{"title":"Graphs with the minimum spectral radius for given independence number","authors":"","doi":"10.1016/j.disc.2024.114265","DOIUrl":null,"url":null,"abstract":"<div><p>Let <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>α</mi></mrow></msub></math></span> be the set of connected graphs with order <em>n</em> and independence number <em>α</em>. The graph with the minimum spectral radius among <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>α</mi></mrow></msub></math></span> is called the minimizer graph. Stevanović in the classical book [Spectral Radius of Graphs, Academic Press, Amsterdam, 2015] pointed out that determining the minimizer graph in <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>α</mi></mrow></msub></math></span> appears to be a tough problem. Recently, Lou and Guo (2022) <span><span>[14]</span></span> proved that the minimizer graph in <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>α</mi></mrow></msub></math></span> must be a tree if <span><math><mi>α</mi><mo>≥</mo><mo>⌈</mo><mfrac><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>⌉</mo></math></span>. In this paper, we further give the structural features for the minimizer graph in detail, and then provide a constructing theorem for it. Thus, theoretically we determine the minimizer graphs in <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>α</mi></mrow></msub></math></span> along with their spectral radius for any given <span><math><mi>α</mi><mo>≥</mo><mo>⌈</mo><mfrac><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>⌉</mo></math></span>. As an application, we determine all the minimizer graphs in <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>α</mi></mrow></msub></math></span> for <span><math><mi>α</mi><mo>=</mo><mi>n</mi><mo>−</mo><mn>5</mn><mo>,</mo><mi>n</mi><mo>−</mo><mn>6</mn></math></span> along with their spectral radius.</p></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7000,"publicationDate":"2024-09-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Discrete Mathematics","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0012365X24003960","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
Abstract
Let be the set of connected graphs with order n and independence number α. The graph with the minimum spectral radius among is called the minimizer graph. Stevanović in the classical book [Spectral Radius of Graphs, Academic Press, Amsterdam, 2015] pointed out that determining the minimizer graph in appears to be a tough problem. Recently, Lou and Guo (2022) [14] proved that the minimizer graph in must be a tree if . In this paper, we further give the structural features for the minimizer graph in detail, and then provide a constructing theorem for it. Thus, theoretically we determine the minimizer graphs in along with their spectral radius for any given . As an application, we determine all the minimizer graphs in for along with their spectral radius.
期刊介绍:
Discrete Mathematics provides a common forum for significant research in many areas of discrete mathematics and combinatorics. Among the fields covered by Discrete Mathematics are graph and hypergraph theory, enumeration, coding theory, block designs, the combinatorics of partially ordered sets, extremal set theory, matroid theory, algebraic combinatorics, discrete geometry, matrices, and discrete probability theory.
Items in the journal include research articles (Contributions or Notes, depending on length) and survey/expository articles (Perspectives). Efforts are made to process the submission of Notes (short articles) quickly. The Perspectives section features expository articles accessible to a broad audience that cast new light or present unifying points of view on well-known or insufficiently-known topics.