{"title":"The Spectral Radius, Maximum Average Degree and Cycles of Consecutive Lengths of Graphs","authors":"Wenqian Zhang","doi":"10.1007/s00373-024-02761-0","DOIUrl":null,"url":null,"abstract":"<p>In this paper, we study the relationship between spectral radius and maximum average degree of graphs. By using this relationship and the previous technique of Li and Ning in (J Graph Theory 103:486–492, 2023), we prove that, for any given positive number <span>\\(\\varepsilon <\\frac{1}{3}\\)</span>, if <i>n</i> is a sufficiently large integer, then any graph <i>G</i> of order <i>n</i> with <span>\\(\\rho (G)>\\sqrt{\\left\\lfloor \\frac{n^{2}}{4}\\right\\rfloor }\\)</span> contains a cycle of length <i>t</i> for all integers <span>\\(t\\in [3,(\\frac{1}{3}-\\varepsilon )n]\\)</span>, where <span>\\(\\rho (G)\\)</span> is the spectral radius of <i>G</i>. This improves the result of Li and Ning (2023).</p>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00373-024-02761-0","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
In this paper, we study the relationship between spectral radius and maximum average degree of graphs. By using this relationship and the previous technique of Li and Ning in (J Graph Theory 103:486–492, 2023), we prove that, for any given positive number \(\varepsilon <\frac{1}{3}\), if n is a sufficiently large integer, then any graph G of order n with \(\rho (G)>\sqrt{\left\lfloor \frac{n^{2}}{4}\right\rfloor }\) contains a cycle of length t for all integers \(t\in [3,(\frac{1}{3}-\varepsilon )n]\), where \(\rho (G)\) is the spectral radius of G. This improves the result of Li and Ning (2023).
本文研究了图的谱半径和最大平均度之间的关系。利用这一关系以及李和宁在《图论》（J Graph Theory 103：486-492, 2023）中的技术，我们证明了对于任何给定的正数 \(\varepsilon <\frac{1}{3}\), 如果 n 是一个足够大的整数，那么任何阶数为 n 的图 G 具有 \(\rho (G)>；\sqrt\{left\floor \frac{n^{2}}{4}\right\rfloor }\) 包含一个长度为 t 的循环，对于所有整数 \(t\in [3,(\frac{1}{3}-\varepsilon )n]\), 其中 \(\rho (G)\) 是 G 的光谱半径。这改进了李和宁（2023）的结果。
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