Introduction to Tropical Geometry states the following theorem about strongly-connected graphs and tropical eigenvalues:
Let A be an n × n-matrix such that G(A) is strongly connected. Then A has precisely one eigenvalue λ(A). It equals the minimum normalized length of a directed cycle.
Considering some sample strongly-connected graph B with the corresponding matrix M(B)
$$ \begin{bmatrix} 4 & 1 \\ 1 & 3 \\ \end{bmatrix} $$
Per the theorem, the sole eigenvalue of M(B) should be $(b_{1,2}+b_{2,1})/2 = 1$. However, what does the corresponding eigenvector mean graphically? What does any vector signify in this context?