Is it true that there is a non-negative eigenvector corresponding the largest eigenvalue for any non-negative adjacency matrix?

Question

I am working on a square matrix that is asymmetric and reducible (not strong connected), say $A$, which is an adjacency matrix (all the elements are either $0$ or $1$).

I am wondering if there is a similar conclusion like Perron-Frobenius theorem that the largest eigenvalue of $A$ and the corresponding eigenvector would be non-negative (not all-zero vector).

Is it possible? If it is, how can I proof it? If not, what else requirements should be proposed?

Thank you.

Answer 1

Take a look at this subsection of the Wikipedia page on the Perron-Frobenius Theorem.