Not very different, but another possible estimate is the following: if you know that $Av\leq tv$ (elementwise) for some nonnegative (elementwise) vector $v$ and real $t$, then $\lambda_\max \leq t$ (proof: left-multiply by the left Perron vector. Irreducibility is needed, otherwise you need $v>0$). This is essentially a "weighted Gershgorin estimate", since Gershgorin's theorem corresponds to the version with $v$ the vector of all ones.

Note that both estimates can be tight, if $v$ is the Perron eigenvector.

You may get other estimates via matrix norms: $\lambda_\max \leq ||A||$. Computing the Euclidean norm is as difficult as computing the maximum eigenvalue, so it is probably unhelpful. This leaves as candidates the Frobenius norm (which is always larger than the Euclidean norm, but at least it is easily computable) and the $1$ norm (which is the Gershgorin bound on $A^T$ --- so yes, I am basically just stating Gershgorin variants and easy stuff in this answer, sorry).