Pub Date : 2024-04-06DOI: 10.1007/s10801-024-01316-z
Abstract
The distance spectral radius of a connected hypergraph is the largest eigenvalue of its distance matrix. In this paper, we determine the k-uniform hypertree with the minimal spectral radius among all k-uniform hypertrees with m edges and diameter d, where (3le dle m-1).
摘要 连通超图的距离谱半径是其距离矩阵的最大特征值。在本文中,我们确定了在所有具有 m 条边和直径 d 的 k 个均匀超图中具有最小谱半径的 k 个均匀超图,其中 (3le dle m-1) .
{"title":"Distance spectral radii of k-uniform hypertrees with fixed diameter","authors":"","doi":"10.1007/s10801-024-01316-z","DOIUrl":"https://doi.org/10.1007/s10801-024-01316-z","url":null,"abstract":"<h3>Abstract</h3> <p>The distance spectral radius of a connected hypergraph is the largest eigenvalue of its distance matrix. In this paper, we determine the <em>k</em>-uniform hypertree with the minimal spectral radius among all <em>k</em>-uniform hypertrees with <em>m</em> edges and diameter <em>d</em>, where <span> <span>(3le dle m-1)</span> </span>.</p>","PeriodicalId":14926,"journal":{"name":"Journal of Algebraic Combinatorics","volume":"37 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-04-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140603459","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-04-04DOI: 10.1007/s10801-024-01307-0
Hayley Bertrand
This work is part of a research program to compute the Hochschild homology groups HH(_*({mathbb {C}}[x_1,ldots ,x_d]/(x_1,ldots ,x_d)^3;{mathbb {C}})) in the case (d = 2) through a lesser-known invariant called Coxeter cohomology, motivated by the isomorphism
$$begin{aligned}text {HH}_i({mathbb {C}}[x_1,ldots ,x_d]/(x_1,ldots ,x_d)^3;{mathbb {C}}) cong sum _{0le j le i} H^j_C left( S_{i+j}, V^{otimes (i+j)}right) end{aligned}$$
provided by Larsen and Lindenstrauss. Here, (H_C^*) denotes Coxeter cohomology, (S_{i+j}) denotes the symmetric group on (i+j) letters, and V is the standard representation of (textrm{GL}_d({mathbb {C}})) on ({mathbb {C}}^d). We compute the Euler characteristic of the Coxeter cohomology (the alternating sum of the ranks of the Coxeter cohomology groups) of several representations of (S_n). In particular, the aforementioned tensor representation, and also several classes of irreducible representations of (S_n). Although the problem and its motivation are algebraic and topological in nature, the techniques used are largely combinatorial.
{"title":"Calculations of the Euler characteristic of the Coxeter cohomology of symmetric groups","authors":"Hayley Bertrand","doi":"10.1007/s10801-024-01307-0","DOIUrl":"https://doi.org/10.1007/s10801-024-01307-0","url":null,"abstract":"<p>This work is part of a research program to compute the Hochschild homology groups HH<span>(_*({mathbb {C}}[x_1,ldots ,x_d]/(x_1,ldots ,x_d)^3;{mathbb {C}}))</span> in the case <span>(d = 2)</span> through a lesser-known invariant called Coxeter cohomology, motivated by the isomorphism </p><span>$$begin{aligned}text {HH}_i({mathbb {C}}[x_1,ldots ,x_d]/(x_1,ldots ,x_d)^3;{mathbb {C}}) cong sum _{0le j le i} H^j_C left( S_{i+j}, V^{otimes (i+j)}right) end{aligned}$$</span><p>provided by Larsen and Lindenstrauss. Here, <span>(H_C^*)</span> denotes Coxeter cohomology, <span>(S_{i+j})</span> denotes the symmetric group on <span>(i+j)</span> letters, and <i>V</i> is the standard representation of <span>(textrm{GL}_d({mathbb {C}}))</span> on <span>({mathbb {C}}^d)</span>. We compute the Euler characteristic of the Coxeter cohomology (the alternating sum of the ranks of the Coxeter cohomology groups) of several representations of <span>(S_n)</span>. In particular, the aforementioned tensor representation, and also several classes of irreducible representations of <span>(S_n)</span>. Although the problem and its motivation are algebraic and topological in nature, the techniques used are largely combinatorial.</p>","PeriodicalId":14926,"journal":{"name":"Journal of Algebraic Combinatorics","volume":"49 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140564887","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-04-03DOI: 10.1007/s10801-024-01311-4
Abstract
A skew morphism of a finite group A is a permutation (varphi ) of A fixing the identity element and for which there is an integer-valued function (pi ) on A such that (varphi (ab)=varphi (a)varphi ^{pi (a)}(b)) for all (a, b in A). A skew morphism (varphi ) of A is smooth if the associated power function (pi ) is constant on the orbits of (varphi ), that is, (pi (varphi (a))equiv pi (a)pmod {|varphi |}) for all (ain A). In this paper, we show that every skew morphism of a cyclic group of order n is smooth if and only if (n=2^en_1), where (0 le e le 4) and (n_1) is an odd square-free number. A partial solution to a similar problem on non-cyclic abelian groups is also given.
