(untitled) (Communications in Mathematical Research, 2021)
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 and the diameter of Γ(Z_n S_3). We show that Γ(Z_n S_3) is connected if and only if n is not a prime number. If Γ(Z_n S_3) is connected then diam (Γ(Z_n S_3)) = 3, while if Γ(Z_n S_3) is disconnected then every connected component of Γ(Z_n S_3) must be a complete graph with same size, and we completely determine the vertice set of every connected component.