{"title":"Super stable tensegrities and the Colin de Verdière number \n \n \n \n ν\n \n \n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23188:jgt23188-math-0001\" wiley:location=\"equation/jgt23188-math-0001.png\"><mrow><mrow><mi>\\unicode{x003BD}</mi></mrow></mrow></math>","authors":"Ryoshun Oba, Shin-ichi Tanigawa","doi":"10.1002/jgt.23188","DOIUrl":null,"url":null,"abstract":"<p>A super stable tensegrity introduced by Connelly in 1982 is a globally rigid discrete structure made from stiff bars and struts connected by cables with tension. We introduce the super stability number of a multigraph as the maximum dimension that a multigraph can be realized as a super stable tensegrity, and show that it equals the Colin de Verdière number <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>ν</mi>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23188:jgt23188-math-0002\" wiley:location=\"equation/jgt23188-math-0002.png\"><mrow><mrow><mi>\\unicode{x003BD}</mi></mrow></mrow></math></annotation>\n </semantics></math> minus one. As a corollary we obtain a combinatorial characterization of multigraphs that can be realized as three-dimensional super stable tensegrities. We also show that, for any fixed <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>d</mi>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23188:jgt23188-math-0003\" wiley:location=\"equation/jgt23188-math-0003.png\"><mrow><mrow><mi>d</mi></mrow></mrow></math></annotation>\n </semantics></math>, there is an infinite family of 3-regular graphs that can be realized as <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>d</mi>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23188:jgt23188-math-0004\" wiley:location=\"equation/jgt23188-math-0004.png\"><mrow><mrow><mi>d</mi></mrow></mrow></math></annotation>\n </semantics></math>-dimensional injective super stable tensegrities.</p>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"108 3","pages":"401-431"},"PeriodicalIF":0.9000,"publicationDate":"2024-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23188","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Graph Theory","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/jgt.23188","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
A super stable tensegrity introduced by Connelly in 1982 is a globally rigid discrete structure made from stiff bars and struts connected by cables with tension. We introduce the super stability number of a multigraph as the maximum dimension that a multigraph can be realized as a super stable tensegrity, and show that it equals the Colin de Verdière number minus one. As a corollary we obtain a combinatorial characterization of multigraphs that can be realized as three-dimensional super stable tensegrities. We also show that, for any fixed , there is an infinite family of 3-regular graphs that can be realized as -dimensional injective super stable tensegrities.
期刊介绍：
The Journal of Graph Theory is devoted to a variety of topics in graph theory, such as structural results about graphs, graph algorithms with theoretical emphasis, and discrete optimization on graphs. The scope of the journal also includes related areas in combinatorics and the interaction of graph theory with other mathematical sciences.
A subscription to the Journal of Graph Theory includes a subscription to the Journal of Combinatorial Designs .
京公网安备 11010802042870号
京ICP备2023020795号-1
分享
求助内容：
应助结果提醒方式：
扫码关注我们
