  We use the \emph{unit-graphs} and the \emph{special unit-digraphs} on matrix
rings to show that every $n \times n$ nonzero matrix over $\Bbb F_q$ can be
written as a sum of two $\operatorname{SL}_n$-matrices when $n>1$. We compute
the eigenvalues of these graphs in terms of Kloosterman sums and study their
spectral properties; and prove that if $X$ is a subset of $\operatorname{Mat}_2
(\Bbb F_q)$ with size $|X| > \frac{2 q^3 \sqrt{q}}{q - 1}$, then $X$ contains
at least two distinct matrices whose difference has determinant $\alpha$ for
any $\alpha \in \Bbb F_q^{\ast}$. Using this result we also prove a sum-product
type result: if $A,B,C,D \subseteq \Bbb F_q$ satisfy $\sqrt[4]{|A||B||C||D|}=
\Omega (q^{0.75})$ as $q \rightarrow \infty$, then $(A - B)(C - D)$ equals all
of $\Bbb F_q$. In particular, if $A$ is a subset of $\Bbb F_q$ with cardinality
$|A| > \frac{3} {2} q^{\frac{3}{4}}$, then the subset $(A - A) (A - A)$ equals
all of $\Bbb F_q$. We also recover a classical result: every element in any
finite ring of odd order can be written as the sum of two units.
