On r-Noncommuting Graph of Finite Rings

Axioms

卷:10

Yilun Shang¹  Rajat Kanti Nath²  Monalisha Sharma²  Parama Dutta³ 

[1] Department of Computer and Information Sciences, Northumbria University, Newcastle NE1 8ST, UK;

[2] Department of Mathematical Sciences, Tezpur University, Sonitpur 784028, India;

[3] Department of Mathematics, Lakhimpur Girls’ College, Lakhimpur 787031, India;

关键词: finite ring;    noncommuting graph;    isoclinism;   

DOI  :  10.3390/axioms10030233

来源: DOAJ

【 摘 要 】

Let R be a finite ring and $r\in R$. The r-noncommuting graph of R, denoted by ${\mathsf{\Gamma }}_R^r$, is a simple undirected graph whose vertex set is R and two vertices x and y are adjacent if and only if $[x,y]\ne r$ and $[x,y]\ne -r$. In this paper, we obtain expressions for vertex degrees and show that ${\mathsf{\Gamma }}_R^r$ is neither a regular graph nor a lollipop graph if R is noncommutative. We characterize finite noncommutative rings such that ${\mathsf{\Gamma }}_R^r$ is a tree, in particular a star graph. It is also shown that ${\mathsf{\Gamma }}_{R_1}^r$ and ${\mathsf{\Gamma }}_{R_2}^{\psi \left(r\right)}$ are isomorphic if $R_1$ and $R_2$ are two isoclinic rings with isoclinism $(\varphi ,\psi )$. Further, we consider the induced subgraph ${\Delta }_R^r$ of ${\mathsf{\Gamma }}_R^r$ (induced by the non-central elements of R) and obtain results on clique number and diameter of ${\Delta }_R^r$ along with certain characterizations of finite noncommutative rings such that ${\Delta }_R^r$ is n-regular for some positive integer n. As applications of our results, we characterize certain finite noncommutative rings such that their noncommuting graphs are n-regular for $n\le 6$.

【 授权许可】

Unknown