Let D be a division ring with center F and n >= 1 a natural number. For S subset of M-n (D) the commuting graph of S, denoted by F(S), is the graph with vertex set S\Z(S) such that distinct vertices a and b are adjacent if and only if ab = ba, In this paper we prove that if n > 2 and A, N, J, T are the sets of all non-invertible, nilpotent, idempotent matrices, and involutions, respectively, then for any division ring D, Gamma(A), Gamma(N), Gamma(J), and Gamma(T) are connected graphs. We show that if n > 2 and U is the set of all upper triangular matrices, then for every algebraic division ring D, Gamma(U) is a connected graph. Also it is shown that if R is the set of all reducible matrices and M-n(D) is algebraic over F, then for n > 2, Gamma(R) is a connected graph. Finally, we prove that for n >= 2, Gamma(M-n(H)) is a connected graph, where H is the ring of real quaternions. (c) 2006 Elsevier Inc. All rights reserved.
