Chirality and Graph Isomorphism

Discrete mathematics can be applied to the chemistry problem of matching molecules together. The problem can be easily modelled usings graphs and can be solved by using graph isomorphism detection techniques.

Two graphs G = (V, E) and H = (U, F) are called isomorphic if there is a one-to-one correspondence between V and U which preserves all adjacencies.

The two graphs on the left are isomorphic. A mapping from the upper graph to the lower graph is ƒ(1) = 2, ƒ(2) = 3, ƒ(3) = 1, and ƒ(4) = 4.
The IUPAC system uses something akin to graph theory to specify the major topological features of molecules. In some cases, the two conventions are mixed. The example below illustrates this situation. The four molecules shown below are all dimethyl pyridine derivatives, but they have been given the common name lutidine. Although they all have the same chemical formula (C₇H₉N), clearly, these molecules are different, given the notion of graph isomorphism with respect to labelled graphs (in this case, actual molecules). The numbers in each of the molucules' names correspond to the positions where the methyl (CH₃) groups are attached relative to the nitrogen (N), which is assigned the number one position. As a result of their different (non-isomorphic) structures, these four molecules interact with the human sense of smell in quite different ways. The first structure, 2,6-lutidine, is judged to have the most horrid smell known to humankind. The other lutidines, while not pleasant, do not fall into this exclusive category of sensory extrema.