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The commuting graph of a ring $\frak{R}$, denoted by
$\Gamma(\frak{R})$, is a graph whose vertices are all non-central
elements of $\frak{R}$ 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
$\Gamma(M_{n}(D))$ 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 $\Gamma(M_{n}(F)\leqslant 6$. We conjecture that if
$\Gamma(M_{n}(F))$ is a connected graph, then diam
$\Gamma(M_{n}(F)\leqslant 5$. 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.
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