in this talk 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.