On the Exponent of a Primitive, Nearly Reducible Matrix

An n × n nonnegative matrix is called nearly reducible provided it is irreducible and the replacement of any positive entry by zero yields a reducible matrix. The purpose of this article is to investigate the exponent (gamma)( A ) of an n × n primitive, nearly reducible matrix A (aperiodic, minimally strong directed graph). We prove that (gamma)( A ) (ge) 6 and that for each n (ge) 4 there exists a matrix A for which equality holds. We also show that (gamma)( A ) (le) n 2 - 4 n + 6 and characterize those matrices for which equality holds. The proofs are carried out in the language of directed graphs.