On the numbers of positive entries of reducible nonnegative matrices

Let RM(n,d) be the class {A∣A is an n×n reducible nonnegative matrix and the greatest common divisor of the lengths of all cycles in D(A) is d}, where D(A) is the associated digraph of A. In this paper we determine the set of numbers of positive entries of A for A∈RM(n,d). We also characterize the reducible nonnegative matrices with the maximum and minimum numbers of positive entries.