{"title":"无 3K3 图形的 Q 指数最大值","authors":"Yanting Zhang, Ligong Wang","doi":"10.1016/j.dam.2024.08.004","DOIUrl":null,"url":null,"abstract":"<div><p>The <span><math><mi>Q</mi></math></span>-index of a graph <span><math><mi>G</mi></math></span> is the largest eigenvalue of its <span><math><mi>Q</mi></math></span>-matrix <span><math><mrow><mi>Q</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><mi>D</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>+</mo><mi>A</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>, where <span><math><mrow><mi>D</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><mi>A</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> are the diagonal matrix of vertex degrees and the adjacency matrix of <span><math><mi>G</mi></math></span>, respectively. Let <span><math><mrow><mn>3</mn><msub><mrow><mi>K</mi></mrow><mrow><mn>3</mn></mrow></msub></mrow></math></span> denote the graph consisting of three vertex-disjoint triangles. A graph is called <span><math><mrow><mn>3</mn><msub><mrow><mi>K</mi></mrow><mrow><mn>3</mn></mrow></msub></mrow></math></span>-free if it does not contain <span><math><mrow><mn>3</mn><msub><mrow><mi>K</mi></mrow><mrow><mn>3</mn></mrow></msub></mrow></math></span> as a subgraph. In this paper, we present a sharp upper bound on the <span><math><mi>Q</mi></math></span>-index of <span><math><mrow><mn>3</mn><msub><mrow><mi>K</mi></mrow><mrow><mn>3</mn></mrow></msub></mrow></math></span>-free graphs of order <span><math><mrow><mi>n</mi><mo>≥</mo><mn>453</mn></mrow></math></span>, and characterize the unique extremal graph which attains the bound.</p></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"358 ","pages":"Pages 448-456"},"PeriodicalIF":1.0000,"publicationDate":"2024-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Maxima of the Q-index for 3K3-free graphs\",\"authors\":\"Yanting Zhang, Ligong Wang\",\"doi\":\"10.1016/j.dam.2024.08.004\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>The <span><math><mi>Q</mi></math></span>-index of a graph <span><math><mi>G</mi></math></span> is the largest eigenvalue of its <span><math><mi>Q</mi></math></span>-matrix <span><math><mrow><mi>Q</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><mi>D</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>+</mo><mi>A</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>, where <span><math><mrow><mi>D</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><mi>A</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> are the diagonal matrix of vertex degrees and the adjacency matrix of <span><math><mi>G</mi></math></span>, respectively. Let <span><math><mrow><mn>3</mn><msub><mrow><mi>K</mi></mrow><mrow><mn>3</mn></mrow></msub></mrow></math></span> denote the graph consisting of three vertex-disjoint triangles. A graph is called <span><math><mrow><mn>3</mn><msub><mrow><mi>K</mi></mrow><mrow><mn>3</mn></mrow></msub></mrow></math></span>-free if it does not contain <span><math><mrow><mn>3</mn><msub><mrow><mi>K</mi></mrow><mrow><mn>3</mn></mrow></msub></mrow></math></span> as a subgraph. In this paper, we present a sharp upper bound on the <span><math><mi>Q</mi></math></span>-index of <span><math><mrow><mn>3</mn><msub><mrow><mi>K</mi></mrow><mrow><mn>3</mn></mrow></msub></mrow></math></span>-free graphs of order <span><math><mrow><mi>n</mi><mo>≥</mo><mn>453</mn></mrow></math></span>, and characterize the unique extremal graph which attains the bound.</p></div>\",\"PeriodicalId\":50573,\"journal\":{\"name\":\"Discrete Applied Mathematics\",\"volume\":\"358 \",\"pages\":\"Pages 448-456\"},\"PeriodicalIF\":1.0000,\"publicationDate\":\"2024-08-21\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Discrete Applied Mathematics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0166218X24003573\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Discrete Applied Mathematics","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0166218X24003573","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
The -index of a graph is the largest eigenvalue of its -matrix , where and are the diagonal matrix of vertex degrees and the adjacency matrix of , respectively. Let denote the graph consisting of three vertex-disjoint triangles. A graph is called -free if it does not contain as a subgraph. In this paper, we present a sharp upper bound on the -index of -free graphs of order , and characterize the unique extremal graph which attains the bound.
期刊介绍：
The aim of Discrete Applied Mathematics is to bring together research papers in different areas of algorithmic and applicable discrete mathematics as well as applications of combinatorial mathematics to informatics and various areas of science and technology. Contributions presented to the journal can be research papers, short notes, surveys, and possibly research problems. The "Communications" section will be devoted to the fastest possible publication of recent research results that are checked and recommended for publication by a member of the Editorial Board. The journal will also publish a limited number of book announcements as well as proceedings of conferences. These proceedings will be fully refereed and adhere to the normal standards of the journal.
Potential authors are advised to view the journal and the open calls-for-papers of special issues before submitting their manuscripts. Only high-quality, original work that is within the scope of the journal or the targeted special issue will be considered.
京公网安备 11010802042870号
京ICP备2023020795号-1
分享
求助内容：
应助结果提醒方式：
扫码关注我们
