On the commuting graph of some non-commutative rings with unity

Let R be a non-commutative ring with unity. The commuting graph of R denoted by 􀀀(R), is a graph with a vertex set R n Z(R) and two vertices a and b are adjacent if and only if ab = ba. In this paper, we investigate non-commutative rings with unity of order pn where p is prime and n 2 f4; 5g. It is shown that, 􀀀(R) is the disjoint unio‎n of complete graphs. Finally, we prove that there are exactly ve commuting graphs of non-commutative rings with unity up to twenty vertices and they are 3K2; 3K4; 7K2;K2 [ 2K6 and 4K2 [K6.