Suppose that $M$ is an $n \times n$ matrix where each entry is a positive integer. Then $M$ is Perron-Frobenius and so has unique largest real eigenvalue $\lambda_{\textrm{PF}}$.

Does an upper bound on $\lambda_{\textrm{PF}}$ give an upper bound on each of the entries of $M$? That is, does $\lambda_{\textrm{PF}}$ being small imply that each entry of $M$ is too?

Note that when $n = 2$: $$ \lambda_{\textrm{PF}} = \frac{1}{2}\left(a + d + \sqrt{a^2 - 2ad + d^2 + 4bc}\right) $$ and so $a, b, c, d \leq \lambda_{\textrm{PF}}^2$.

Additionally, if we remove the requirement that the entries of $M$ be integers then the answer is no as for every $k \geq 1$ the matrix $$ \left( \begin{matrix} 1 & k \\ k^{-1} & 1 \end{matrix} \right) $$ has Perron-Frobenius eigenvalue 2.