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arXiv · 1907.10350

On r-noncommuting graph of finite rings

Rajat Kanti Nath·Monalisha Sharma·Parama Dutta·Yilun Shang
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Abstract

Let $R$ be a finite ring and $r\in R$. The $r$-noncommuting graph of $R$, denoted by $\Gamma_R^r$, is a simple undirected graph whose vertex set is $R$ and two vertices $x$ and $y$ are adjacent if and only if $[x,y] \neq r$ and $-r$. In this paper, we study several properties of $\Gamma_R^r$. We show that $\Gamma_R^r$ is not a regular graph, a lollipop graph and complete bipartite graph. Further, we consider an induced subgraph of $\Gamma_R^r$ (induced by the non-central elements of $R$) and obtained some characterizations of $R$.

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Rajat Kanti Nath, Monalisha Sharma, Parama Dutta, Yilun Shang. 2019-07-24. On r-noncommuting graph of finite rings. https://arxiv.org/abs/1907.10350

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