$A$ is a non-negative, integer, irreducible, $m$ by $m$ matrix. It is well known (Perron-Frobenius) that $A$ has a positive eigen value (denote it by $\lambda$) with a positive eigen vector ($x$). It is needed to prove that:

$x_{max}/x_{min} \le \lambda^{(m-1)}$

$x_{max}$ denotes the largest element of $x$, $x_{min}$ denotes the smallest element of $x$.

I proved that when considering A as an adjacency matrix, where there is an edge from state $u$ to state $v$, the following holds:

$x_v \le \lambda x_u$

(it was the hint in this question). I don't see how to proceed from here.