math. 1992,45, 97. Simon, J. A Topological Approach to the Stereochemistry ofNonrigid Molecules, in King, R. B. Rouvray, D. H. Eds., Graph Theory and Topology in Chemistry, Elsevier Amsterdam, 1987, pp. 43-75. 

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