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| Paper IPM / M / 8278 |
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| Abstract: | |||||
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The commuting graph of a ring \frakR, denoted by
Γ(\frakR), is a graph whose vertices are all non-central
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(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)\leqslant 6. We conjecture that if
Γ(Mn(F)) is a connected graph, then diam
Γ(Mn(F)\leqslant 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.
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