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