Some cycle indices Top Introduction

Introduction

Fullerenes are a new form of carbon atoms discovered a few years ago. In 1985 the first C60 clusters were produced and it was assumed [16] that its 60 carbon atoms are placed at the vertices of a truncated icosahedron (which is commonly called a foot ball or soccer ball.) The truncated icosahedron is one of the 14 Archimedean solids already known to the ancient Greeks. Five years later [15][19] the original hypotheses on the structure of C60 could be confirmed. In the meantime fullerenes play an important role in chemistry and many other forms of fullerenes were detected.

In my talk today I will present some examples how combinatorics under finite group action can be applied in order to determine the number of essentially different colourings of the soccer ball. From the chemical point of view this means isomer enumeration. Furthermore I will introduce a multi-dimensional cycle index which represents the simultaneous action of the symmetry group of a fullerene on its sets of vertices, edges and faces. Finally I will describe the so called leapfrog principle, which was invented by Fowler [6][9]. It can be used for the construction of a fullerene C3v from a fullerene Cv having the same or even a bigger symmetry group than Cv.


harald.fripertinger@kfunigraz.ac.at

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