Abstract

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 0 s in an irreducible, totally nonnegative ( 0 , 1 ) -matrix of order n is ( n - 1 ) 2 and characterize those matrices with this number of 0 s. We also show that the minimum Perron value of an irreducible, totally nonnegative ( 0 , 1 ) -matrix of order n equals 2 + 2 cos 2 π n + 2 and characterize those matrices with this Perron value.

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