Topological Symmetry Groups of Complete Bipartite Graphs

The symmetries of complex molecular structures can be modeled by the topological symmetry group of the underlying embedded graph. It is therefore important to understand which topological symmetry groups can be realized by particular abstract graphs. This question has been answered for complete graphs [7]; it is natural next to consider complete bipartite graphs. In previous work we classified the complete bipartite graphs that can realize topological symmetry groups isomorphic to A(4), S-4 or A(5) [12]; in this paper we determine which complete bipartite graphs have an embedding in S-3 whose topological symmetry group is isomorphic to Z(m), D-m, Z(r) x Z(s) or (Z(r) x (Z)s) proportional to Z(2).