  Let $R$ be a finite ring with identity. The unit graph (unitary Cayley graph)
of $R$ is the graph with vertex set $R$, where two distinct vertices $x$ and
$y$ are adjacent exactly whenever $x+y$ is a unit in $R$ ($x-y$ is a unit in
$R$). Here, we study independent sets of unit graphs of matrix rings over
finite fields and use them to characterize all finite rings for which the unit
graph is well-covered or Cohen-Macaulay. Moreover, we show that the unit graph
of $R$ is well-covered if and only if the unitary Cayley graph of $R$ is
well-covered and the characteristic of $R/J(R)$
