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

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The nonsolvability by radicals of generic 3-connected planar Laman graphs. / Power, Stephen C.; Owen, J. C.

In: Transactions of the American Mathematical Society, Vol. 359, No. 5, 2007, p. 2269-2303.

Research output: Contribution to journalJournal articlepeer-review

Harvard

Power, SC & Owen, JC 2007, 'The nonsolvability by radicals of generic 3-connected planar Laman graphs.', Transactions of the American Mathematical Society, vol. 359, no. 5, pp. 2269-2303. https://doi.org/10.1090/S0002-9947-06-04049-9

APA

Power, S. C., & Owen, J. C. (2007). The nonsolvability by radicals of generic 3-connected planar Laman graphs. Transactions of the American Mathematical Society, 359(5), 2269-2303. https://doi.org/10.1090/S0002-9947-06-04049-9

Vancouver

Power SC, Owen JC. The nonsolvability by radicals of generic 3-connected planar Laman graphs. Transactions of the American Mathematical Society. 2007;359(5):2269-2303. https://doi.org/10.1090/S0002-9947-06-04049-9

Author

Power, Stephen C. ; Owen, J. C. / The nonsolvability by radicals of generic 3-connected planar Laman graphs. In: Transactions of the American Mathematical Society. 2007 ; Vol. 359, No. 5. pp. 2269-2303.

Bibtex

@article{d074253f0f0f4f7a9c727bb9f23913ba,
title = "The nonsolvability by radicals of generic 3-connected planar Laman graphs.",
abstract = "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.",
author = "Power, {Stephen C.} and Owen, {J. C.}",
note = "Copyright 2006, American Mathematical Society RAE_import_type : Journal article RAE_uoa_type : Pure Mathematics",
year = "2007",
doi = "10.1090/S0002-9947-06-04049-9",
language = "English",
volume = "359",
pages = "2269--2303",
journal = "Transactions of the American Mathematical Society",
issn = "0002-9947",
publisher = "American Mathematical Society",
number = "5",

}

RIS

TY - JOUR

T1 - The nonsolvability by radicals of generic 3-connected planar Laman graphs.

AU - Power, Stephen C.

AU - Owen, J. C.

N1 - Copyright 2006, American Mathematical Society RAE_import_type : Journal article RAE_uoa_type : Pure Mathematics

PY - 2007

Y1 - 2007

N2 - 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.

AB - 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.

U2 - 10.1090/S0002-9947-06-04049-9

DO - 10.1090/S0002-9947-06-04049-9

M3 - Journal article

VL - 359

SP - 2269

EP - 2303

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 5

ER -