Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
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ON COMMUTING GRAPHS OF GROUP RING ZnQ8

  • Chen, Jianlong (Department of Mathematics Southeast University) ;
  • Gao, Yanyan (Department of Mathematics Southeast University) ;
  • Tang, Gaohua (School of Mathematical Sciences Guangxi Education University)
  • Received : 2010.09.21
  • Published : 2012.01.31
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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 $Z_nQ_8$. The main result is that $\Gamma(Z_nQ_8)$ is connected if and only if n is not a prime. If $\Gamma(Z_nQ_8)$ is connected, then diam($Z_nQ_8$)= 3, while $\Gamma(Z_nQ_8)$ is disconnected then every connected component of $\Gamma(Z_nQ_8)$ must be a complete graph with a same size. Further, we obtain the degree of every vertex in $\Gamma(Z_nQ_8)$, the maximum degree and the minimum degree of $\Gamma(Z_nQ_8)$.

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References

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