Ambarzumian type theorems

Consider the eigenvalue problem \(\left. \begin{array}{cc} -y''+q(x)y=\lambda y\\ y'(0)=y'(1)=0 \end{array} \right\}.\) If $q=0$ then the eigenvalues are $\lambda_n=n^2\pi^2$ ($n\ge 0$). What is surprising is that the converse is also true; this is Ambarzumian's theorem. We discuss some similar statements on graphs and in a PDE setting.