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

Citation Format:
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 $\mathbb{Z}_n$.

Fulltext View|Download
Keywords: generalized nonbraid graph; ring

Article Metrics:

Article Info
Section: FUNDAMENTAL MATHEMATICS AND APPLICATIONS
Language : EN
Recent articles
SOME NEW REMARKS ON POWER SUMS ANALISIS PERBANDINGAN METODE ARIMA DAN DOUBLE EXPONENTIAL SMOOTHING DARI BROWN PADA PERAMALAN INFLASI DI INDONESIA ESTIMASI PARAMETER MODEL ARCH MENGGUNAKAN METODE BAYES More recent articles
  1. Cayley, “Desiderata and suggestions: No. 2. the theory of groups: Graphical representation,” American Journal of Mathematics, vol. 1, no. 2, pp. 174–176, 1878.[Online]. Available: http://www.jstor.org/stable/2369306
  2. A. Abdollahi, S. Akbari, and H. Maimani, “Non-commuting graph of a group,” Journal of Algebra, vol. 298, no. 2, pp. 468–492, 2006. [Online]. Available: https://www.sciencedirect.com/science/article/pii/S002186930600113X
  3. J. C. M. Pezzott, “Groups whose non-commuting graph on a transversal is planar or toroidal,” Journal of Algebra and its Applications, vol. 21, no. 10, 2022
  4. S. Mukhtar, M. Salman, A. D. Maden, and M. Ur Rehman, “Metric properties of non-commuting graph associated to two groups,” International Journal of Foundations of Computer Science, vol. 33, no. 6-7, 2022
  5. M. Salman, T. Noreen, M. U. Rehman, J. Cao, and M. Z. Abbas, “Non-commuting graph of the dihedral group determined by hosoya parameters,” Alexandria Engineering Jour-nal, vol. 61, no. 5, p. 3709 – 3717, 2022
  6. F. Ali, B. A. Rather, M. Sarfraz, A. Ullah, N. Fatima, and W. K. Mashwani, “Certain topological indices of non-commuting graphs for finite non-abelian groups,” Molecules, vol. 27, no. 18, 2022
  7. J. Kalita and S. Paul, “On the spectra of non-commuting graphs of certain class of groups,” Asian-European Journal of Mathematics, 2022
  8. F. Kakeri, A. Erfanian, and F. Mansoori, “Generalization of the non-commuting graph of a group via a normal subgroup,” ScienceAsia, vol. 42, p. 231, 06 2016
  9. A. Erfanian, K. Khashyarmanesh, and K. Nafar, “Non-commuting graphs of rings,” Dis-crete Mathematics, Algorithms and Applications, vol. 07, no. 03, p. 1550027, 2015
  10. S. Akbari, M. Ghandehari, M. Hadian, and A. Mohammadian, “On commuting graphs of semisimple rings,” Linear Algebra and its Applications, vol. 390, pp. 345–355, 2004. [On-line]. Available: https://www.sciencedirect.com/science/article/pii/S0024379504002393
  11. J. Dutta, D. K. Basnet, and R. K. Nath, “On generalized non-commuting graph of a finite ring,” Algebra Colloquium, vol. 25, no. 1, p. 149 – 160, 2018
  12. E. Cahyati, R. Fadhiilah, A. Candra, and I. Wijayanti, “Non-braid graph of ring Zn,” Jurnal Teori dan Aplikasi Matematika, vol. 6, no. 1, pp. 106–116, 2021

Last update:

No citation recorded.

Last update:

No citation recorded.