On the diameters of commuting graphs

Yhe commuting graph of a ring R, denoted by Gamma(R), is a graph whose vertices are all non-central elements of R and two distinct vertices x and y are adjacent if and only if xy = yx. Let D be a division ring and n >= 3. In this paper we investigate the diameters of Gamma(M-n(D)) and determine the diameters of some induced subgraphs of Gamma(M-n(D)), such as the induced subgraphs on the set of all non-scalar non-invertible, nilpotent, idempotent, and involution matrices in M-n(D). For every field F, it is shown that if Gamma(M-n(F)) is a connected graph, then diam Gamma(M-n(F)) <= 6. We conjecture that if Gamma(M-n(F)) is a connected graph, then diam Gamma(M-n(F)) <= 5. We show that if F is an algebraically closed field or n is a prime number and Gamma(M-n(F)) is a connected graph, then diam Gamma(M-n(F)) = 4. Finally, we present some applications to the structure of pairs of idempotents which may prove of independent interest. (c) 2006 Elsevier Inc. All rights reserved.