Commuting graphs of full matrix rings over finite fields

Let R be a non-commutative ring and Z(R) be its center. The commuting graph of R is defined to be the graph Γ(R) whose vertex set is R⧹Z(R) and two distinct vertices are joint by an edge whenever they commute. Let F be a finite field, n⩾2 an arbitrary integer and R be a ring with identity such that Γ(R)≅Γ(Mn(F)), where Mn(F) is the ring of n×n matrices over F. Here we prove that |R|=|Mn(F)|. We also show that if |F| is prime and n=2, then R≅M2(F).