Abstract 有限群 A 的偏斜变形是 A 的一个固定同元素的置换(permutation (varphi)),对于这个置换,A 上有一个整数值函数 (pi),使得 (varphi(ab)=varphi(a)varphi ^{pi (a)}(b)) for all (a, b in A) 。如果相关的幂函数 (pi ) 在 (varphi ) 的轨道上是常数,即 (pi (varphi (a))equiv pi (a)pmod {|varphi |}) for all (ain A) ,那么 A 的倾斜变形 (varphi ) 是平稳的。在本文中,我们证明了当且仅当 (n=2^en_1) ,其中 (0 le e le 4) 和 (n_1) 是奇数无平方数时,阶数为 n 的循环群的每个倾斜态都是光滑的。此外,还给出了非循环无性系群类似问题的部分解。
{"title":"Classification of cyclic groups underlying only smooth skew morphisms","authors":"","doi":"10.1007/s10801-024-01311-4","DOIUrl":"https://doi.org/10.1007/s10801-024-01311-4","url":null,"abstract":"<h3>Abstract</h3> <p>A skew morphism of a finite group <em>A</em> is a permutation <span> <span>(varphi )</span> </span> of <em>A</em> fixing the identity element and for which there is an integer-valued function <span> <span>(pi )</span> </span> on <em>A</em> such that <span> <span>(varphi (ab)=varphi (a)varphi ^{pi (a)}(b))</span> </span> for all <span> <span>(a, b in A)</span> </span>. A skew morphism <span> <span>(varphi )</span> </span> of <em>A</em> is smooth if the associated power function <span> <span>(pi )</span> </span> is constant on the orbits of <span> <span>(varphi )</span> </span>, that is, <span> <span>(pi (varphi (a))equiv pi (a)pmod {|varphi |})</span> </span> for all <span> <span>(ain A)</span> </span>. In this paper, we show that every skew morphism of a cyclic group of order <em>n</em> is smooth if and only if <span> <span>(n=2^en_1)</span> </span>, where <span> <span>(0 le e le 4)</span> </span> and <span> <span>(n_1)</span> </span> is an odd square-free number. A partial solution to a similar problem on non-cyclic abelian groups is also given.</p>","PeriodicalId":14926,"journal":{"name":"Journal of Algebraic Combinatorics","volume":"69 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140597569","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-04-02DOI: 10.1007/s10801-024-01312-3
Yuefeng Yang, Qing Zeng, Kaishun Wang
Weakly distance-regular digraphs are a natural directed version of distance-regular graphs. In Wang and Suzuki (Discrete Math 264:225–236, 2003), the third author and Suzuki proposed a question when an orientation of a distance-regular graph defines a weakly distance-regular digraph. In this paper, we initiate this project and classify all commutative weakly distance-regular digraphs whose underlying graphs are Hamming graphs, folded n-cubes and Doob graphs, respectively.
