Chemical graph theory, mathematical objects

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). 

Common conventions of graph theory are followed here (see, e.g.. Refs. 8-10), but a rigid distinction is not always maintained between chemical and mathematical contexts. So the term graph is used interchangeably in both its strict sense as a mathematical object and as a shorthand term for its realization as an actual (usually carbon) molecule. Some equivalent pairs of terms are treated as being synonymous atom = vertex, bond = edge, valency = degree, and so on. 

Here, atoms are represented by small cirdes, and chemical bonds by lines which link the corresponding aides. In gr h theory the objects designated by small cirdes are called neritces, points or nodes the use of the latter term is not to be recommended, however, because in quantum chemistry, surfaces with wave functions having zero values are called nodes. The linking lines are denoted as e es, Itnes, or when directed, as ores. Thus, the structures defined in mathematics as graphs consist... 

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