Let $A=(a_{i,j})$ be a square matrix with non-negative entries. (Assume $A$ is symmetric, if it helps.) Let $v$ be a Perron-Frobenius eigenvector. What do we need to assume about $A$ in order to have useful bounds on how "flat", or close to uniform, $v$ is?

One easy bound (for $v$ having strictly positive entries, as is implied by standard conditions) is given by $$\frac{\max_i v_i}{\min_i v_i} \leq \frac{|A|}{\min_i \sum_j a_{i,j}} \leq \frac{\max_i \sum_j a_{i,j}}{\min_i \sum_j a_{i,j}}.$$

Let $A=(a_{i,j})$ be a square matrix with non-negative entries. (Assume $A$ is symmetric, if it helps.) Let $v$ be a Perron-Frobenius eigenvector. What do we need to assume about $A$ in order to have useful bounds on how "flat", or close to uniform, $v$ is?

One easy bound (for $v$ having strictly positive entries, as is implied by standard conditions) is given by $$\frac{\max_i v_i}{\min_i v_i} \leq \frac{|A|}{\min_i \sum_j a_{i,j}} \leq \frac{\max_i \sum_j a_{i,j}}{\min_i \sum_j a_{i,j}}.$$ EDIT: Actually, I no longer remember quite how I proved this "easy bound", which may not even be true.