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 F_q can be written as a sum of two 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 Mat₂(F_q) with size |X|>2q³√q/(q−1), then X contains at least two distinct matrices whose difference has determinant α for any α ∈ F∗_q. Using this result, we also prove a sum-product type result: if A,B,C,D ⊆ F_q satisfy ⁴√|A||B||C||D| = Ω(q^(0.75)) as q → ∞, then (A−B)(C−D) equals all of F∗_q. In particular, if A is a subset of F_q with cardinality |A| > 3/2q^(3/4), then the subset (A−A)(A−A) equals all of F_q. 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.