Ramezani, F., Vatandoost, E. (2016). On the commuting graph of some non-commutative rings with unity. Journal of Linear and Topological Algebra (JLTA), 05(04), 289-294.

F. Ramezani; E. Vatandoost. "On the commuting graph of some non-commutative rings with unity". Journal of Linear and Topological Algebra (JLTA), 05, 04, 2016, 289-294.

Ramezani, F., Vatandoost, E. (2016). 'On the commuting graph of some non-commutative rings with unity', Journal of Linear and Topological Algebra (JLTA), 05(04), pp. 289-294.

Ramezani, F., Vatandoost, E. On the commuting graph of some non-commutative rings with unity. Journal of Linear and Topological Algebra (JLTA), 2016; 05(04): 289-294.

On the commuting graph of some non-commutative rings with unity

^{}Department of Basic science , Imam Khomeini International University, Qazvin, Iran

Abstract

Let $R$ be a non-commutative ring with unity. The commuting graph of $R$ denoted by $\Gamma(R)$, is a graph with a vertex set $R\setminus Z(R)$ and two vertices $a$ and $b$ are adjacent if and only if $ab=ba$. In this paper, we investigate non-commutative rings with unity of order $p^n$ where $p$ is prime and $n \in \lbrace 4,5 \rbrace$. It is shown that, $\Gamma(R)$ is the disjoint union of complete graphs. Finally, we prove that there are exactly five commuting graphs of non-commutative rings with unity up to twenty vertices and they are $3K_2,3K_4,7K_2, K_2 \cup 2K_6$ and $4K_2 \cup K_6$.

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