{"title":"线图的特征值多重性","authors":"Sarula Chang , Jianxi Li , Yirong Zheng","doi":"10.1016/j.laa.2024.08.021","DOIUrl":null,"url":null,"abstract":"<div><p>Let <span><math><mi>m</mi><mo>(</mo><mi>G</mi><mo>,</mo><mi>λ</mi><mo>)</mo></math></span>, <span><math><mi>c</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> and <span><math><mi>p</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> be the multiplicity of an eigenvalue <em>λ</em>, the cyclomatic number and the number of pendant vertices of a connected graph <em>G</em>, respectively. Yang et al. (2023) <span><span>[10]</span></span> proved that <span><math><mi>m</mi><mo>(</mo><mi>L</mi><mo>(</mo><mi>T</mi><mo>)</mo><mo>,</mo><mi>λ</mi><mo>)</mo><mo>≤</mo><mi>p</mi><mo>(</mo><mi>T</mi><mo>)</mo><mo>−</mo><mn>1</mn></math></span> for any tree <em>T</em>, and characterized all trees <em>T</em> with <span><math><mi>m</mi><mo>(</mo><mi>L</mi><mo>(</mo><mi>T</mi><mo>)</mo><mo>,</mo><mi>λ</mi><mo>)</mo><mo>=</mo><mi>p</mi><mo>(</mo><mi>T</mi><mo>)</mo><mo>−</mo><mn>1</mn></math></span>, where <span><math><mi>L</mi><mo>(</mo><mi>T</mi><mo>)</mo></math></span> is the line graph of <em>T</em>. In this paper, we extend their result from a tree <em>T</em> to any graph <span><math><mi>G</mi><mo>≠</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, and prove that <span><math><mi>m</mi><mo>(</mo><mi>L</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>,</mo><mi>λ</mi><mo>)</mo><mo>≤</mo><mn>2</mn><mi>c</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>+</mo><mi>p</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>−</mo><mn>1</mn></math></span> for any graph <span><math><mi>G</mi><mo>≠</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>. Moreover, all graphs <em>G</em> with <span><math><mi>m</mi><mo>(</mo><mi>L</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>,</mo><mo>−</mo><mn>1</mn><mo>)</mo><mo>=</mo><mn>2</mn><mi>c</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>+</mo><mi>p</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>−</mo><mn>1</mn></math></span> are completely characterized.</p></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"703 ","pages":"Pages 47-62"},"PeriodicalIF":1.0000,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"The eigenvalue multiplicity of line graphs\",\"authors\":\"Sarula Chang , Jianxi Li , Yirong Zheng\",\"doi\":\"10.1016/j.laa.2024.08.021\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>Let <span><math><mi>m</mi><mo>(</mo><mi>G</mi><mo>,</mo><mi>λ</mi><mo>)</mo></math></span>, <span><math><mi>c</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> and <span><math><mi>p</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> be the multiplicity of an eigenvalue <em>λ</em>, the cyclomatic number and the number of pendant vertices of a connected graph <em>G</em>, respectively. Yang et al. (2023) <span><span>[10]</span></span> proved that <span><math><mi>m</mi><mo>(</mo><mi>L</mi><mo>(</mo><mi>T</mi><mo>)</mo><mo>,</mo><mi>λ</mi><mo>)</mo><mo>≤</mo><mi>p</mi><mo>(</mo><mi>T</mi><mo>)</mo><mo>−</mo><mn>1</mn></math></span> for any tree <em>T</em>, and characterized all trees <em>T</em> with <span><math><mi>m</mi><mo>(</mo><mi>L</mi><mo>(</mo><mi>T</mi><mo>)</mo><mo>,</mo><mi>λ</mi><mo>)</mo><mo>=</mo><mi>p</mi><mo>(</mo><mi>T</mi><mo>)</mo><mo>−</mo><mn>1</mn></math></span>, where <span><math><mi>L</mi><mo>(</mo><mi>T</mi><mo>)</mo></math></span> is the line graph of <em>T</em>. In this paper, we extend their result from a tree <em>T</em> to any graph <span><math><mi>G</mi><mo>≠</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, and prove that <span><math><mi>m</mi><mo>(</mo><mi>L</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>,</mo><mi>λ</mi><mo>)</mo><mo>≤</mo><mn>2</mn><mi>c</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>+</mo><mi>p</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>−</mo><mn>1</mn></math></span> for any graph <span><math><mi>G</mi><mo>≠</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>. Moreover, all graphs <em>G</em> with <span><math><mi>m</mi><mo>(</mo><mi>L</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>,</mo><mo>−</mo><mn>1</mn><mo>)</mo><mo>=</mo><mn>2</mn><mi>c</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>+</mo><mi>p</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>−</mo><mn>1</mn></math></span> are completely characterized.</p></div>\",\"PeriodicalId\":18043,\"journal\":{\"name\":\"Linear Algebra and its Applications\",\"volume\":\"703 \",\"pages\":\"Pages 47-62\"},\"PeriodicalIF\":1.0000,\"publicationDate\":\"2024-09-03\",\"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/S0024379524003550\",\"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/S0024379524003550","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
设 m(G,λ)、c(G) 和 p(G) 分别为连通图 G 的特征值 λ 的倍率、循环数和挂顶点数。Yang 等人（2023）[10] 证明了对于任意树 T，m(L(T),λ)≤p(T)-1，并表征了 m(L(T),λ)=p(T)-1 的所有树 T，其中 L(T) 是 T 的线图。本文将他们的结果从树 T 扩展到任何图 G≠Cn，并证明对于任何图 G≠Cn，m(L(G),λ)≤2c(G)+p(G)-1。此外，m(L(G),-1)=2c(G)+p(G)-1 的所有图形 G 都是完全表征的。
Let , and be the multiplicity of an eigenvalue λ, the cyclomatic number and the number of pendant vertices of a connected graph G, respectively. Yang et al. (2023) [10] proved that for any tree T, and characterized all trees T with , where is the line graph of T. In this paper, we extend their result from a tree T to any graph , and prove that for any graph . Moreover, all graphs G with are completely characterized.
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.
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