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

Subject Areas : Combinatorics, Graph theory

E. Vatandoost ¹ (Faculty of Basic Science, Imam Khomeini International University, Qazvin, Iran)
F. Ramezani ² (Faculty of Basic Science, Imam Khomeini International University, Qazvin, Iran)
A. Bahraini ³ (Department of Mathematics, Islamic Azad University, Central Tehran Branch, Tehran, Iran)

Received: 2014-05-17 Accepted : 2014-08-04 Published : 2014-03-01

Keywords: Commuting graph, non-commutative ring, non-connected graph, algebraic graph,

Abstract :

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.

References:

[1] A. Abdollahi, Commuting graphs of full matrix rings over finite fields, Linear Algebra Appl. 422 (2008), 654–658.

[2] S. Akbari, M. Ghandehari, M. Hadian, and A. Mohammadian, On commuting graphs of semisimple rings, Linear Algebra and its Applications, 390 (2004), 345-355.

[3] S. Akbari, A. Mohammadian, H. Radjavi, and P. Raja, On the diameters of commuting graphs, Linear Algebra and its Applications, 418 (2006), 161-176.

[4] S. Akbari and P. Raja, Commuting graphs of some subsets in simple rings, Linear Algebra and its Applications, 416 (2006), 1038-1047.

[5] N. Biggs, Algebraic Graph Theory, Cambridge University Press, Cambridge, 1993.

[6] D. M. Cvetkovi´c, M. Doob, and H. Sachs, Spectra of graphs - Theory and applications, 3rd edition, Johann Ambrosius Barth Verlag, Heidelberg-Leipzig, 1995.

[7] J. B. Derr, G. F. Orr, and Paul S. Peck, Noncommutative rings of order p 4 , Journal of Pure and Applied Algebra, 97 (1994), 109-116.

[8] K. E. Eldridge, Orders for finite noncommutative rings with unity, The American Mathematical Monthly, 75(5). (May, 1968), 512-514.

[9] G. R. Omidi, E. Vatandoost, On the commuting graph of rings, Journal of Algebra and Its Applications. 10 (2011), 521–527.