Let R be a non-commutative ring with unity. The commuting graph of R denoted by (R), is a graph with vertex set RnZ(R) and two vertices a and b are adjacent i ab = ba. In this paper, we consider the commuting graph of non-commutative rings of order pq and p2q with Z(R) = 0 and non-commutative rings with unity of order p3q. It is proved that CR(a) is a commutative ring for every 0 ̸= a 2 R n Z(R). Also it is shown that if a; b 2 R n Z(R) and ab ̸= ba, then CR(a) CR(b) = Z(R). We show that the commuting graph (R) is the disjoint union of k copies of the complete graph and so is not a connected graph.