Some Reflections on the Topological Structure of Covalent Molecules
D. H. Rouvray University of the Witwatersrand Johannesburg. South Africa
I t is well-known that the number of structural formulas which may be assigned to a given molecule is limited by the number of chemically meaningful ways in which its structural units may be fitted together. Several factors determine whether a structural formula is meaningful, but one of the most fundamental is that every atom in the structure satisfy the usual rules of valence. We assume throughout that every atom in a molecule has a fixed, integral value of valence. Our method then involves a description of the possible topological structures a chemical species may adopt, subiect to the constraints imposed bv the valence. We consider here only covalently -bonded species, though our methods are readilv aeneralizable to other kinds of species. Once the topol& of a chemical species is kno-m, we have an overall oicture of the molecular structure. This picture, however, fails in several important respects, for it provides no information about factors such as the types of bonding present in the molecule, the bond angles, the bond lengths, or the orientation in space of the component parts of the molecule. Thus, a topological representation yields very little information of a purely chemical nature. In fact, the only useful chemical information i t gives is whether a given pair of atoms in the structure is bonded together or not. This information, however, is not without some important applications. Depiction of the topology of a molecule is easily achieved by use of a mathematical device known as a graph. Reference is made here t o the to~oloeical . not to the . - e r a ~ h and Cartesian graph frequently encountered in the display of scientific data. A toooloeical e r a ~ hconsists of a set of points (known as ueitices), an2 set of lines (known as edges), which connect pairs of the vertices together. In symbols, any graph may be represented by the pair
Figure 1. Topological graphs representing the molecules of (a1 ethylene, (b) benzene, and (c) cubane. All three molecules may be represented by twodimensbnal graphs
a
c = (X,i;,
(1)
where X is a set of points, and r is an operator which generates the lines in the graph by mapping points of set X into other points of the set. A graph thus provides us with all the neighborhood relations for a given vertex, i.e. the number of edges terminating on that vertex. Several illustrations of graphs are to be seen in Figures 1-3. The study of graphs forms a mathematical discipline known as graph theory (1-4), deriving originally from the work of Euler (5) on networks in the mid 1730's. The idea that graphs could be used to depict chemical species is ap~ a r e n t l vdue to the mathematician Svlvester 16). thoueh the first publication of the idea was byanother mathematician Cayley (7). Cayley first used graphs in his enumeration of the alkane structural isomers (8). Since then graphs have been used in numerous guises, such as in the study of phase equilibria (9).reaction kinetics (lo), bonding theory (11, IZ), and in the investigation of additive phenomena (13). Reviews of many of the more recent chemical applications of graphs have been presented by the author (14,15). Graphs are especially convenient for the representation of organic molecules, or, in general, for any species which is covalently bonded. When used for this purpose, graphs are referred to as chemical graphs. In such graphs the vertices represent the time-averaged position of each of the atoms, and the edges represent the covalent bonds existing between pairs of atoms. Graphs depicting the molecules of 788 /
Journal of Chemical Education
Figure 2. Diagrams of (a) a loop graph. (b) a cyclic graph. and (c) a bee
ethylene, benzene, and cubane are shown in Figure 1. All three graphs are two-dimensional; the equivalent graph for cubane shows that this is so even for this threedimensional structure. Such a two-dimensional representation for cubane is possible only because in graphs no account is taken of bond lengths or bond angles. We consider now some of the mathematical consequences of representing chemical species by graphs.
Figure 3. Topological graph far the coronene molecule with the hydragen atoms omitted. This graph canteins seven independent cycles.
Graphical Terminology
We start by presenting some further graph-theoretical terminology pertinent to the ensuing discussion of chemical topology. A fuller account of this aspect of graph theory is t o be found in the review of Essam and Fisher (16). If a vertex of? topological graph is mapped onto itself by the operator r, the graph is said t o possess a loop a t that vertex. A graph possessed of a sequence of edges that form a
continuous circuit is termed a cyclic graph or cycle. Any graph containing neither loops nor cycles is known as a tree. Examples of these three types of graph are illustrated in Figure 2. A graph having more than one edge connecting a given pair of vertices is referred to as a multigraph. An example of a multigraph is the graph for the ethylene molecule shown in Figure 1. The numher of edges terminating on a vertex is known as the degree of that vertex. Thus, all vertices for the cyclic graph in Figure 2 are of degree two, whereas those of the tree graph in the same figure are of degrees one, two, and four. The number of cycles contained within a graph is of considerable mathematical interest. Use of the term cycle here means independent cycle, i.e. a cycle which contains a t least one edge not contained in any other cycle in the graph. Inspection of the graph of the coronene molecule, illustrated with the hydrogen atoms omitted in Figure 3, reveals that this graph contains seven independent cycles. The numher of independent cycles in a graph may also be defined as the number of edges which must be removed to form a tree graph. For any graph the number of independent cycles may he calculated directly from the numher of edges and vertices it contains. This is achieved hy use of a parameter characteristic of the graph in question, known as the cyclomatic number. The cyclomatic number p is defined as follows
Table 1. A Listing of the Cyelomaic Number and Special Name for Cenain Graphs Having Specific Values of n,. the Number of Edger Cyclomstic Number
Value of
me
