{"title":"准阿贝尔半凯利图上的完美状态转移","authors":"Shixin Wang, Majid Arezoomand, Tao Feng","doi":"10.1007/s10801-023-01288-6","DOIUrl":null,"url":null,"abstract":"<p>Perfect state transfer on graphs has attracted extensive attention due to its application in quantum information and quantum computation. A graph is a semi-Cayley graph over a group <i>G</i> if it admits <i>G</i> as a semiregular subgroup of the full automorphism group with two orbits of equal size. A semi-Cayley graph <i>SC</i>(<i>G</i>, <i>R</i>, <i>L</i>, <i>S</i>) is called quasi-abelian if each of <i>R</i>, <i>L</i> and <i>S</i> is a union of some conjugacy classes of <i>G</i>. This paper establishes necessary and sufficient conditions for a quasi-abelian semi-Cayley graph to have perfect state transfer. As a corollary, it is shown that if a quasi-abelian semi-Cayley graph over a finite group <i>G</i> has perfect state transfer between distinct vertices <i>g</i> and <i>h</i>, and <i>G</i> has a faithful irreducible character, then <span>\\(gh^{-1}\\)</span> lies in the center of <i>G</i> and <span>\\(gh=hg\\)</span>; in particular, <i>G</i> cannot be a non-abelian simple group. We also characterize quasi-abelian Cayley graphs over arbitrary groups having perfect state transfer, which is a generalization of previous works on Cayley graphs over abelian groups, dihedral groups, semi-dihedral groups and dicyclic groups.</p>","PeriodicalId":14926,"journal":{"name":"Journal of Algebraic Combinatorics","volume":"106 1","pages":""},"PeriodicalIF":0.6000,"publicationDate":"2024-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Perfect state transfer on quasi-abelian semi-Cayley graphs\",\"authors\":\"Shixin Wang, Majid Arezoomand, Tao Feng\",\"doi\":\"10.1007/s10801-023-01288-6\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>Perfect state transfer on graphs has attracted extensive attention due to its application in quantum information and quantum computation. A graph is a semi-Cayley graph over a group <i>G</i> if it admits <i>G</i> as a semiregular subgroup of the full automorphism group with two orbits of equal size. A semi-Cayley graph <i>SC</i>(<i>G</i>, <i>R</i>, <i>L</i>, <i>S</i>) is called quasi-abelian if each of <i>R</i>, <i>L</i> and <i>S</i> is a union of some conjugacy classes of <i>G</i>. This paper establishes necessary and sufficient conditions for a quasi-abelian semi-Cayley graph to have perfect state transfer. As a corollary, it is shown that if a quasi-abelian semi-Cayley graph over a finite group <i>G</i> has perfect state transfer between distinct vertices <i>g</i> and <i>h</i>, and <i>G</i> has a faithful irreducible character, then <span>\\\\(gh^{-1}\\\\)</span> lies in the center of <i>G</i> and <span>\\\\(gh=hg\\\\)</span>; in particular, <i>G</i> cannot be a non-abelian simple group. We also characterize quasi-abelian Cayley graphs over arbitrary groups having perfect state transfer, which is a generalization of previous works on Cayley graphs over abelian groups, dihedral groups, semi-dihedral groups and dicyclic groups.</p>\",\"PeriodicalId\":14926,\"journal\":{\"name\":\"Journal of Algebraic Combinatorics\",\"volume\":\"106 1\",\"pages\":\"\"},\"PeriodicalIF\":0.6000,\"publicationDate\":\"2024-01-20\",\"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/s10801-023-01288-6\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Algebraic Combinatorics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s10801-023-01288-6","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
摘要
图上的完美状态转移因其在量子信息和量子计算中的应用而受到广泛关注。如果一个图允许 G 作为全自形群的半圆子群,且有两个大小相等的轨道,那么这个图就是群 G 上的半 Cayley 图。如果 R、L 和 S 中的每一个都是 G 的某些共轭类的联合,则半 Cayley 图 SC(G, R, L, S) 被称为准阿贝尔图。作为推论,本文证明了如果一个有限群 G 上的准阿贝尔半凯利图在不同顶点 g 和 h 之间具有完美的状态转移,并且 G 具有忠实的不可还原性,那么 \(gh^{-1}\) 位于 G 的中心,并且 \(gh=hg\) ;特别地,G 不可能是一个非阿贝尔简单群。我们还描述了具有完美状态转移的任意群上的准阿贝尔 Cayley 图的特征,这是对以前关于无性群、二面群、半二面群和二环群上的 Cayley 图的研究的推广。
Perfect state transfer on quasi-abelian semi-Cayley graphs
Perfect state transfer on graphs has attracted extensive attention due to its application in quantum information and quantum computation. A graph is a semi-Cayley graph over a group G if it admits G as a semiregular subgroup of the full automorphism group with two orbits of equal size. A semi-Cayley graph SC(G, R, L, S) is called quasi-abelian if each of R, L and S is a union of some conjugacy classes of G. This paper establishes necessary and sufficient conditions for a quasi-abelian semi-Cayley graph to have perfect state transfer. As a corollary, it is shown that if a quasi-abelian semi-Cayley graph over a finite group G has perfect state transfer between distinct vertices g and h, and G has a faithful irreducible character, then \(gh^{-1}\) lies in the center of G and \(gh=hg\); in particular, G cannot be a non-abelian simple group. We also characterize quasi-abelian Cayley graphs over arbitrary groups having perfect state transfer, which is a generalization of previous works on Cayley graphs over abelian groups, dihedral groups, semi-dihedral groups and dicyclic groups.
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.