Automorphism Group of Generalized Cayley Graph of Upper Triangular Matrices

Abstract

Let F q be a finite field with q elements, n(≥2) a positive integer, and T n (q) the semigroup of all n × n upper triangular matrices over F q . The generalized Cayley graph GCay(T n (q)) of T n (q) is a directed graph with vertex set T n (q), in which there is a directed edge from a vertex A to a distinct vertex B if and only if B = XAY for some X, Y ∈ T n (q). The main result of this article proves that a bijective map σ is an automorphism of GCay(T n (q)) if and only if, for any vertex A of GCay(T n (q)), either σ(A) = P A AQ A or σ(A) = P A JA t JQ A , where A t denotes the transpose of A, , and P A and Q A are invertible upper triangular matrices depending on A.