On the diameters of commuting graphs

The commuting graph of a ring R, denoted by Γ(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 Γ(Mn(D)) and determine the diameters of some induced subgraphs of Γ(Mn(D)), such as the induced subgraphs on the set of all non-scalar non-invertible, nilpotent, idempotent, and involution matrices in Mn(D). For every field F, it is shown that if Γ(Mn(F)) is a connected graph, then diam Γ(Mn(F)) ⩽ 6. We conjecture that if Γ(Mn(F)) is a connected graph, then diam Γ(Mn(F)) ⩽ 5. We show that if F is an algebraically closed field or n is a prime number and Γ(Mn(F)) is a connected graph, then diam Γ(Mn(F)) = 4. Finally, we present some applications to the structure of pairs of idempotents which may prove of independent interest.