Nonnegative primitive matrices with exponent 2

AbstractA nonnegative matrix M with zero trace is primitive if for some positive integer k, Mk is positive. The exponent exp(M) of the primitive matrix is the smallest such k. By treating the digraph G whose adjacency matrix is the primitive matrix M, we will show that the minimum number of positive entries of M is 3n−3 when exp(M)=2. We will also show that for a symmetric n×n matrix M if exp(M)=2, the minimum number of positive entries of M is 3n−2 or 3n−3 according to n