弱距离规则数图是距离规则图的自然有向版本。在 Wang and Suzuki (Discrete Math 264:225-236, 2003)一文中,第三作者和铃木提出了一个问题:距离规则图的定向何时定义为弱距离规则数图?在本文中,我们启动了这一项目,并对所有底图分别是汉明图、折叠 n 立方图和 Doob 图的交换弱距离正则数图进行了分类。
{"title":"Weakly distance-regular digraphs whose underlying graphs are distance-regular, I","authors":"Yuefeng Yang, Qing Zeng, Kaishun Wang","doi":"10.1007/s10801-024-01312-3","DOIUrl":"https://doi.org/10.1007/s10801-024-01312-3","url":null,"abstract":"<p>Weakly distance-regular digraphs are a natural directed version of distance-regular graphs. In Wang and Suzuki (Discrete Math 264:225–236, 2003), the third author and Suzuki proposed a question when an orientation of a distance-regular graph defines a weakly distance-regular digraph. In this paper, we initiate this project and classify all commutative weakly distance-regular digraphs whose underlying graphs are Hamming graphs, folded <i>n</i>-cubes and Doob graphs, respectively.</p>","PeriodicalId":14926,"journal":{"name":"Journal of Algebraic Combinatorics","volume":"62 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140597571","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-04-01DOI: 10.1007/s10801-024-01310-5
S. Anukumar Kathirvel, Peter J. Cameron, T. Tamizh Chelvam
Let G be a finite group with identity e and ( H ne {e}) be a subgroup of G. The generalized non-coprime graph (varGamma _{G,H}) of ( G ) with respect to (H) is the simple undirected graph with (G setminus {e }) as the vertex set and two distinct vertices ( x ) and ( y) are adjacent if and only if (gcd (|x|,|y|) ne 1) and either (x in H) or (y in H), where |x| is the order of (xin G). In this paper, we study certain graph theoretical properties of generalized non-coprime graphs of finite groups, concentrating on cyclic groups. More specifically, we obtain necessary and sufficient conditions for the generalized non-coprime graph of a cyclic group to be in the class of stars, paths, triangle-free, complete bipartite, complete, split, claw-free, chordal or perfect graphs. Then we show that widening the class of groups to all finite nilpotent groups gives us no new graphs, but we give as an example of contrasting behaviour the class of EPPO groups (those in which all elements have prime power order). We conclude with a connection to the Gruenberg–Kegel graph.
让 G 是一个有限群,其特征是 e,而 H 是 G 的一个子群。当且仅当((gcd (|x|. |y|)ne 1) 和((gcd (|x|. |y|)ne 1) 中的任意一个)且((gcd (|x|. |y|)ne 1) 和((gcd (|x|. |y|)ne 1) 中的任意一个)时,(gcd (|x|. |y|)ne 1) 是(gcd (|x|、|)且(x 在 H 中)或(y 在 H 中)是相邻的,其中 |x| 是(x 在 G 中)的阶。)在本文中,我们研究了有限群的广义非彗星图的某些图论性质,主要集中在循环群上。更具体地说,我们得到了循环群的广义非彗星图属于星图、路径图、无三角形图、完全双方图、完全图、分裂图、无爪图、弦图或完美图类的必要条件和充分条件。然后我们证明,将群的类别扩大到所有有限零能群,并不会得到新的图形,但我们举出了 EPPO 群(所有元素都具有素幂阶的群)这类图形作为对比行为的例子。最后,我们将其与格伦伯格-凯格尔图联系起来。
{"title":"Generalized non-coprime graphs of groups","authors":"S. Anukumar Kathirvel, Peter J. Cameron, T. Tamizh Chelvam","doi":"10.1007/s10801-024-01310-5","DOIUrl":"https://doi.org/10.1007/s10801-024-01310-5","url":null,"abstract":"<p>Let <i>G</i> be a finite group with identity <i>e</i> and <span>( H ne {e})</span> be a subgroup of <i>G</i>. The generalized non-coprime graph <span>(varGamma _{G,H})</span> of <span>( G )</span> with respect to <span>(H)</span> is the simple undirected graph with <span>(G setminus {e })</span> as the vertex set and two distinct vertices <span>( x )</span> and <span>( y)</span> are adjacent if and only if <span>(gcd (|x|,|y|) ne 1)</span> and either <span>(x in H)</span> or <span>(y in H)</span>, where |<i>x</i>| is the order of <span>(xin G)</span>. In this paper, we study certain graph theoretical properties of generalized non-coprime graphs of finite groups, concentrating on cyclic groups. More specifically, we obtain necessary and sufficient conditions for the generalized non-coprime graph of a cyclic group to be in the class of stars, paths, triangle-free, complete bipartite, complete, split, claw-free, chordal or perfect graphs. Then we show that widening the class of groups to all finite nilpotent groups gives us no new graphs, but we give as an example of contrasting behaviour the class of EPPO groups (those in which all elements have prime power order). We conclude with a connection to the Gruenberg–Kegel graph.\u0000</p>","PeriodicalId":14926,"journal":{"name":"Journal of Algebraic Combinatorics","volume":"25 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140597289","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-03-29DOI: 10.1007/s10801-024-01313-2
Marian Aprodu, Gavril Farkas, Claudiu Raicu, Alessio Sammartano, Alexander I. Suciu
Each connected graded, graded-commutative algebra A of finite type over a field (Bbbk ) of characteristic zero defines a complex of finitely generated, graded modules over a symmetric algebra, whose homology graded modules are called the (higher) Koszul modules of A. In this note, we investigate the geometry of the support loci of these modules, called the resonance schemes of the algebra. When (A=Bbbk langle Delta rangle ) is the exterior Stanley–Reisner algebra associated to a finite simplicial complex (Delta ), we show that the resonance schemes are reduced. We also compute the Hilbert series of the Koszul modules and give bounds on the regularity and projective dimension of these graded modules. This leads to a relationship between resonance and Hilbert series that generalizes a known formula for the Chen ranks of a right-angled Artin group.
