on the commuting graph of non-commutative rings of order pnq

Let r be a non-commutative ring with unity. 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 iff 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 ∈ r z(r). also it is shown that if a, b ∈ r z(r) and ab ̸= ba, then cr(a) ∩ cr(b) = z(r). we show that the commuting graph γ(r) is the disjoint union of k copies of the complete graph and so is not a connected graph.