Topological Symmetry Groups of Complete Bipartite Graphs

Abstract

The symmetries of complex molecular structures can be modeled by the {\em 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 $(\Z_r \x \Z_s) \ltimes \Z_2$.