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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 $(mathbb{Z}_r times mathbb{Z}_s) ltimes mathbb{Z}_2$.
A weight system is a function on chord diagrams that satisfies the so-called four-term relations. Vassilievs theory of finite-order knot invariants describes these invariants in terms of weight systems. In particular, there is a weight system corresp
This note gives a detailed proof of the following statement. Let $din mathbb{N}$ and $m,n ge d + 1$, with $m + n ge binom{d+2}{2} + 1$. Then the complete bipartite graph $K_{m,n}$ is generically globally rigid in dimension $d$.
We classify all groups which can occur as the topological symmetry group of some embedding of the Heawood graph in $S^3$.
We describe a very simple condition that is necessary for the universal rigidity of a complete bipartite framework $(K(n,m),p,q)$. This condition is also sufficient for universal rigidity under a variety of weak assumptions, such as general position.
Let $mathrm{rex}(n, F)$ denote the maximum number of edges in an $n$-vertex graph that is regular and does not contain $F$ as a subgraph. We give lower bounds on $mathrm{rex}(n, F)$, that are best possible up to a constant factor, when $F$ is one of