Topological Symmetry Groups of Complete Bipartite Graphs

Kathleen Hake; Blake Mellor; Matthew Pittluck

Topological Symmetry Groups of Complete Bipartite Graphs

Kathleen Hake, Blake Mellor, Matthew Pittluck

Loyola Marymount University

Research output: Contribution to journal › Article

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 A₄ , S₄ or A₅ [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) ⋉ \Z₂.

Original language	Undefined/Unknown
Journal	Mathematics, Statistics and Data Science Faculty Works
State	Published - Jun 1 2016

Access to Document

http://digitalcommons.lmu.edu/math_fac/184

Cite this

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title = "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 A4 , S4 or A5 [12]; in this paper we determine which complete bipartite graphs have an embedding in S 3 whose topological symmetry group is isomorphic to \Zm, Dm, \Zr\x\Zs or ( \Zr\x \Zs) ⋉ \Z2.",

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N2 - 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 A4 , S4 or A5 [12]; in this paper we determine which complete bipartite graphs have an embedding in S 3 whose topological symmetry group is isomorphic to \Zm, Dm, \Zr\x\Zs or ( \Zr\x \Zs) ⋉ \Z2.

AB - 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 A4 , S4 or A5 [12]; in this paper we determine which complete bipartite graphs have an embedding in S 3 whose topological symmetry group is isomorphic to \Zm, Dm, \Zr\x\Zs or ( \Zr\x \Zs) ⋉ \Z2.

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