Let R be a non-commutative ring. The commuting graph of R denoted by Γ(R), is a graph with vertex set RZ(R) and two vertices a and b are adjacent if ab = ba. It has been shown that the diameter of Γ(R) c is less than 3. For a finite ring R we show that the diameter of Γ(R)c is one if and only if R is the non-commutative ring on 4 elements. Also we characterize all rings where the complements of their commuting graphs are planar. Moreover, we identify the commuting graphs of rings of order pi for i = 2, 3 and prime number p.
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