{"title":"Spectral characterization of the complete graph removing a cycle","authors":"Muhuo Liu , Xiaofeng Gu , Haiying Shan , Zoran Stanić","doi":"10.1016/j.jcta.2024.105868","DOIUrl":null,"url":null,"abstract":"<div><p>A graph is determined by its spectrum if there is not another graph with the same spectrum. Cámara and Haemers proved that the graph <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>∖</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span>, obtained from the complete graph <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> with <em>n</em> vertices by deleting all edges of a cycle <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> with <em>k</em> vertices, is determined by its spectrum for <span><math><mi>k</mi><mo>∈</mo><mo>{</mo><mn>3</mn><mo>,</mo><mn>4</mn><mo>,</mo><mn>5</mn><mo>}</mo></math></span>, but not for <span><math><mi>k</mi><mo>=</mo><mn>6</mn></math></span>. In this paper, we show that <span><math><mi>k</mi><mo>=</mo><mn>6</mn></math></span> is the unique exception for the spectral determination of <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>∖</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span>.</p></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":null,"pages":null},"PeriodicalIF":0.9000,"publicationDate":"2024-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Combinatorial Theory Series A","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0097316524000074","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
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Abstract
A graph is determined by its spectrum if there is not another graph with the same spectrum. Cámara and Haemers proved that the graph , obtained from the complete graph with n vertices by deleting all edges of a cycle with k vertices, is determined by its spectrum for , but not for . In this paper, we show that is the unique exception for the spectral determination of .
期刊介绍：
The Journal of Combinatorial Theory publishes original mathematical research concerned with theoretical and physical aspects of the study of finite and discrete structures in all branches of science. Series A is concerned primarily with structures, designs, and applications of combinatorics and is a valuable tool for mathematicians and computer scientists.
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