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 Gamma(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 Gamma(R) congruent to Gamma(M-n(F)), where M-n(F) is the ring of n x n matrices over F. Here we prove that |R| = |M-n(F)|. We also show that if |F| is prime and n = 2, then R congruent to M-2(F). (C) 2008 Elsevier Inc. All rights reserved.