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

Abstract

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.