Unit-graphs and special unit-digraphs on matrix rings

We use the unit-graphs and the special unit-digraphs on matrix rings to show that every n×n nonzero matrix over 𝔽q can be written as a sum of two SLn-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 Mat2⁡(𝔽q) with size |X|>2q3√q/(q−1), then X contains at least two distinct matrices whose difference has determinant α for any α∈𝔽q∗. Using this result, we also prove a sum-product type result: if A,B,C,D⊆Fq satisfy 4√|A||B||C||D|=Ω(q0.75) as q→∞, then (A−B)(C−D) equals all of 𝔽q∗. In particular, if A is a subset of 𝔽q with cardinality |A|>3/2⁢q3/4, then the subset (A−A)(A−A) equals all of 𝔽q. We derive some identities involving character sums of the entries of 2×2 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.