Independent sets and the prediction of addition patterns for higher fullerenes

In the search for simple rationalisations of fullerene reactivity, several concepts taken from graph theory have proved useful. One is the independence number, which gives a mathematical embodiment to the chemical notion of steric strain. The independence number, I(G), of a graph G is the maximum number of its vertices that can be marked simultaneously such that no two marked vertices are adjacent. For icosahedral C60 the independence number is 24, and this fact has been used 1 to rationalise the structure of the end product of the bromination of [60]fullerene, i.e. C60Br24. 2 The argument goes as follows. Assume that the cage takes up the maximum number of exo addends compatible with two rules, one steric and one electronic. Rule (i): no two brominated carbon atoms should be adjacent. Rule (ii): the final addition pattern should contain only closed-shell π subsystems. In more formal mathematical language, rules (i) and (ii) are obeyed by those maximum independent sets (if any such exist) in which all components of the graph of unmarked vertices are even in size and have adjacency spectra split into equal halves across a non-zero gap. In C60 the independence number I(C60) = 24 predicts the stoichiometry, and of the ~3 × 10 conceivable isomers of C60Br24 only 1085 are maximum independent sets. Of these, all but one are radicals. In fact, for C60, a weaker version of rule (ii) would achieve the same reduction: we can simply forbid isolated bare carbon atoms (unmarked vertices). The sole survivor under the simultaneous action of rule (i) and of either version of rule (ii) is the experimental Th isomer, in which the 36 bare carbon centres are arranged as 18 isolated double bonds. The present work is concerned with the extension of these very simple rules to the prediction of the so-far uncharacterised products of bromination of higher fullerenes. A new invariant related to the independence number is required.