ON COMMUTING GRAPHS OF GROUP RING ZnQ8

Title & Authors
ON COMMUTING GRAPHS OF GROUP RING ZnQ8
Chen, Jianlong; Gao, Yanyan; Tang, Gaohua;

Abstract
The commuting graph of an arbitrary ring R, denoted by $\small{{\Gamma}(R)}$, 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 $\small{Z_nQ_8}$. The main result is that $\small{\Gamma(Z_nQ_8)}$ is connected if and only if n is not a prime. If $\small{\Gamma(Z_nQ_8)}$ is connected, then diam($\small{Z_nQ_8}$)= 3, while $\small{\Gamma(Z_nQ_8)}$ is disconnected then every connected component of $\small{\Gamma(Z_nQ_8)}$ must be a complete graph with a same size. Further, we obtain the degree of every vertex in $\small{\Gamma(Z_nQ_8)}$, the maximum degree and the minimum degree of $\small{\Gamma(Z_nQ_8)}$.
Keywords
group ring;commuting graph;connected component;diameter of a graph;
Language
English
Cited by
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