Suppose I have a tridiagonal square matrix $A$ of some nice form, for which I know the eigenvalues $\lambda_1<\dots<\lambda_n$. $A$ is also essentially nonnegative (nonnegative everywhere except main diagonal).

What can I say about the sum of $A$ and a diagonal matrix $B=\text{diag}(x_1,\dots,x_n)$, for which I know only about sum of it's elements?

Some equalities, maybe inequalities with eigenvalues of the first matrix and values of the second? Special interest is in the maximal eigenvalue