ON COMMUTING GRAPHS OF GROUP RING ZnQ8 Chen, Jianlong; Gao, Yanyan; Tang, Gaohua;
The commuting graph of an arbitrary ring R, denoted by , is a graph whose vertices are all non-central elements of R, and two distinct vertices a and b are adjacent if and only if ab = ba. In this paper, we investigate the connectivity, the diameter, the maximum degree and the minimum degree of the commuting graph of group ring . The main result is that is connected if and only if n is not a prime. If is connected, then diam()= 3, while is disconnected then every connected component of must be a complete graph with a same size. Further, we obtain the degree of every vertex in , the maximum degree and the minimum degree of .
group ring;commuting graph;connected component;diameter of a graph;
A. Abdollahi, Commuting graphs of full matrix rings over finite fields, Linear Algebra Appl. 428 (2008), no. 11-12, 2947-2954.
S. Akbari, M. Ghandehari, M. Hadian, and A. Mohammadian, On commuting graphs of semisimple rings, Linear Algebra Appl. 390 (2004), 345-355.
S. Akbari, A. Mohammadian, H. Radjavi, and P. Raja, On the diameters of commuting graphs, Linear Algebra Appl. 418 (2006), no. 1, 161-176.
S. Akbari and P. Raja, Commuting graphs of some subsets in simple rings, Linear Algebra Appl. 416 (2006), no. 2-3, 1038-1047.
G. Karpilovsky, Unit Group of Classical Rings, Clarendon Press, Oxford, 1988.
T. Y. Lam, A First Course in Noncommutative Rings, Springer Verlag, New York, 1991.
C. P. Milies and S. K. Sehgal, An Introduction to Group Rings, Kluwer Academic Publishers, 2002.
C. D. Pan and C. B. Pan, Elementary Number Theory, Second edition, Beijing University Publishing Company, Beijing, 2005.
D. S. Passman, The Algebraic Structure of Group Rings, Wiley - Interscience, John Wiley Sons, New York, 1977.
G. H. Tang and H. D. Su, The properties of zero-divisors graph of $Z_n$[i], J. Guangxi Norm. Univ. Natur. Sci. Ed. 3 (2007), 32-35.
Y. J. Wei, G. H. Tang, and H. D. Su, The commuting graph of the quaternion algebra over residue classes of integers, Ars Combin. 95 (2010), 113-127.