{"title":"Cayley 和图中的子群完全码","authors":"Xiaomeng Wang, Lina Wei, Shou-Jun Xu, Sanming Zhou","doi":"10.1007/s10623-024-01405-x","DOIUrl":null,"url":null,"abstract":"<p>Let <span>\\(\\Gamma \\)</span> be a graph with vertex set <i>V</i>, and let <i>a</i>, <i>b</i> be nonnegative integers. An (<i>a</i>, <i>b</i>)-regular set in <span>\\(\\Gamma \\)</span> is a nonempty proper subset <i>D</i> of <i>V</i> such that every vertex in <i>D</i> has exactly <i>a</i> neighbours in <i>D</i> and every vertex in <span>\\(V \\setminus D\\)</span> has exactly <i>b</i> neighbours in <i>D</i>. In particular, a (1, 1)-regular set is called a total perfect code. Let <i>G</i> be a finite group and <i>S</i> a square-free subset of <i>G</i> closed under conjugation. The Cayley sum graph <span>\\(\\textrm{CayS}(G,S)\\)</span> of <i>G</i> is the graph with vertex set <i>G</i> such that two vertices <i>x</i>, <i>y</i> are adjacent if and only if <span>\\(xy \\in S\\)</span>. A subset (respectively, subgroup) <i>D</i> of <i>G</i> is called an (<i>a</i>, <i>b</i>)-regular set (respectively, subgroup (<i>a</i>, <i>b</i>)-regular set) of <i>G</i> if there exists a Cayley sum graph of <i>G</i> which admits <i>D</i> as an (<i>a</i>, <i>b</i>)-regular set. We obtain two necessary and sufficient conditions for a subgroup of a finite group <i>G</i> to be a total perfect code in a Cayley sum graph of <i>G</i>. We also obtain two necessary and sufficient conditions for a subgroup of a finite abelian group <i>G</i> to be a total perfect code of <i>G</i>. We classify finite abelian groups whose all non-trivial subgroups of even order are total perfect codes of the group, and as a corollary we obtain that a finite abelian group has the property that every non-trivial subgroup is a total perfect code if and only if it is isomorphic to an elementary abelian 2-group. We prove that, for a subgroup <i>H</i> of a finite abelian group <i>G</i> and any pair of positive integers (<i>a</i>, <i>b</i>) within certain ranges depending on <i>H</i>, <i>H</i> is an (<i>a</i>, <i>b</i>)-regular set of <i>G</i> if and only if it is a total perfect code of <i>G</i>. Finally, we give a classification of subgroup total perfect codes of a cyclic group, a dihedral group and a generalized quaternion group.</p>","PeriodicalId":11130,"journal":{"name":"Designs, Codes and Cryptography","volume":null,"pages":null},"PeriodicalIF":1.4000,"publicationDate":"2024-04-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Subgroup total perfect codes in Cayley sum graphs\",\"authors\":\"Xiaomeng Wang, Lina Wei, Shou-Jun Xu, Sanming Zhou\",\"doi\":\"10.1007/s10623-024-01405-x\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>Let <span>\\\\(\\\\Gamma \\\\)</span> be a graph with vertex set <i>V</i>, and let <i>a</i>, <i>b</i> be nonnegative integers. An (<i>a</i>, <i>b</i>)-regular set in <span>\\\\(\\\\Gamma \\\\)</span> is a nonempty proper subset <i>D</i> of <i>V</i> such that every vertex in <i>D</i> has exactly <i>a</i> neighbours in <i>D</i> and every vertex in <span>\\\\(V \\\\setminus D\\\\)</span> has exactly <i>b</i> neighbours in <i>D</i>. In particular, a (1, 1)-regular set is called a total perfect code. Let <i>G</i> be a finite group and <i>S</i> a square-free subset of <i>G</i> closed under conjugation. The Cayley sum graph <span>\\\\(\\\\textrm{CayS}(G,S)\\\\)</span> of <i>G</i> is the graph with vertex set <i>G</i> such that two vertices <i>x</i>, <i>y</i> are adjacent if and only if <span>\\\\(xy \\\\in S\\\\)</span>. A subset (respectively, subgroup) <i>D</i> of <i>G</i> is called an (<i>a</i>, <i>b</i>)-regular set (respectively, subgroup (<i>a</i>, <i>b</i>)-regular set) of <i>G</i> if there exists a Cayley sum graph of <i>G</i> which admits <i>D</i> as an (<i>a</i>, <i>b</i>)-regular set. We obtain two necessary and sufficient conditions for a subgroup of a finite group <i>G</i> to be a total perfect code in a Cayley sum graph of <i>G</i>. We also obtain two necessary and sufficient conditions for a subgroup of a finite abelian group <i>G</i> to be a total perfect code of <i>G</i>. We classify finite abelian groups whose all non-trivial subgroups of even order are total perfect codes of the group, and as a corollary we obtain that a finite abelian group has the property that every non-trivial subgroup is a total perfect code if and only if it is isomorphic to an elementary abelian 2-group. We prove that, for a subgroup <i>H</i> of a finite abelian group <i>G</i> and any pair of positive integers (<i>a</i>, <i>b</i>) within certain ranges depending on <i>H</i>, <i>H</i> is an (<i>a</i>, <i>b</i>)-regular set of <i>G</i> if and only if it is a total perfect code of <i>G</i>. Finally, we give a classification of subgroup total perfect codes of a cyclic group, a dihedral group and a generalized quaternion group.