Automorphisms of the zero-divisor graph of the full matrix ring

Abstract

The zero-divisor graph of a non-commutative ring R, written as , is a directed graph with vertex set of all non-zero zero-divisors of R, and there is a directed edge from a vertex x to a distinct vertex y if and only if . Let M(n, q) (resp., T(n, q)) be the ring of all matrices (resp., upper triangular matrices) over a finite field . Recently, Wang (Linear Algebra Appl. 2015;465:214–220) determined the automorphisms of the zero-divisor graph of T(n, q). In this paper, we determine the automorphisms of , extending the result due to L. Wang from T(n, q) to M(n, q). Since the case is trivial, and the case has been examined in Ma et al. (J. Korean Math. Soc. 2016;53:519–532), we just determine the automorphisms of in the case. We show that a bijective map on is an automorphism of if and only if there exist invertible matrices and a such that for any , where , and depend on A, and .