{"title":"Minimal graphs with eigenvalue multiplicity of n − d","authors":"Yuanshuai Zhang , Dein Wong , Wenhao Zhen","doi":"10.1016/j.laa.2024.06.015","DOIUrl":null,"url":null,"abstract":"<div><p>For a connected graph <em>G</em> with order <em>n</em>, let <span><math><mi>e</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> be the number of its distinct eigenvalues and <em>d</em> be the diameter. We denote by <span><math><msub><mrow><mi>m</mi></mrow><mrow><mi>G</mi></mrow></msub><mo>(</mo><mi>μ</mi><mo>)</mo></math></span> the eigenvalue multiplicity of <em>μ</em> in <em>G</em>. It is well known that <span><math><mi>e</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≥</mo><mi>d</mi><mo>+</mo><mn>1</mn></math></span>, which shows <span><math><msub><mrow><mi>m</mi></mrow><mrow><mi>G</mi></mrow></msub><mo>(</mo><mi>μ</mi><mo>)</mo><mo>≤</mo><mi>n</mi><mo>−</mo><mi>d</mi></math></span> for any real number <em>μ</em>. A graph is called <span><math><mi>m</mi><mi>i</mi><mi>n</mi><mi>i</mi><mi>m</mi><mi>a</mi><mi>l</mi></math></span> if <span><math><mi>e</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>=</mo><mi>d</mi><mo>+</mo><mn>1</mn></math></span>. In 2013, Wong et al. characterize all minimal graphs with <span><math><msub><mrow><mi>m</mi></mrow><mrow><mi>G</mi></mrow></msub><mo>(</mo><mn>0</mn><mo>)</mo><mo>=</mo><mi>n</mi><mo>−</mo><mi>d</mi></math></span>. In this paper, by applying the star complement theory, we prove that if <em>G</em> is not a path and <span><math><msub><mrow><mi>m</mi></mrow><mrow><mi>G</mi></mrow></msub><mo>(</mo><mi>μ</mi><mo>)</mo><mo>=</mo><mi>n</mi><mo>−</mo><mi>d</mi></math></span>, then <span><math><mi>μ</mi><mo>∈</mo><mo>{</mo><mn>0</mn><mo>,</mo><mo>−</mo><mn>1</mn><mo>}</mo></math></span>. Furthermore, we completely characterize all minimal graphs with <span><math><msub><mrow><mi>m</mi></mrow><mrow><mi>G</mi></mrow></msub><mo>(</mo><mo>−</mo><mn>1</mn><mo>)</mo><mo>=</mo><mi>n</mi><mo>−</mo><mi>d</mi></math></span>.</p></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":null,"pages":null},"PeriodicalIF":1.0000,"publicationDate":"2024-06-17","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/S0024379524002696","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
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Abstract
For a connected graph G with order n, let be the number of its distinct eigenvalues and d be the diameter. We denote by the eigenvalue multiplicity of μ in G. It is well known that , which shows for any real number μ. A graph is called if . In 2013, Wong et al. characterize all minimal graphs with . In this paper, by applying the star complement theory, we prove that if G is not a path and , then . Furthermore, we completely characterize all minimal graphs with .
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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