Stability of eigenvalues of quantum graphs with respect to magnetic   perturbation and the nodal count of the eigenfunctions
  We prove an analogue of the magnetic nodal theorem on quantum graphs: the number of zeros $\phi$ of the $n$-th eigenfunction of the Schr\"odinger operator on a quantum graph is related to the stability of the $n$-th eigenvalue of the perturbation of the operator by magnetic potential. More precisely, we consider the $n$-th eigenvalue as a function of the magnetic perturbation and show that its Morse index at zero magnetic field is equal to $\phi - (n-1)$. 
G. Berkolaiko and T. Weyand
math-ph math.MP math.SP