Let $A=(a_{kl})$ be the matrix in $M_n(\mathbb{C})$ given by $a_{kk}=0$ and $a_{kl}=\frac12$ if $k\neq l$. Let $D$ be the diagonal matrix in $M_n(\mathbb{C})$ with $D=\operatorname{diag}(n,\frac{2n+1}{4},,\cdots,\frac{2n+1}{4})$.
Q. Any suggestion to find the minimal polynomial/eigenvalues of $A+D$ ?
The vectors with first entry 0 and total sum 0 are eigenvectors with eigenvalue $2n-1/4$.
As the matrix is symmetric the orthogonal of the above is stable under the operator. This orthogonal consists of the vectors having last n-1 coordinates equal.
This is a 2-dimensional matrix which you can easily diagonalize. In total you get what Neil Strickland suggested in the comments by experiments (by the way I agree this is a bit easy for MO).