An improvement of the Dulmage-Mendelsohn theorem

An n × n nonnegative matrix A is called primitive if for some positive integer k, every entry in the matrix Ak is positive or, in notation, Ak ⪢ 0. The exponent of primitivity of A is defined to be γ(A) = min{k ϵ Z+: Ak ⪢ 0}, where Z+ denotes the set of positive integers. The well known Dulmage-Mendelsohn theorem is that γ(A) ⩽ n + s(n − 2), where s is the shortest circuit in D(A), the directed graph of A. In this paper we prove that γ(A) ⩽ D + 1 + s(D − 1), where D is the diameter of D(A).