The commuting graph of a ring \frakR, denoted by
Γ(\frakR), is a graph whose vertices are all noncentral
elements of \frakR and two distinct vertices x and y are
adjacent if and only if xy = yx. Let D be a division ring and
n \geqslant 3. In this paper we investigate the diameters of
Γ(M_{n}(D)) and determine the diameters of some induced
subgraphs of Γ(M_{n}(D)), such as the induced subgraphs on
the set of all nonscalar noninvertible, nilpotent, idempotent,
and involution matrices in (M_{n}(D)). For every field F, it
is shown that if Γ(M_{n}(F)) is a connected graph, then
diam Γ(M_{n}(F)\leqslant 6. We conjecture that if
Γ(M_{n}(F)) is a connected graph, then diam
Γ(M_{n}(F)\leqslant 5. We show that if F is an
algebraically closed field or n is a prime number and
Γ(M_{n}(F)) is a connected graph, then diam
Γ(M_{n}(F)=4. Finally, we present some applications to the
structure of pairs of idempotents which may prove of independent
interest.
Download TeX format
