On the commuting graph of some non-commutative rings with unity
الموضوعات :
F. Ramezani
1
(Department of Basic science , Imam Khomeini International University, Qazvin, Iran)
E. Vatandoost
2
(Department of Basic science , Imam Khomeini International University, Qazvin, Iran)
الکلمات المفتاحية: Commuting graphs, non-commutative rings, non-connected graphs,
ملخص المقالة :
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 commutinggraphs 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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