费德勒向量在树木上的极值

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications Pub Date : 2024-09-27 DOI:10.1016/j.laa.2024.09.014

Roy R. Lederman , Stefan Steinerberger

{"title":"费德勒向量在树木上的极值","authors":"Roy R. Lederman , Stefan Steinerberger","doi":"10.1016/j.laa.2024.09.014","DOIUrl":null,"url":null,"abstract":"<div><div>Let <em>G</em> be a tree on <em>n</em> vertices and let <span><math><mi>L</mi><mo>=</mo><mi>D</mi><mo>−</mo><mi>A</mi></math></span> denote the Laplacian matrix on <em>G</em>. The second-smallest eigenvalue <span><math><msub><mrow><mi>λ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo><mo>></mo><mn>0</mn></math></span>, also known as the algebraic connectivity, as well as the associated eigenvector have been of substantial interest. We investigate the question of when the maxima and minima of an associated eigenvector are assumed at the endpoints of the longest path in <em>G</em>. Our results also apply to more general graphs that ‘behave globally’ like a tree but can exhibit more complicated local structure. The crucial new ingredient is a reproducing formula for eigenvectors of graphs.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":null,"pages":null},"PeriodicalIF":1.0000,"publicationDate":"2024-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Extreme values of the Fiedler vector on trees\",\"authors\":\"Roy R. Lederman , Stefan Steinerberger\",\"doi\":\"10.1016/j.laa.2024.09.014\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><div>Let <em>G</em> be a tree on <em>n</em> vertices and let <span><math><mi>L</mi><mo>=</mo><mi>D</mi><mo>−</mo><mi>A</mi></math></span> denote the Laplacian matrix on <em>G</em>. The second-smallest eigenvalue <span><math><msub><mrow><mi>λ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo><mo>></mo><mn>0</mn></math></span>, also known as the algebraic connectivity, as well as the associated eigenvector have been of substantial interest. We investigate the question of when the maxima and minima of an associated eigenvector are assumed at the endpoints of the longest path in <em>G</em>. Our results also apply to more general graphs that ‘behave globally’ like a tree but can exhibit more complicated local structure. The crucial new ingredient is a reproducing formula for eigenvectors of graphs.</div></div>\",\"PeriodicalId\":18043,\"journal\":{\"name\":\"Linear Algebra and its Applications\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.0000,\"publicationDate\":\"2024-09-27\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Linear Algebra and its Applications\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0024379524003768\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Linear Algebra and its Applications","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0024379524003768","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}

引用次数: 0

摘要

让 G 是 n 个顶点上的树，让 L=D-A 表示 G 上的拉普拉斯矩阵。第二最小特征值 λ2(G)>0（也称为代数连通性）以及相关特征向量一直备受关注。我们研究了相关特征向量的最大值和最小值何时假定位于 G 中最长路径端点的问题。我们的结果也适用于更一般的图，这些图的 "全局行为 "类似于树，但可能表现出更复杂的局部结构。关键的新要素是图形特征向量的重现公式。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

查看原文本刊更多论文

Extreme values of the Fiedler vector on trees

Let G be a tree on n vertices and let

L = D - A

denote the Laplacian matrix on G. The second-smallest eigenvalue

λ_{2} (G) > 0

, also known as the algebraic connectivity, as well as the associated eigenvector have been of substantial interest. We investigate the question of when the maxima and minima of an associated eigenvector are assumed at the endpoints of the longest path in G. Our results also apply to more general graphs that ‘behave globally’ like a tree but can exhibit more complicated local structure. The crucial new ingredient is a reproducing formula for eigenvectors of graphs.

求助全文

通过发布文献求助，成功后即可免费获取论文全文。去求助

来源期刊

Linear Algebra and its Applications 数学-应用数学

CiteScore

2.20

自引率

9.10%

发文量

333

审稿时长

13.8 months

期刊介绍： Linear Algebra and its Applications publishes articles that contribute new information or new insights to matrix theory and finite dimensional linear algebra in their algebraic, arithmetic, combinatorial, geometric, or numerical aspects. It also publishes articles that give significant applications of matrix theory or linear algebra to other branches of mathematics and to other sciences. Articles that provide new information or perspectives on the historical development of matrix theory and linear algebra are also welcome. Expository articles which can serve as an introduction to a subject for workers in related areas and which bring one to the frontiers of research are encouraged. Reviews of books are published occasionally as are conference reports that provide an historical record of major meetings on matrix theory and linear algebra.