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Generalized Non-Braid Graphs of Rings

*Era Setya Cahyati  -  Dept. of Mathematics, Gadjah Mada University, Indonesia, Indonesia
Rambu Maya Imung Maharani  -  Departement of Mathematics UGM
Sri Nurhayati  -  Department of Mathematics UGM
Yeni Susanti  -  Depatment of Mathematics UGM

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Abstract

In this paper, we introduce the definition of generalized non-braid graph of a given ring. Let $R$ be a ring and let $k$ be a natural number. By generalized braider of $R$ we mean the set $B^k(R):=\{x \in R~|~\forall
y \in R,~ (xyx)^k = (yxy)^k\}$. The generalized non-braid graph of $R$, denoted by $G_k(\Upsilon_R)$, is a simple undirected graph with vertex set $R\backslash B^k(R)$ and two distinct vertices $x$ and $y$ are adjacent if and only if $(xyx)^k \neq (yxy)^k$. In particular, we investigate some properties of generalized non-braid graph $G_k(\Upsilon_{\mathbb{Z}_n})$ of the ring $\mathbb{Z}_n$.

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Keywords: generalized nonbraid graph; ring

Article Metrics:

  1. Abdollahi, A., Akbari, S., Maimani, H.R., 2006, Non-commuting graph of a group, Journal of Algebra 298,
  2. –492
  3. Cahyati, E.S, Fadhiilah, R.A, & Candra, A.D Wijayanti, I.E., 2021, Non-Braid Graph of Ring $mathbb{Z}_n$. Jurnal Teori dan Aplikasi Matematika. Vol. 6. No. 1. pp. 106-116
  4. Cayley, A., 1878, {Desiderata and suggestions: No. 2. The theory of groups: Graphical representation}, {it Am. J. Math.}, textbf{1}, 174--176
  5. Erfanian, A., Khashyarmanesh, K., & Nafar, Kh., 2015, Non-commuting Graphs of Rings. Discrete Mathematics, Algorithms and Applications. Vol. 7. No. 3. 1550027
  6. Erfanian, A., Kakeri, F., & Mansoori, F., 2016, Generalization of the Non-Commuting graph of a Group via a Normal Subbgroup. Science Asia 42, 231-235

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