On commuting graphs of group ring Z nQ 8

Abstract

The commuting graph of an arbitrary ring R, denoted by Γ(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 Γ(Z nQ 8) is connected if and only if n is not a prime. If Γ(Z nQ 8) is connected, then diam(Z nQ 8)= 3, while Γ(Z nQ 8) is disconnected then every connected component of Γ(Z nQ 8) must be a complete graph with a same size. Further, we obtain the degree of every vertex in Γ(Z nQ 8), the maximum degree and the minimum degree of Γ(Z nQ 8). © 2012 The Korean Mathematical Society.