Finite 4-geodesic-transitive graphs with bounded girth

IF 0.6 3区 数学 Q3 MATHEMATICS
Wei Jin, Li Tan
{"title":"Finite 4-geodesic-transitive graphs with bounded girth","authors":"Wei Jin, Li Tan","doi":"10.1007/s10801-024-01358-3","DOIUrl":null,"url":null,"abstract":"<p>Praeger and the first author in Jin and Praeger (J Combin Theory Ser A 178:105349, 2021) asked the following problem: classify <i>s</i>-geodesic-transitive graphs of girth <span>\\(2s-1\\)</span> or <span>\\(2s-2\\)</span>, where <span>\\(s=4,5,6,7,8\\)</span>. In this paper, we study the <span>\\(s=4\\)</span> case, that is, study the family of finite (<i>G</i>, 4)-geodesic-transitive graphs of girth 6 or 7 for some group <i>G</i> of automorphisms. A reduction result on this family of graphs is first given. Let <i>N</i> be a normal subgroup of <i>G</i> which has at least 3 orbits on the vertex set. We show that such a graph <span>\\(\\Gamma \\)</span> is a cover of its quotient <span>\\(\\Gamma _N\\)</span> modulo the <i>N</i>-orbits and either <span>\\(\\Gamma _N\\)</span> is (<i>G</i>/<i>N</i>, <i>s</i>)-geodesic-transitive where <span>\\(s=\\min \\{4,\\textrm{diam}(\\Gamma _N)\\}\\ge 3\\)</span>, or <span>\\(\\Gamma _N\\)</span> is a (<i>G</i>/<i>N</i>, 2)-arc-transitive strongly regular graph. Next, using the classification of 2-arc-transitive strongly regular graphs, we determine all the (<i>G</i>, 4)-geodesic-transitive covers <span>\\(\\Gamma \\)</span> when <span>\\(\\Gamma _N\\)</span> is strongly regular.</p>","PeriodicalId":14926,"journal":{"name":"Journal of Algebraic Combinatorics","volume":"32 1","pages":""},"PeriodicalIF":0.6000,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Algebraic Combinatorics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s10801-024-01358-3","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0

Abstract

Praeger and the first author in Jin and Praeger (J Combin Theory Ser A 178:105349, 2021) asked the following problem: classify s-geodesic-transitive graphs of girth \(2s-1\) or \(2s-2\), where \(s=4,5,6,7,8\). In this paper, we study the \(s=4\) case, that is, study the family of finite (G, 4)-geodesic-transitive graphs of girth 6 or 7 for some group G of automorphisms. A reduction result on this family of graphs is first given. Let N be a normal subgroup of G which has at least 3 orbits on the vertex set. We show that such a graph \(\Gamma \) is a cover of its quotient \(\Gamma _N\) modulo the N-orbits and either \(\Gamma _N\) is (G/Ns)-geodesic-transitive where \(s=\min \{4,\textrm{diam}(\Gamma _N)\}\ge 3\), or \(\Gamma _N\) is a (G/N, 2)-arc-transitive strongly regular graph. Next, using the classification of 2-arc-transitive strongly regular graphs, we determine all the (G, 4)-geodesic-transitive covers \(\Gamma \) when \(\Gamma _N\) is strongly regular.

具有有界周长的有限 4- 大地遍历图
Praeger 和第一作者在 Jin and Praeger (J Combin Theory Ser A 178:105349, 2021) 中提出了以下问题:分类周长为 \(2s-1\) 或 \(2s-2\) 的 s 节点变换图,其中 \(s=4,5,6,7,8\).在本文中,我们研究的是\(s=4\)的情况,也就是研究对于某个自动形群 G 而言周长为 6 或 7 的有限(G,4)-大地遍历图形族。首先给出这个图形族的还原结果。让 N 是顶点集上至少有 3 个轨道的 G 的正则子群。我们证明这样的图\(\Gamma \)是它的商\(\Gamma _N\)的覆盖,并且\(\Gamma _N\)是(G/N、s=min \{4,\textrm{diam}(\Gamma _N)\}ge 3\), 或者 \(\Gamma _N\) 是一个(G/N, 2)弧遍历强规则图。接下来,利用2-弧-传递强正则图的分类,我们确定了当\(\Gamma _N\)是强正则图时所有的(G,4)-大地-传递盖\(\Gamma \)。
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
CiteScore
1.50
自引率
12.50%
发文量
94
审稿时长
6-12 weeks
期刊介绍: The Journal of Algebraic Combinatorics provides a single forum for papers on algebraic combinatorics which, at present, are distributed throughout a number of journals. Within the last decade or so, algebraic combinatorics has evolved into a mature, established and identifiable area of mathematics. Research contributions in the field are increasingly seen to have substantial links with other areas of mathematics. The journal publishes papers in which combinatorics and algebra interact in a significant and interesting fashion. This interaction might occur through the study of combinatorial structures using algebraic methods, or the application of combinatorial methods to algebraic problems.
×
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