On commuting graphs of group ring $Z_{n}Q_{8}$
Commun. Korean Math. Soc. 2012 Vol. 27, No. 1, 57-68
Printed March 1, 2012
Jianlong Chen, Yanyan Gao, and Gaohua Tang
Southeast University, Southeast University, Guangxi Education University
Abstract : The commuting graph of an arbitrary ring $R$, denoted by $\Gamma(R)$, 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 $Z_{n}Q_{8}$. The main result is that $\Gamma(Z_{n}Q_{8})$ is connected if and only if $n$ is not a prime. If $\Gamma(Z_{n}Q_{8})$ is connected, then diam($Z_{n}Q_{8}$)= 3, while $\Gamma(Z_{n}Q_{8})$ is disconnected then every connected component of $\Gamma(Z_{n}Q_{8})$ must be a complete graph with a same size. Further, we obtain the degree of every vertex in $\Gamma(Z_{n}Q_{8})$, the maximum degree and the minimum degree of $\Gamma(Z_{n}Q_{8})$.
Keywords : group ring, commuting graph, connected component, diameter of a graph
MSC numbers : 16S34, 20C05, 05C12, 05C40
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