A note on automorphisms of the zero-divisor graph of upper triangular matrices

Abstract

Let Fq be a finite field with q elements, n(≥3) a positive integer, T(n,q) the set of all n×n upper triangular matrices over Fq. In [13], the zero-divisor graph of T(n,q), written as T, is defined to be a graph with all nonzero zero-divisors in T(n,q) as vertices, and there is a directed edge from a vertex X to a vertex Y if and only if XY=0. The subgraph of T induced by all rank one matrices in T(n,q) is denoted by R. Wong et al. (2014) in [13] determined the automorphisms of R and left the automorphisms of T unsolved. In this note, we solve this problem.