On the Exponent of a Primitive, Nearly Reducible Matrix. II
Jeffrey A. Ross · SIAM Journal on Algebraic and Discrete Methods · 1982
A 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 $n \times n$ primitive, nearly reducible matrix A. Our principal result is that $\gamma ( A )\leqq n + s ( n - 3 )$, where s is the length of a shortest circuit in the directed graph associated with A. It is an easy application of this result to find gaps in the exponent set of $n \times n$ primitive, nearly reducible matrices. We also show that for integers n, k satisfying $n\geqq k - 1\geqq 5$ there exists an $n \times n$ primitive, nearly reducible matrix with exponent k. The proofs are carried out by means of directed graphs.