The nonsolvability by radicals of generic 3-connected planar Laman graphs.

Power, Stephen C. and Owen, J. C. (2007) The nonsolvability by radicals of generic 3-connected planar Laman graphs. Transactions of the American Mathematical Society, 359 (5). pp. 2269-2303. ISSN 1088-6850

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We show that planar embeddable -connected Laman graphs are generically non-soluble. A Laman graph represents a configuration of points on the Euclidean plane with just enough distance specifications between them to ensure rigidity. Formally, a Laman graph is a maximally independent graph, that is, one that satisfies the vertex-edge count together with a corresponding inequality for each subgraph. The following main theorem of the paper resolves a conjecture of Owen (1991) in the planar case. Let be a maximally independent -connected planar graph, with more than 3 vertices, together with a realisable assignment of generic distances for the edges which includes a normalised unit length (base) edge. Then, for any solution configuration for these distances on a plane, with the base edge vertices placed at rational points, not all coordinates of the vertices lie in a radical extension of the distance field.

Item Type:
Journal Article
Journal or Publication Title:
Transactions of the American Mathematical Society
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Copyright 2006, American Mathematical Society RAE_import_type : Journal article RAE_uoa_type : Pure Mathematics
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01 Apr 2008 12:49
Last Modified:
20 Mar 2024 00:26