{"title":"超稳定张拉整体和柯林常数ν \\unicode{x003BD}","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":"{\"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}","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
摘要
超稳定张拉整体是Connelly于1982年提出的一种整体刚性离散结构，由刚性杆和支柱通过张力电缆连接而成。引入多图的超稳定数作为多图可以实现为超稳定张拉整体的最大维数；并显示它等于Colin de verdi数字ν <；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>< \unicode{x003BD}</ mrow></mrow></ mrow></mrow></ mrow></math>；- 1。作为一个推论，我们得到了多重图的组合表征，可以实现为三维超稳定张拉整体。我们还表明，对于任何固定的d<； 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>< d</ mrow></mrow></ mrow></mrow></math>；，存在一个无限的3正则图家族，可以实现为d<； 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>< d</ mrow></mrow></ mrow></mrow></ mrow></math>；维注入超稳定张拉整体。
Super stable tensegrities and the Colin de Verdière number
ν
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 .
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