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?