Let $\mathcal{F}$ be the family of symmetric, nonnegative matrices of order $n$ whose diagonal entries are zero. Let $\rho(A)$ be the largest eigenvalue of $A$. Suppose $\min\{\rho(A):A\in \mathcal{F}\}=\rho(B).$
Now consider a new family $\mathcal{S}=\{\left(\begin{array}{cc}A&x\\x^T&0\end{array}\right): A\in \mathcal{F} , x=(0,0,\dots,0,2)\}$. If we want to get minimum over the largest eigenvalues of matrices in $\mathcal{S}$. Then how to proceed. I started with calculating characteristic polynomial of a matrix in $\mathcal{S}$ and but could not relate it to the characteristic polynomial of a matrix in $\mathcal{F}.$
Any suggestion please?
