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The commuting graph of a ring $R$, denoted by
$\mathnormal{\Gamma}(R)$, is a graph whose vertices are all
noncentral elements of $R$ and two distinct vertices are joint by an edge whenever
they commute. It is conjectured that
if $R$ is a ring with identity such that $\mathnormal{\Gamma}(R)\simeq\mathnormal{\Gamma}(M_n(F))$,
for a finite field $F$ and $n\geq2$, then $R\simeq M_n(F)$.
Here we prove this conjecture when $n=2$.}
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