{"title":"Some stability results for spectral extremal problems of graphs with bounded matching number","authors":"Shixia Jiang , Xiying Yuan , Yanni Zhai","doi":"10.1016/j.laa.2024.12.018","DOIUrl":null,"url":null,"abstract":"<div><div>For a set of graphs <span><math><mi>H</mi></math></span>, a graph is called <span><math><mi>H</mi></math></span>-free if it does not contain any member of <span><math><mi>H</mi></math></span> as a subgraph. The maximum value of spectral radius among all <span><math><mi>H</mi></math></span>-free graphs of order <em>n</em> is denoted by <span><math><mrow><mi>spex</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo></math></span>, and the set of corresponding extremal graphs is denoted by <span><math><mrow><mi>SPEX</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo></math></span>. In this paper, we give a stability result for graphs in <span><math><mrow><mi>SPEX</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo></math></span> when <span><math><mrow><mi>spex</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo><mo>≥</mo><msqrt><mrow><mi>s</mi><mo>(</mo><mi>n</mi><mo>−</mo><mi>s</mi><mo>)</mo></mrow></msqrt></math></span> and <span><math><mrow><mi>ex</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo><mo>≤</mo><mi>s</mi><mi>n</mi></math></span>. As an application, we may give some characterizations for the graphs in <span><math><mrow><mi>SPEX</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mo>{</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>s</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>,</mo><mi>H</mi><mo>}</mo><mo>)</mo></math></span>, where <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>s</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span> is a matching with <span><math><mi>s</mi><mo>+</mo><mn>1</mn></math></span> edges and <em>H</em> is any non-bipartite graph.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"708 ","pages":"Pages 513-524"},"PeriodicalIF":1.0000,"publicationDate":"2024-12-20","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/S0024379524004907","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 set of graphs , a graph is called -free if it does not contain any member of as a subgraph. The maximum value of spectral radius among all -free graphs of order n is denoted by , and the set of corresponding extremal graphs is denoted by . In this paper, we give a stability result for graphs in when and . As an application, we may give some characterizations for the graphs in , where is a matching with edges and H is any non-bipartite graph.
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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