On Representation and Characterization of Fullerene C60

An operation on trivalent graphs leads from the truncated cube to buckminsterfullerene, and C60 is the only fullerene with disjoint pentagons which can be obtained by this method. The construction and the proof emphasize maximal independent sets that contains two fifths of vertices of trivalent graphs. In the case of C60 these sets define the structure of experimentally obtained bromofullerene C60Br24. The construction and characterization is a result of work on the conjectures of the computer program Graffiti about stable fullerenes. 1. Let G = (V,E) be a graph with the vertex set V and the edge set E. A graph aligning or simply aligning is a system (G, l) where G is a graph and l is an adjacency relation of a graph whose vertex set is the set of edges of G, i.e., (E, l) is itself a graph. Two edges of G adjacent in (E, l) will be called aligned. Given an aligning (G, l) we define Gl to be the graph with the vertex set V ∪ E and the edge set defined as follows: v is adjacent to u if and only if v is a vertex of G incident with the edge u of G, or v and u are aligned edges of G Throughout this paper we shall assume that our graphs are simple, trivalent, and that every edge x of G is aligned with a unique edge y different from x. Such alignings will be called perfect. This special case is often easy to visualize. The graph Gl is obtained from G first by the subdivision of its edges, i.e., by placing a new vertex on each of them, and then linking pairs of new vertices residing on aligned edges of G. By a fullerene we shall mean here a trivalent planar graph in which every face has five or six sides, though this term is also used for more general structures. In mid-sixties David Jones suggested that shell-shaped, spherical, hollow trivalent graphs can be realized as carbon molecules and he conjectured that the stability of these molecules would require at least several hundred atoms. The work of Curl, Kroto and Smalley led eventually in 1985 to the synthesis of the first stable form with 60 atoms buckminsterfullerene. Chemists think that pentagonal faces of stable fullerenes must be disjoint. Using the Euler polyhedral formula one can show that the number of pentagonal faces of fullerenes is exactly 12, so C60 is the smallest candidate for a stable molecule. Many properties of C60, some of which 1991 Mathematics Subject Classification. Primary 05C99; Secondary F8299,F94B99.