I'm interested in getting spectral radius $\rho$ of the following $d\times d$ matrix
$$A=\text{diag}(s)+s1'$$
Where diag indicates diagonal matrix, and $1$ is a vector of 1's. For example, for $d=3$, it is the following
$$A= \left( \begin{array}{ccc} s_1 & 0 & 0 \\ 0 & s_2 & 0 \\ 0 & 0 & s_3 \\ \end{array} \right) + \left( \begin{array}{ccc} s_1 & s_1 & s_1 \\ s_2 & s_2 & s_2 \\ s_3 & s_3 & s_3 \\ \end{array} \right) $$
The following seems to be a way to bound on $\rho$ using $\infty$-norm of $s$ and its dual norm, but it gets loose for large $d$, is there a better estimate?
$$\rho\le \|s\|_\infty + \|s\|_1$$
Didn't see a nice analytical form for large $d$, but for small $d$, a sequence of tighter bounds bounds may be obtained by using the fact that
$$\rho^n \le \|A^n\|_*$$
for any norm $\|\cdot\|_*$
For instance, for $n=2$ and L1 matrix norm (max-column-norm), this gives
$$\rho^2 \le \|s\|_\infty^2+\|s\odot s\|_1+\|s\|_\infty \|s\|_1+\|s\|_1^2$$