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 γ(r)‎, ‎is a graph with a vertex set‎ ‎r z(r) and two vertices a and b are adjacent if and only if‎ ab=ba‎. ‎in this paper‎, ‎we investigate noncommutative rings with unity of order p^n where p is prime and n ∈ {4,5}‎. it is shown that‎, γ(r) is the disjoint union of complete graphs‎. finally‎, ‎we prove that there are exactly five commuting‎ ‎graphs of noncommutative rings with unity up to twenty vertices and they are 3k_2,3k_4,7k_2‎, ‎k_2 u 2k_6 and 4k_2 u k_6‎.