  In this article the well known "Perron-Frobenius theory" is investigated
involving the higher rank numerical range $\Lambda_{k}(A)$ of an irreducible
and entrywise nonnegative matrix $A$ and extending the notion of elements of
maximum modulus in $\Lambda_{k}(A)$. Further, an application of this theory to
the $\Lambda_{k}(L(\lambda))$ of a Perron polynomial $L(\lambda)$ is elaborated
via its companion matrix $C_{L}$.
