^{}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 $4K_2 \cup K_6$.

[1] A. Abdollahi, Commuting graphs of full matrix rings over nite elds. Linear Algebra and its Applications. 428 (11) (2008), 2947-2954. [2] S. Akbari, M. Ghandehari, M. Hadian, A. Mohammadian, On commuting graphs of semisimple rings. Linear algebra and its applications. 390 (2004), 345-355. [3] S. Akbari, A. Mohammadian, H. Radjavi, P. Raja, On the diameters of commuting graphs. Linear algebra and its applications. 418(1) (2006), 161{176 [4] S. Akbari, P. Raja, Commuting graphs of some subsets in simple rings. Linear algebra and its applications. 416 (2) (2006), 1038-1047. [5] N. Biggs, Algebraic Graph Theory. Cambridge University Press, Cambridge, 1993. [6] K. E. Eldridge, Orders for nite noncommutative rings with unity. American Mathematical Monthly. (1968), 512-514. [7] G. R. Omidi, E. Vatandoost, On the commuting graph of rings. Journal of Algebra and Its Applications. 10 (03) (2011), 521-527. [8] E. Vatandoost, F. Ramezani , A. Bahraini, On the commuting graph of non-commutative rings of order pnq. J. Linear. Topol. Algebra, 03 (01) (2014), 1-6.