On the exponent of a primitive digraph

We characterize the equality case of the upper bound $\text{γ(D) ? n + s(n ? 2)}$ for the exponent of a primitive digraph in the case s ? 2, where n is the number of the vertices of the digraph D and s is the length of the shortest elementary circuit of D. We also answer a question about the equality case when s = 1. The exponent of an n × n primitive nonnegative matrix A is the same as the exponent of the associated digraph D(A) of A. So every theorem in this paper can be translated into a theorem about nonnegative matrices.