{"title":"Well-covered unitary Cayley graphs of matrix rings over finite fields and applications","authors":"Shahin Rahimi, Ashkan Nikseresht","doi":"10.1016/j.ffa.2024.102428","DOIUrl":null,"url":null,"abstract":"<div><p>Suppose that <em>F</em> is a finite field and <span><math><mi>R</mi><mo>=</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>F</mi><mo>)</mo></math></span> is the ring of <em>n</em>-square matrices over <em>F</em>. Here we characterize when the Cayley graph of the additive group of <em>R</em> with respect to the set of invertible elements of <em>R</em>, called the unitary Cayley graph of <em>R</em>, is well-covered. Then we apply this to characterize all finite rings with identity whose unitary Cayley graph is well-covered or Cohen-Macaulay.</p></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"96 ","pages":"Article 102428"},"PeriodicalIF":1.2000,"publicationDate":"2024-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Finite Fields and Their Applications","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S1071579724000674","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
Suppose that F is a finite field and is the ring of n-square matrices over F. Here we characterize when the Cayley graph of the additive group of R with respect to the set of invertible elements of R, called the unitary Cayley graph of R, is well-covered. Then we apply this to characterize all finite rings with identity whose unitary Cayley graph is well-covered or Cohen-Macaulay.
假设 F 是有限域，R=Mn(F) 是 F 上的 n 方矩阵环。在此，我们将描述 R 的加法群关于 R 的可逆元素集的 Cayley 图（称为 R 的单元 Cayley 图）何时被很好地覆盖。然后，我们将其应用于表征所有具有同一性的有限环，这些有限环的单元 Cayley 图都是井盖图或 Cohen-Macaulay 图。
期刊介绍：
Finite Fields and Their Applications is a peer-reviewed technical journal publishing papers in finite field theory as well as in applications of finite fields. As a result of applications in a wide variety of areas, finite fields are increasingly important in several areas of mathematics, including linear and abstract algebra, number theory and algebraic geometry, as well as in computer science, statistics, information theory, and engineering.
For cohesion, and because so many applications rely on various theoretical properties of finite fields, it is essential that there be a core of high-quality papers on theoretical aspects. In addition, since much of the vitality of the area comes from computational problems, the journal publishes papers on computational aspects of finite fields as well as on algorithms and complexity of finite field-related methods.
The journal also publishes papers in various applications including, but not limited to, algebraic coding theory, cryptology, combinatorial design theory, pseudorandom number generation, and linear recurring sequences. There are other areas of application to be included, but the important point is that finite fields play a nontrivial role in the theory, application, or algorithm.
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