1Faculty of Basic Science, Imam Khomeini International University, Qazvin, Iran.
2Department of Mathematics, Islamic Azad University, Central Tehran Branch, Tehran, Iran.
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.