Totally nonnegative (0,1)-matrices

We investigate (0,1)-matrices which are totally nonnegative and therefore which have all of their eigenvalues equal to nonnegative real numbers. Such matrices are characterized by four forbidden submatrices (of orders 2 and 3). We show that the maximum number of 0s in an irreducible, totally nonnegative (0,1)-matrix of order n is (n-1)2 and characterize those matrices with this number of 0s. We also show that the minimum Perron value of an irreducible, totally nonnegative (0,1)-matrix of order n equals and characterize those matrices with this Perron value.