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 .

Keywords:

• Directed graphs
• zero-divisor graphs
• automorphisms
• matrix rings

AMS Subject Classifications:

• 05C20
• 05C60

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

The first author was supported by the Research Innovation Program for College Graduates of Jiangsu Province [grant number KYZZ0371]; partially supported by the NNSF of China [grant number 11401570]; NSF of Jiangsu Province [grant number BK20140177]. The second author was supported by the National Natural Science Foundation of China [grant number 11571360].