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 ϕ of the n th eigenfunction of the Schrödinger 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 ϕ −( n −1).