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; 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 $A_5$; in this paper we determine which complete bipartite graphs have an embedding in $S^3$ whose topological symmetry group is isomorphic to ${\mathbb{Z}}_m$, $D_m$, ${\mathbb{Z}}_r\times {\mathbb{Z}}_s$ or $({\mathbb{Z}}_r\times {\mathbb{Z}}_s)⋉{\mathbb{Z}}_2$.

arXiv comment26 pages, minor revisions; this is the final version accepted by Tokyo Journal of Mathematics