{"title":"The absolute values of the perfect matching derangement graph's eigenvalues almost follow the lexicographic order of partitions","authors":"","doi":"10.1016/j.disc.2024.114188","DOIUrl":null,"url":null,"abstract":"<div><p>In 2013, Ku and Wong showed that for any partitions <em>μ</em> and <span><math><msup><mrow><mi>μ</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span> of a positive integer <em>n</em> with the same first part <em>u</em> and the lexicographic order <span><math><mi>μ</mi><mo>◃</mo><msup><mrow><mi>μ</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span>, the eigenvalues <span><math><msub><mrow><mi>ξ</mi></mrow><mrow><mi>μ</mi></mrow></msub></math></span> and <span><math><msub><mrow><mi>ξ</mi></mrow><mrow><msup><mrow><mi>μ</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow></msub></math></span> of the derangement graph <span><math><msub><mrow><mi>Γ</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> have the property <span><math><mo>|</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>μ</mi></mrow></msub><mo>|</mo><mo>≤</mo><mo>|</mo><msub><mrow><mi>ξ</mi></mrow><mrow><msup><mrow><mi>μ</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow></msub><mo>|</mo></math></span>, where the equality holds if and only if <span><math><mi>u</mi><mo>=</mo><mn>3</mn></math></span> and all other parts are less than 3. In this article, we obtain an analogous conclusion on the eigenvalues of the perfect matching derangement graph <span><math><msub><mrow><mi>M</mi></mrow><mrow><mn>2</mn><mi>n</mi></mrow></msub></math></span> of <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>2</mn><mi>n</mi></mrow></msub></math></span> by finding a new recurrence formula for the eigenvalues of <span><math><msub><mrow><mi>M</mi></mrow><mrow><mn>2</mn><mi>n</mi></mrow></msub></math></span>.</p></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7000,"publicationDate":"2024-07-30","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/S0012365X24003194","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
In 2013, Ku and Wong showed that for any partitions μ and of a positive integer n with the same first part u and the lexicographic order , the eigenvalues and of the derangement graph have the property , where the equality holds if and only if and all other parts are less than 3. In this article, we obtain an analogous conclusion on the eigenvalues of the perfect matching derangement graph of by finding a new recurrence formula for the eigenvalues of .
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.