Exponents of primitive directed Toeplitz graphs

Given disjoint non-empty subsets S and T of {1, ... , n - 1}, a digraph D with the vertex set {1, 2, ... , n} is called a directed Toeplitz graph provided the arc i -> j occurs if and only if i - j is an element of S or j - i is an element of T. We investigate strong connectivity and primitivity of directed Toeplitz graphs. We prove that any primitive directed Toeplitz graph with n >= 6 vertices has exponent at least 3 and for each n >= 6, there is a primitive directed Toeplitz graph of order n which has exponent 3. By Wielandt's result, we know that exp(D) <= (n - 1)(2) + 1 for a primitive digraph D of order n. We characterize the primitive directed Toeplitz graph, for which the upper bound it attained.