Unit-graphs and special unit-digraphs on matrix rings

Abstract

We use the unit-graphs and the 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 {\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 we prove that if X is a subset of {\operatorname{Mat}_{2}({\mathbb{F}}_{q})} with size {\lvert X\rvert>2q^{3}\sqrt{q}/(q-1)} , then X contains at least two distinct matrices whose difference has determinant α 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]{\lvert A\rvert\lvert B\rvert\lvert C\rvert\lvert D\rvert}=\Omega(q^{% 0.75})} as {q\rightarrow\infty} , then {(A-B)(C-D)} equals all of {{\mathbb{F}}_{q}^{\ast}} . In particular, if A is a subset of {{\mathbb{F}}_{q}} with cardinality {\lvert A\rvert>\frac{3}{2}q^{3/4}} , then the subset {(A-A)(A-A)} equals all of {{\mathbb{F}}_{q}} . We derive some identities involving character sums of the entries of {2\times 2} matrices over finite fields. We also recover a classical result: every element in any finite ring of odd order can be written as the sum of two units.