Incidence and distance matrices

Abstract

It is now recognised that many aspects of chemistry are, in some considerable measure, topologically determined. For example, graph theory has been applied to problems in Valence Theory an aspect which has been explored by several authors.1,2,3*4 Perhaps the earliest application of formal graph theory to a chemistry problem was Cayley’s work in determining the isomer counts of the paraffinic hydrocarbons.’ A much more recent application also concerns isomers and is the subject of the present communication. Fluxional (stereochemically non-rigid) molecules, undergo rapid molecular rearrangements. All the isomers have the same equilibrium geometry and differ only in atomic permutations, but on a nuclear magnetic resonance time scale the properties of individual atoms may be a time average of those associated with mote than one position. Although is not always possible to associate a unique permutation with a unique permutational mechanism each species has a definite number of permutational isomer?j’ each permutational mechanism leading to a characteristic pattern of interconversions between these isomers. The various mechanisms are conveniently distinguished by the different shortest path sequences by which any given isomer is converted into another. The shortest paths are detailed in a so-called distance matrix of the system, examples of which have been given by several authors. However there appears to be in the chemical literature no details of the derivation of such matrices. It is the purpose of the present communication to remedy this ommision and to propose some innovations that will increase the information content of them.