A bound on the exponent of a primitive matrix using Boolean rank Original Research Article

We present a bound on the exponent exp(A) of an n × n primitive matrix A in terms of its boolean rank b = b(A); namely exp(A) ≤ (b − 1)2 + 2. Further, we show that for each 2 ≤ b ≤ n − 1, there is an n × n primitive matrix A with b(A) = b such that exp(A) = (b − 1)2+ 2, and we explicitly describe all such matrices. The new bound is compared with a well-known bound of Dulmage and Mendelsohn, and with a conjectured bound of Hartwig and Neumann. Several open problems are posed.