From Graph Theory to Chemistry

We provide here elementary definitions that we later use to count, enumerate, and sample molecules. Rather than a formal mathematical presentation, examples and illustrations are given. 

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In this book, we do not strictly distinguish between terms from graph theory and those from chemistry. Thus, there are bonds as well as lines in a graph just as in a molecular graph. We decided to use the chemical term bond with its connotation of a possible multiple bond since a corresponding term seems to be missing in graph theory. 

Graphs are used in mathematics to describe a variety of problems and situations [.37. The methods of graph theoi y analyze graphs and the problems modeled by them, The transfer of models and abstractions from other sciences (computer science, chemistry, physics, economics, sociology, etc.) to graph theory makes it possible to process them mathematically because of the easily understandable basics of graph theory. 

The class of the empirical descriptors is a fuzzy, not well-defined class. In principle, empirical descriptors are those not defined on the basis of a general theory such as, for example, quantum chemistry or graph theory. Rather they are defined by practical rules derived from chemical experience, e.g. considering specific or local structural factors present in the molecules, often sets of congeneric compounds. As a consequence, in most cases, empirical descriptors represent limited subsets of compounds and cannot be extended to classes of compounds different from those for which they were defined. Empirical descriptors have not to be confused with experimentally derived descriptors even if it is well known that several of them are empirically derived. 

Although several molecular quantities were defined from the beginiming of quantum chemistry and graph theory, the term molecular descriptor has become popular with the development of structure-property correlation models. The - Platt number [Platt, 1947] and - Wiener index [Wiener, 1947c], defined in 1947, are sometimes referred to as the first molecular descriptors. 

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