特征为零的域(Bbbk )上的每个有限类型的连通分级、分级交换代数 A 定义了一个对称代数上有限生成的分级模块复数,其同调分级模块被称为 A 的(高等)Koszul 模块。当 (A=Bbbk langle Delta rangle )是与有限单纯复数 (Delta )相关的外部斯坦利-雷斯纳代数时,我们证明共振方案是还原的。我们还计算了科斯祖尔模块的希尔伯特数列,并给出了这些分级模块的正则性和投影维数的边界。这导致了共振与希尔伯特数列之间的关系,而共振与希尔伯特数列概括了已知的直角阿尔丁群的陈等级公式。
{"title":"Higher resonance schemes and Koszul modules of simplicial complexes","authors":"Marian Aprodu, Gavril Farkas, Claudiu Raicu, Alessio Sammartano, Alexander I. Suciu","doi":"10.1007/s10801-024-01313-2","DOIUrl":"https://doi.org/10.1007/s10801-024-01313-2","url":null,"abstract":"<p>Each connected graded, graded-commutative algebra <i>A</i> of finite type over a field <span>(Bbbk )</span> of characteristic zero defines a complex of finitely generated, graded modules over a symmetric algebra, whose homology graded modules are called the <i>(higher) Koszul modules</i> of <i>A</i>. In this note, we investigate the geometry of the support loci of these modules, called the <i>resonance schemes</i> of the algebra. When <span>(A=Bbbk langle Delta rangle )</span> is the exterior Stanley–Reisner algebra associated to a finite simplicial complex <span>(Delta )</span>, we show that the resonance schemes are reduced. We also compute the Hilbert series of the Koszul modules and give bounds on the regularity and projective dimension of these graded modules. This leads to a relationship between resonance and Hilbert series that generalizes a known formula for the Chen ranks of a right-angled Artin group.\u0000</p>","PeriodicalId":14926,"journal":{"name":"Journal of Algebraic Combinatorics","volume":"97 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140597095","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-03-27DOI: 10.1007/s10801-024-01308-z
Mohsen Aliabadi, Shira Zerbib
We formulate and prove matroid analogues of results concerning matchings in groups. A matching in an abelian group ((G,+)) is a bijection (f:Arightarrow B) between two finite subsets A, B of G satisfying (a+f(a)notin A) for all (ain A). A group G has the matching property if for every two finite subsets (A,B subset G) of the same size with (0 notin B), there exists a matching from A to B. In Losonczy (Adv Appl Math 20(3):385–391, 1998) it was proved that an abelian group has the matching property if and only if it is torsion-free or cyclic of prime order. Here we consider a similar question in a matroid setting. We introduce an analogous notion of matching between matroids whose ground sets are subsets of an abelian group G, and we obtain criteria for the existence of such matchings. Our tools are classical theorems in matroid theory, group theory and additive number theory.
