Answer by Rodrigo de Azevedo for Spectral radius of a rank-1 perturbation

To complement Christian's comment, since the spectral radius is upper-bounded by the spectral norm,

$$\begin{aligned} \rho \left( {\bf A} + {\bf u} {\bf v}^\top \right) &\leq \left\| {\bf A} + {\bf u} {\bf v}^\top \right\|_2 \\ &\leq \left\| {\bf A} \right\|_2 + \left\| {\bf u} {\bf v}^\top \right\|_2 \\ &= \left\| {\bf A} \right\|_2 + \left\| {\bf u} \right\|_2 \left\| {\bf v} \right\|_2 \end{aligned}$$