Topological Symmetry Groups of Complete Bipartite Graphs

TL;DR

This paper investigates which complete bipartite graphs can be embedded in three-dimensional space with specific topological symmetry groups, extending previous classifications to include cyclic, dihedral, and certain product groups.

Contribution

It extends the classification of topological symmetry groups of complete bipartite graphs to include cyclic, dihedral, and certain product groups, building on prior work for symmetric groups.

Findings

Classified bipartite graphs with cyclic symmetry groups

Identified bipartite graphs with dihedral symmetry groups

Extended understanding of symmetry realizations in 3D embeddings

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 $(\mathbb{Z}_r \times \mathbb{Z}_s) \ltimes \mathbb{Z}_2$.