Cayley Digraphs of Matrix Rings over Finite Fields

Abstract

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 ${\mathbb{F}}_q$ can be written as a sum of two ${\mathrm{SL}}_n$-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 ${\mathrm{Mat}}_2({\mathbb{F}}_q)$ with size $|X|>\frac{2q^3\sqrt{q}}{q-1}$, then $X$ contains at least two distinct matrices whose difference has determinant $\alpha$ for any $\alpha \in {\mathbb{F}}_q^{\ast }$. Using this result we also prove a sum-product type result: if $A,B,C,D\subseteq {\mathbb{F}}_q$ satisfy $\sqrt[4]{|A||B||C||D|}=\Omega (q^{0.75})$ as $q\to \infty$, then $(A-B)(C-D)$ equals all of ${\mathbb{F}}_q$. In particular, if $A$ is a subset of ${\mathbb{F}}_q$ with cardinality $|A|>\frac{3}{2}{q}^{\frac{3}{4}}$, then the subset $(A-A)(A-A)$ equals all of ${\mathbb{F}}_q$. We also recover a classical result: every element in any finite ring of odd order can be written as the sum of two units.

19 pages