On extremal matrices of second largest exponent by Boolean rank

Abstract

Let b = b( A) be the Boolean rank of an n × n primitive Boolean matrix A and exp( A) be the exponent of A. Then exp( A) ⩽ ( b − 1) 2 + 2, and the matrices for which equality occurs have been determined in [D.A. Gregory, S.J. Kirkland, N.J. Pullman, A bound on the exponent of a primitive matrix using Boolean rank, Linear Algebra Appl. 217 (1995) 101–116]. In this paper, we show that for each 3 ⩽ b ⩽ n − 1, there are n × n primitive Boolean matrices A with b( A) = b such that exp( A) = ( b − 1) 2 + 1, and we explicitly describe all such matrices.