Exponents of primitive companion matrices

Abstract

A nonnegative matrix $A$ is primitive if for some positive integer $m$ all entries in $A^m$ are positive. The smallest such $m$ is called the exponent of $A$ and written $\exp (A)$. For the class of primitive companion matrices $X$, we find $\exp (A)$ for certain $A \in X$. We find certain values of $m$ for which there is an $n \times n$ primitive companion matrix (for given $n$) with exponent $m$. We also propose open problems for further research.