</p>\",\"PeriodicalId\":11130,\"journal\":{\"name\":\"Designs, Codes and Cryptography\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.4000,\"publicationDate\":\"2024-04-29\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Designs, Codes and Cryptography\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1007/s10623-024-01405-x\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"COMPUTER SCIENCE, THEORY & METHODS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Designs, Codes and Cryptography","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s10623-024-01405-x","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"COMPUTER SCIENCE, THEORY & METHODS","Score":null,"Total":0}
引用次数: 0
摘要
让 \(\Gamma\) 是一个有顶点集 V 的图,让 a, b 是非负整数。(a, b) -regular set in \(\Gamma \)是 V 的一个非空适当子集 D,使得 D 中的每个顶点在 D 中都有恰好 a 个邻居,并且 \(V \setminus D\) 中的每个顶点在 D 中都有恰好 b 个邻居。让 G 是一个有限群,S 是 G 在共轭作用下封闭的无平方子集。G 的 Cayley 和图(textrm{CayS}(G,S)\)是具有顶点集 G 的图,当且仅当\(xy \in S\) 时,两个顶点 x、y 相邻。如果存在一个 G 的 Cayley 和图,而这个 Cayley 和图接纳 D 作为 G 的(a, b)-正则集合,那么 G 的一个子集(分别是子群)D 称为 G 的(a, b)-正则集合(分别是子群(a, b)-正则集合)。我们得到了有限群 G 的一个子群在 G 的 Cayley 和图中是全完全码的两个必要条件和充分条件。我们对所有偶数阶非琐分组都是该组的全完全码的有限无边群进行了分类,并由此推论,当且仅当一个有限无边群与一个基本无边 2 群同构时,它才具有每个非琐分组都是全完全码的性质。最后,我们给出了循环群、二面群和广义四元组的子群全完美码的分类。
Let \(\Gamma \) be a graph with vertex set V, and let a, b be nonnegative integers. An (a, b)-regular set in \(\Gamma \) is a nonempty proper subset D of V such that every vertex in D has exactly a neighbours in D and every vertex in \(V \setminus D\) has exactly b neighbours in D. In particular, a (1, 1)-regular set is called a total perfect code. Let G be a finite group and S a square-free subset of G closed under conjugation. The Cayley sum graph \(\textrm{CayS}(G,S)\) of G is the graph with vertex set G such that two vertices x, y are adjacent if and only if \(xy \in S\). A subset (respectively, subgroup) D of G is called an (a, b)-regular set (respectively, subgroup (a, b)-regular set) of G if there exists a Cayley sum graph of G which admits D as an (a, b)-regular set. We obtain two necessary and sufficient conditions for a subgroup of a finite group G to be a total perfect code in a Cayley sum graph of G. We also obtain two necessary and sufficient conditions for a subgroup of a finite abelian group G to be a total perfect code of G. We classify finite abelian groups whose all non-trivial subgroups of even order are total perfect codes of the group, and as a corollary we obtain that a finite abelian group has the property that every non-trivial subgroup is a total perfect code if and only if it is isomorphic to an elementary abelian 2-group. We prove that, for a subgroup H of a finite abelian group G and any pair of positive integers (a, b) within certain ranges depending on H, H is an (a, b)-regular set of G if and only if it is a total perfect code of G. Finally, we give a classification of subgroup total perfect codes of a cyclic group, a dihedral group and a generalized quaternion group.
期刊介绍:
Designs, Codes and Cryptography is an archival peer-reviewed technical journal publishing original research papers in the designated areas. There is a great deal of activity in design theory, coding theory and cryptography, including a substantial amount of research which brings together more than one of the subjects. While many journals exist for each of the individual areas, few encourage the interaction of the disciplines.
The journal was founded to meet the needs of mathematicians, engineers and computer scientists working in these areas, whose interests extend beyond the bounds of any one of the individual disciplines. The journal provides a forum for high quality research in its three areas, with papers touching more than one of the areas especially welcome.
The journal also considers high quality submissions in the closely related areas of finite fields and finite geometries, which provide important tools for both the construction and the actual application of designs, codes and cryptographic systems. In particular, it includes (mostly theoretical) papers on computational aspects of finite fields. It also considers topics in sequence design, which frequently admit equivalent formulations in the journal’s main areas.
Designs, Codes and Cryptography is mathematically oriented, emphasizing the algebraic and geometric aspects of the areas it covers. The journal considers high quality papers of both a theoretical and a practical nature, provided they contain a substantial amount of mathematics.