Automorphisms of the Zero-Divisor Graph Over 2 × 2 Matrices

Abstract

Abstract. The zero-divisor graph of a noncommutative ring R, denoted by Γ(R), is a graph whose vertices are nonzero zero-divisors of R, and there is a directed edge from a vertex x to a distinct vertex y if and only if xy = 0. Let R = M 2 (Fq) be the 2 × 2 matrix ring over a finite field Fq. In this article, we investigate the automorphism group of Γ(R).

ON ZERO DIVISOR GRAPH OF MATRIX RING $M_n(\mathbf{Z}_p)$

“…Anderson and Badawi [1] defined total graph of commutative ring. The total graph T (Γ(R)) of a commutative ring R is an undirected graph with R as vertex set and any two elements x, y ∈ R, x is adjacent to y if and only if x + y is a zero divisor of R. For more results on zero divisor graphs, readers may refer to [7], [13], [15], [16], [18].…”

Automorphisms of left Ideal relation graph over full matrix ring

“…Recently, the automorphisms of the zero-divisor graph over the matrix ring attracted the attention of researchers (see [9,12,17,20,24]). Also, Feng Xu et al [21], determined all the automorphisms of the intersection graph of ideals over a matrix ring.…”

Compressed zero-divisor graphs of noncommutative rings

“…The vertex set of both graphs is the set of all nonzero zero-divisors of R. In Γ(R) there is a directed edge x → y if and only if x = y and xy = 0, while in Γ(R) there is an undirected edge x − y if and only if x = y and either xy = 0 or yx = 0. Properties of these graphs have been studied in [4,5,14,23,27]. Most attention has been devoted to matrix rings either over fields or over general commutative unital rings, but other ring such as group rings, polynomial rings etc.…”

“…We remark that for the compressed zero-divisor graph Γ E , the group of automorphisms of Γ E (M 2 (F q )) was described in [23].…”