On the Exponent of a Primitive, Nearly Reducible Matrix

Abstract
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 γ(A) of an n × n primitive, nearly reducible matrix A (aperiodic, minimally strong directed graph). We prove that γ(A) ≥ 6 and that for each n ≥ 4 there exists a matrix A for which equality holds. We also show that γ(A) ≤ n2 − 4n + 6 and characterize those matrices for which equality holds. The proofs are carried out in the language of directed graphs.

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