On extremal matrices of second largest exponent by Boolean rank

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.