Graphs with the minimum spectral radius for given independence number

IF 0.7 3区 数学 Q2 MATHEMATICS
Yarong Hu , Qiongxiang Huang , Zhenzhen Lou
{"title":"Graphs with the minimum spectral radius for given independence number","authors":"Yarong Hu ,&nbsp;Qiongxiang Huang ,&nbsp;Zhenzhen Lou","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":"348 2","pages":"Article 114265"},"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 Gn,α be the set of connected graphs with order n and independence number α. The graph with the minimum spectral radius among Gn,α 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 Gn,α appears to be a tough problem. Recently, Lou and Guo (2022) [14] proved that the minimizer graph in Gn,α must be a tree if αn2. 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 Gn,α along with their spectral radius for any given αn2. As an application, we determine all the minimizer graphs in Gn,α for α=n5,n6 along with their spectral radius.

给定独立数时具有最小谱半径的图形
设 Gn,α 是阶数为 n 且独立数为 α 的连通图集合,Gn,α 中谱半径最小的图称为最小图。Stevanović 在其经典著作[Spectral Radius of Graphs, Academic Press, Amsterdam, 2015]中指出,确定 Gn,α 中的最小图似乎是一个难题。最近,Lou 和 Guo(2022)[14] 证明,如果α≥⌈n2⌉,则 Gn,α 中的最小图一定是树。本文进一步详细给出了最小图的结构特征,并给出了其构造定理。因此,我们从理论上确定了 Gn,α 中的最小化图,以及任意给定 α≥⌈n2⌉ 时它们的谱半径。作为应用,我们确定了 α=n-5,n-6 时 Gn,α 中的所有最小图及其谱半径。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
求助全文
约1分钟内获得全文 求助全文
来源期刊
Discrete Mathematics
Discrete Mathematics 数学-数学
CiteScore
1.50
自引率
12.50%
发文量
424
审稿时长
6 months
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.
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术官方微信