Upper bound on largest eigenvalue of a sub-stochastic transition matrix

Question

I would like to find an upper bound on the largest eigenvalue,$\lambda$, of a sub-stochastic transition matrix on set $S$, $P_{S}$, in terms of the stationary distribution of $P$, named $\pi$. Is it possible to get the upper bound of $\pi(S)$?I mean, if we consider eigenvector 1, then

$\lambda \leq \pi(S)$

I appreciate any input.