我们提出并证明了有关群中匹配结果的类比矩阵。一个无阶梯群((G,+))中的匹配是 G 的两个有限子集 A、B 之间的双投影(f:A/rightarrow B) 满足所有 (a/in A)的 (a+f(a)notin A )。Losonczy (Adv Appl Math 20(3):385-391, 1998)证明,如果且只有当无孪生群是无扭的或素阶循环群时,无孪生群才具有匹配属性。在此,我们考虑在矩阵环境中的类似问题。我们引入了一个类似的矩阵之间匹配的概念,这些矩阵的基集是一个无边群 G 的子集,我们还得到了这种匹配存在的标准。我们的工具是矩阵理论、群论和加数理论中的经典定理。
{"title":"Matchings in matroids over abelian groups","authors":"Mohsen Aliabadi, Shira Zerbib","doi":"10.1007/s10801-024-01308-z","DOIUrl":"https://doi.org/10.1007/s10801-024-01308-z","url":null,"abstract":"<p>We formulate and prove matroid analogues of results concerning matchings in groups. A matching in an abelian group <span>((G,+))</span> is a bijection <span>(f:Arightarrow B)</span> between two finite subsets <i>A</i>, <i>B</i> of <i>G</i> satisfying <span>(a+f(a)notin A)</span> for all <span>(ain A)</span>. A group <i>G</i> has the matching property if for every two finite subsets <span>(A,B subset G)</span> of the same size with <span>(0 notin B)</span>, there exists a matching from <i>A</i> to <i>B</i>. In Losonczy (Adv Appl Math 20(3):385–391, 1998) it was proved that an abelian group has the matching property if and only if it is torsion-free or cyclic of prime order. Here we consider a similar question in a matroid setting. We introduce an analogous notion of matching between matroids whose ground sets are subsets of an abelian group <i>G</i>, and we obtain criteria for the existence of such matchings. Our tools are classical theorems in matroid theory, group theory and additive number theory.</p>","PeriodicalId":14926,"journal":{"name":"Journal of Algebraic Combinatorics","volume":"2 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140316727","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-03-02DOI: 10.1007/s10801-024-01297-z
Jing Jian Li, Xiao Qian Zhang, Jin-Xin Zhou
A bipartite graph (Gamma ) is a bi-Cayley graph over a group H if (Hleqslant textrm{Aut}Gamma ) acts regularly on each part of (Gamma ). A bi-Cayley graph (Gamma ) is said to be a normal bi-Cayley graph over H if (Hunlhd textrm{Aut}Gamma ), and bi-primitive if the bipartition preserving subgroup of (textrm{Aut}Gamma ) acts primitively on each part of (Gamma ). In this paper, a classification is given for 2-arc-transitive bi-Cayley graphs which are bi-primitive and non-normal.
{"title":"Bi-primitive 2-arc-transitive bi-Cayley graphs","authors":"Jing Jian Li, Xiao Qian Zhang, Jin-Xin Zhou","doi":"10.1007/s10801-024-01297-z","DOIUrl":"https://doi.org/10.1007/s10801-024-01297-z","url":null,"abstract":"<p>A bipartite graph <span>(Gamma )</span> is a <i>bi-Cayley graph</i> over a group <i>H</i> if <span>(Hleqslant textrm{Aut}Gamma )</span> acts regularly on each part of <span>(Gamma )</span>. A bi-Cayley graph <span>(Gamma )</span> is said to be a <i>normal bi-Cayley graph over H</i> if <span>(Hunlhd textrm{Aut}Gamma )</span>, and <i>bi-primitive</i> if the bipartition preserving subgroup of <span>(textrm{Aut}Gamma )</span> acts primitively on each part of <span>(Gamma )</span>. In this paper, a classification is given for 2-arc-transitive bi-Cayley graphs which are bi-primitive and non-normal.</p>","PeriodicalId":14926,"journal":{"name":"Journal of Algebraic Combinatorics","volume":"177 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-03-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140018431","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-03-02DOI: 10.1007/s10801-024-01298-y
Xiaomeng Wang, Shou-Jun Xu, Sanming Zhou
Let (Gamma = (V, E)) be a graph and a, b 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. A (0, 1)-regular set is called a perfect code, an efficient dominating set, or an independent perfect dominating set. A subset D of a group G is called an (a, b)-regular set of G if it is an (a, b)-regular set in some Cayley graph of G, and an (a, b)-regular set in a Cayley graph of G is called a subgroup (a, b)-regular set if it is also a subgroup of G. In this paper, we study (a, b)-regular sets in Cayley graphs with a focus on (0, k)-regular sets, where (k ge 1) is an integer. Among other things, we determine when a non-trivial proper normal subgroup of a group is a (0, k)-regular set of the group. We also determine all subgroup (0, k)-regular sets of dihedral groups and generalized quaternion groups. We obtain necessary and sufficient conditions for a hypercube or the Cartesian product of n copies of the cycle of length p to admit (0, k)-regular sets, where p is an odd prime. Our results generalize several known results from perfect codes to (0, k)-regular sets.
