Let $R$ be a non-commutative ring with unity. The commuting graph of $R$ denoted by $Gamma(R)$, is a graph with vertex set $RZ(R)$ and two vertices $a$ and $b$ are adjacent iff $ab=ba$. In this paper, we consider the commuting graph of non-commutative rings of order pq and $p^2q$ with Z(R) = 0 and non-commutative rings with unity of order $p^3q$. It is proved that $C_R(a)$ is a commutative ring for every $0neq a in RZ(R)$. Also it is shown that if $a,bin RZ(R)$ and $abneq ba$, then $C_R(a)cap C_R(b)= Z(R)$. We show that the commuting graph $Gamma(R)$ is the disjoint union of $k$ copies of the complete graph and so is not a connected graph.