GENERALIZED NON-BRAID GRAPHS OF RINGS - Cahyati, Era Setya; Maharani, Rambu Maya Imung; Nurhayati, Sri; Susanti, Yeni (Journal of Fundamental Mathematics and Applications (JFMA), 2022)
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 $G_k(Υ_R)$, 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 $(xyx)^k ≠ (yxy)^k$. In particular, we investigate some properties of generalized non-braid graph $G_k(Υ_{\mathbb{Z}_n})$ of the ring $\mathbb{Z}_n$.