Automorphism group of the one-divisor graph over the semigroup of upper triangular matrices

Let F-q be a finite field with q elements, n >= 2 a positive integer, T(n,q) the semigroup of all n x n upper triangular matrices over F-q under matrix multiplication, T*(n,q) the group of all invertible matrices in T(n,q), PT*(n,q) the quotient group of T*(n,q) by its center. The one-divisor graph of T(n,q), written as T-n(q), is defined to be a directed graph with T(n,q) as vertex set, and there is a directed edge from A is an element of T(n,q) to B is an element of T(n,q) if and only if r(AB) = 1, i.e. A and B are, respectively, a left divisor and a right divisor of a rank one matrix in T(n,q). The definition of T-n(q) is motivated by the definition of zero-divisor graph T of T(n,q), which has vertex set of all nonzero zero-divisors in T(n,q) and there is a directed edge from a vertex A to a vertex B if and only if AB = 0, i.e. r(AB) = 0. The automorphism group of zero-divisor graph T of T(n,q) was recently determined by Wang [A note on automorphisms of the zero-divisor graph of upper triangular matrices, Lin. Alg. Appl. 465 (2015) 214-220.]. In this paper, we characterize the automorphism group of one-divisor graph T-n(q) of T(n,q), proving that Aut(T-n(q)) similar or equal to PT*(n,q) x Aut(F-q) x H, where Aut(F-q) is the automorphism group of field F-q, H is a direct product of some symmetric groups. Besides, an application of automorphisms of T-n(q) is given in this paper.