Topological Symmetry Groups of Complete Bipartite Graphs

Kathleen Hake; Blake Mellor; Matthew Pittluck

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arXiv · 1412.7255

Topological Symmetry Groups of Complete Bipartite Graphs

Kathleen Hake·Blake Mellor·Matthew Pittluck

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

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BibTeX RIS

Kathleen Hake, Blake Mellor, Matthew Pittluck. 2014-12-23. Topological Symmetry Groups of Complete Bipartite Graphs. https://arxiv.org/abs/1412.7255

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