让 (Gamma = (V, E)) 是一个图,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。一个 (0, 1) -regular set 被称为完美编码、有效支配集或独立完美支配集。群 G 的子集 D 如果是 G 的某个 Cayley 图中的 (a, b) 不规则集合,则称为 G 的 (a, b) 不规则集合;G 的 Cayley 图中的 (a, b) 不规则集合如果也是 G 的子群,则称为子群 (a, b) 不规则集合。本文将研究 Cayley 图中的(a, b)-正则集合,重点是(0, k)-正则集合,其中(k)是整数。其中,我们确定了一个群的非琐碎适当正则子群何时是该群的(0,k)-正则集合。我们还确定了二面体群和广义四元组的所有子群(0,k)-正则集合。我们获得了长度为 p 的超立方体或循环的 n 个副本的笛卡儿积接纳 (0, k) 不规则集合的必要条件和充分条件,其中 p 是奇素数。我们的结果将完美码的几个已知结果推广到了(0,k)-规则集。
{"title":"On regular sets in Cayley graphs","authors":"Xiaomeng Wang, Shou-Jun Xu, Sanming Zhou","doi":"10.1007/s10801-024-01298-y","DOIUrl":"https://doi.org/10.1007/s10801-024-01298-y","url":null,"abstract":"<p>Let <span>(Gamma = (V, E))</span> be a graph and <i>a</i>, <i>b</i> 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>. A (0, 1)-regular set is called a perfect code, an efficient dominating set, or an independent perfect dominating set. A subset <i>D</i> of a group <i>G</i> is called an (<i>a</i>, <i>b</i>)-regular set of <i>G</i> if it is an (<i>a</i>, <i>b</i>)-regular set in some Cayley graph of <i>G</i>, and an (<i>a</i>, <i>b</i>)-regular set in a Cayley graph of <i>G</i> is called a subgroup (<i>a</i>, <i>b</i>)-regular set if it is also a subgroup of <i>G</i>. In this paper, we study (<i>a</i>, <i>b</i>)-regular sets in Cayley graphs with a focus on (0, <i>k</i>)-regular sets, where <span>(k ge 1)</span> is an integer. Among other things, we determine when a non-trivial proper normal subgroup of a group is a (0, <i>k</i>)-regular set of the group. We also determine all subgroup (0, <i>k</i>)-regular sets of dihedral groups and generalized quaternion groups. We obtain necessary and sufficient conditions for a hypercube or the Cartesian product of <i>n</i> copies of the cycle of length <i>p</i> to admit (0, <i>k</i>)-regular sets, where <i>p</i> is an odd prime. Our results generalize several known results from perfect codes to (0, <i>k</i>)-regular sets.</p>","PeriodicalId":14926,"journal":{"name":"Journal of Algebraic Combinatorics","volume":"261 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-03-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140018476","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-02-26DOI: 10.1007/s10801-024-01306-1
Yanliang Cheng, Yong Shao, Lingli Zeng
We first determine the structure of the power digraphs of completely 0-simple semigroups, and then some properties of their power graphs are given. As the main result in this paper, using Cameron and Ghosh’s theorem about power graphs of abelian groups, we obtain a characterization that two (G^{0})-normal completely 0-simple orthodox semigroups S and T with abelian group (mathcal {H})-classes are isomorphic based on their power graphs. We also present an algorithm to determine that S and T are isomorphic or not.
我们首先确定了完全 0 简单半群的幂图结构,然后给出了它们的幂图的一些性质。作为本文的主要结果,我们利用卡梅隆和戈什关于无边际群幂图的定理,得到了两个具有无边际群((mathcal {H})类的 (G^{0})-normal 完全 0-simple 正交半群 S 和 T 基于它们的幂图是同构的。我们还提出了一种判定 S 和 T 是否同构的算法。
{"title":"Power graphs of a class of completely 0-simple semigroups","authors":"Yanliang Cheng, Yong Shao, Lingli Zeng","doi":"10.1007/s10801-024-01306-1","DOIUrl":"https://doi.org/10.1007/s10801-024-01306-1","url":null,"abstract":"<p>We first determine the structure of the power digraphs of completely 0-simple semigroups, and then some properties of their power graphs are given. As the main result in this paper, using Cameron and Ghosh’s theorem about power graphs of abelian groups, we obtain a characterization that two <span>(G^{0})</span>-normal completely 0-simple orthodox semigroups <i>S</i> and <i>T</i> with abelian group <span>(mathcal {H})</span>-classes are isomorphic based on their power graphs. We also present an algorithm to determine that <i>S</i> and <i>T</i> are isomorphic or not.</p>","PeriodicalId":14926,"journal":{"name":"Journal of Algebraic Combinatorics","volume":"22 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140006981","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}