Nodal count of graph eigenfunctions via magnetic perturbation

We establish a connection between the stability of an eigenvalue under a magnetic perturbation and the number of zeros of the corresponding eigenfunction. Namely, we consider an eigenfunction of discrete Laplacian on a graph and count the number of edges where the eigenfunction changes sign (has a “zero”). It is known that the [math] -th eigenfunction has [math] such zeros, where the “nodal surplus” [math] is an integer between 0 and the first Betti number of the graph.¶ We then perturb the Laplacian with a weak magnetic field and view the [math] -th eigenvalue as a function of the perturbation. It is shown that this function has a critical point at the zero field and that the Morse index of the critical point is equal to the nodal surplus [math] of the [math] -th eigenfunction of the unperturbed graph.