{"title":"关于六价半弧透二面体","authors":"Mi-Mi Zhang","doi":"10.1016/j.disc.2024.114180","DOIUrl":null,"url":null,"abstract":"<div><p>A graph is <em>half-arc-transitive</em> if its full automorphism group acts transitively on its vertex set and edge set, but not arc set. A half-arc-transitive graph is <em>half-arc-regular</em> if its full automorphism group acts regularly on its edges. A graph is said to be a <em>bi-Cayley graph</em> over a group <em>H</em> if it admits <em>H</em> as a semiregular automorphism group with two vertex-orbits. A bi-Cayley graph over a dihedral group is called <em>bi-dihedrant</em>. In this paper, it is shown that the smallest valency of half-arc-transitive bi-dihendrants is 6, and then a classification is given of connected half-arc-regular bi-dihedrants of valency 6. This work together with the result in <span><span>[8, Theorem 6.7]</span></span> completes the classification of edge-regular bi-dihedrants of valency 6.</p></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"347 11","pages":"Article 114180"},"PeriodicalIF":0.7000,"publicationDate":"2024-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On hexavalent half-arc-transitive bi-dihedrants\",\"authors\":\"Mi-Mi Zhang\",\"doi\":\"10.1016/j.disc.2024.114180\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>A graph is <em>half-arc-transitive</em> if its full automorphism group acts transitively on its vertex set and edge set, but not arc set. A half-arc-transitive graph is <em>half-arc-regular</em> if its full automorphism group acts regularly on its edges. A graph is said to be a <em>bi-Cayley graph</em> over a group <em>H</em> if it admits <em>H</em> as a semiregular automorphism group with two vertex-orbits. A bi-Cayley graph over a dihedral group is called <em>bi-dihedrant</em>. In this paper, it is shown that the smallest valency of half-arc-transitive bi-dihendrants is 6, and then a classification is given of connected half-arc-regular bi-dihedrants of valency 6. This work together with the result in <span><span>[8, Theorem 6.7]</span></span> completes the classification of edge-regular bi-dihedrants of valency 6.</p></div>\",\"PeriodicalId\":50572,\"journal\":{\"name\":\"Discrete Mathematics\",\"volume\":\"347 11\",\"pages\":\"Article 114180\"},\"PeriodicalIF\":0.7000,\"publicationDate\":\"2024-07-25\",\"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/S0012365X2400311X\",\"RegionNum\":3,\"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":"Discrete Mathematics","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0012365X2400311X","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
A graph is half-arc-transitive if its full automorphism group acts transitively on its vertex set and edge set, but not arc set. A half-arc-transitive graph is half-arc-regular if its full automorphism group acts regularly on its edges. A graph is said to be a bi-Cayley graph over a group H if it admits H as a semiregular automorphism group with two vertex-orbits. A bi-Cayley graph over a dihedral group is called bi-dihedrant. In this paper, it is shown that the smallest valency of half-arc-transitive bi-dihendrants is 6, and then a classification is given of connected half-arc-regular bi-dihedrants of valency 6. This work together with the result in [8, Theorem 6.7] completes the classification of edge-regular bi-dihedrants of valency 6.
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.