On the commuting graph of non-commutative rings of order $p^nq$

محورهای موضوعی : Combinatorics, Graph theory

E. Vatandoost ¹ , F. Ramezani ² , A. Bahraini ³

1 - Faculty of Basic Science, Imam Khomeini International University, Qazvin, Iran
2 - Faculty of Basic Science, Imam Khomeini International University, Qazvin, Iran
3 - Department of Mathematics, Islamic Azad University, Central Tehran Branch, Tehran, Iran

تاریخ دریافت : 1393/02/27 تاریخ پذیرش : 1393/05/13 تاریخ انتشار : 1392/12/10

کلید واژه: Commuting graph, non-commutative ring, non-connected graph, algebraic graph,

چکیده مقاله :

Let $R$ be a non-commutative ring with unity. The commuting graph of $R$ denotedby $\Gamma(R)$, is a graph with vertex set $R\Z(R)$ and two vertices $a$ and $b$ are adjacent iff $ab=ba$.In this paper, we consider the commuting graph of non-commutative rings of order pq and $p^2q$with Z(R) = 0 and non-commutative rings with unity of order $p^3q$. It is proved that $C_R(a)$is a commutative ring for every $0\neq a \in R\Z(R)$. Also it is shown that if $a,b\in R\Z(R)$and $ab\neq ba$, then $C_R(a)\cap C_R(b)= Z(R)$. We show that the commuting graph $\Gamma(R)$ is thedisjoint union of $k$ copies of the complete graph and so is not a connected graph.

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