DOI: https://doi.org/10.14710/jfma.v5i2.14152

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, Indonesia
Sri Nurhayati  -  Department of Mathematics UGM, Indonesia
Yeni Susanti  -  Depatment of Mathematics UGM, Indonesia

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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

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Section: FUNDAMENTAL MATHEMATICS AND APPLICATIONS
Language : EN
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