Complete Graphs whose Topological Symmetry Groups are Polyhedral

Characterizing the symmetries of a molecule is an important step in predicting its chemical behavior. Chemists have long used the group of rigid symmetries, known as the point group, as a means of representing the symmetries of a molecule. However, molecules which are flexible or partially flexible may have symmetries which are not included in the point group. Jon Simon [11] introduced the concept of the topological symmetry group in order to study symmetries of such non-rigid molecules. The topological symmetry group provides a way to classify, not only the symmetries of molecular graphs, but the symmetries of any graph embedded in S3 .