{"title":"去除一个周期的完整图谱特征","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":"{\"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}","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}
引用次数: 0
摘要
如果不存在另一个具有相同频谱的图，则该图由其频谱决定。Cámara 和 Haemers 证明，通过删除具有 k 个顶点的循环 Ck 的所有边，从具有 n 个顶点的完整图 Kn 得到的图 Kn∖Ck，在 k∈{3,4,5} 时由其谱决定，但在 k=6 时则不是。在本文中，我们将证明 k=6 是 Kn∖Ck 的谱确定性的唯一例外。
Spectral characterization of the complete graph removing a cycle
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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