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All Journal JTAM (Jurnal Teori dan Aplikasi Matematika) Journal of Fundamental Mathematics and Applications (JFMA)
Cahyati, Era Setya
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Decision Sciences, Operations Research & Management Mathematics

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GENERALIZED NON-BRAID GRAPHS OF RINGS Cahyati, Era Setya; Maharani, Rambu Maya Imung; Nurhayati, Sri; Susanti, Yeni
Journal of Fundamental Mathematics and Applications (JFMA) Vol 5, No 2 (2022)
Publisher : Diponegoro University

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.14710/jfma.v5i2.14152

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 $\mathbb{Z}_n$.
Non-Braid Graphs of Ring Zn Cahyati, Era Setya; Fadhiilah, Rizka 'Abid; Candra Bp, Ananditya Dwi; Wijayanti, Indah Emilia
JTAM (Jurnal Teori dan Aplikasi Matematika) Vol 6, No 1 (2022): January
Publisher : Universitas Muhammadiyah Mataram

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.31764/jtam.v6i1.5559

Abstract

The research in graph theory has been widened by combining it with ring. In this paper, we introduce the definition of a non-braid graph of a ring.  The non-braid graph of a ring R, denoted by YR, is a simple graph with a vertex set R\B(R), where B(R) is the set of x in R such that  xyx=yxy for all y in R.  Two distinct vertices x and y are adjacent if and only if xyx not equal to yxy.  The method that we use to observe the non-braid graphs of Zn is by seeing the adjacency of the vertices and its braider.  The main objective of this paper is to prove the completeness and connectedness of the non-braid graph of ring Zn. We prove that if n is a prime number, the non-braid graph of Zn is a complete graph. For all n greater than equal to 3,  the non-braid graph of Zn is a connected graph.
Co-Authors Candra Bp, Ananditya Dwi Fadhiilah, Rizka 'Abid Indah Emilia Wijayanti Maharani, Rambu Maya Imung Sri Nurhayati Yeni Susanti