buckyball - the graph of truncated icosahedron is also the graph of truncated aligned cube shown on the right. This representation of the new form of carbon is a result of my work on a conjecture of computer program Graffiti that in stable n-atom fullerenes, the independence number is greater or equal to n/2 - 6. The uniqueness of the construction among IP-fullerenes (dodecahedron is another example) is based on the structure of very special 24-element "pentagonal" independent sets shown below.

Picture by: Craig Larson

Drawing by: Tomaz Pisanski and Matjaz Zawersnik

Pentagonal independent set determine the structure of bromofullerene C_60Br_24 and very likely also of the fullerol C_60(OH)_24, which was reported to have taxol-like effects. Taxol is a rare drug for lungs, breast and ovarian cancers. This is intriguing in our context. Many viruses have icosahedral symmetry studied by Coxeter along with Fuller's constructions. The pentagonal independent sets of the buckyball may be related to Golay code; the soccer ball with pentagonal independent set seems to be naturally suited for a geometric and hopefully a physical model of the Miracle Octad Generator - a method to compute with Golay code. The latter is the best device to transmit information to minimize the chance of unrecoverable errors. The new representation of the buckyball is a special case of a general construction: alignments are special involutions of line-graphs and the possible relation to Golay code may be an indication of structural representations of some sporadic simple groups as automorphisms of certain geometric and related to platonic, cubic graphs, probably including generalized Petersen graphs. Some of these ideas are discussed in greater details in my "C60" paper published in DIMACS volume "Graphs and Discovery"

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