Chemistry and discrete mathematics

Abstract

Chemistry is rightly classified as one of the mathematical sciences; for it is very clear how certain branches of continuous mathematics can play an important role in the study of chemical processes. It is, perhaps, not quite so clear to what extent discrete mathematics lends itself to chemical application. Nevertheless, in the 19th century ~ not all that long after the ideas about valency and the nature of chemical compounds had gained general comprehension ~ some mathematicians had realized that there were many combinatorial problems inherent in these ideas. Arthur Cayley, in particular, was led to some pioneering work in the enumeration of trees by considering the general problem of determining the numbers of isomers of certain kinds (see [4, 51). Cayley’s results were extended by others, see, for example [2] and [3], but with the publication of Polya’s important 1937 paper [ 121 the solution of many such problems was greatly simplified. By the use of Polya’s “Hauptsatz”, the main result in this long paper, the solutions to a broad range of enumeration problems became largely a matter of routine. By such means the enumeration of a wide class of acyclic compounds was achieved. See, for example, [13]. The obvious next step in this direction was the enumeration of chemical compounds that were not acyclic, but which contained ring structures of one kind or another. This was a problem of a higher order of difficulty altogether. In its most general form this was the following problem: given the number of atoms of each kind occurring in a chemical compound, determine the number of possible isomers, i.e. the number of ways in which these atoms can be connected together in a manner consistent with their valencies. In graph-theoretical terms this is a variation on the problem of determining the number of graphs with a given degree sequence. This is a problem to which no satisfactory practical solution has yet been found. So although Polya and others have given solutions for special cases - those in which the molecule consists of some fairly simple cyclic structure to which are added acyclic side-chains (the alkyl derivatives of benzene provide a simple example) - the general isomer enumeration problem remains a largely open combinatorial problem. In the absence of a general formula for the number of isomers, it is natural to look for an algorithm that will determine (usually using a computer) the number of isomers for any given instance of the problem. Algorithms by Farrell [6], and by James and Riha [9] have done this for the graph-theoretical problem, and the program DENDRAL