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 $n$-th eigenfunction has $n-1+s$ such zeros, where the "nodal surplus" $s$ is an integer between 0 and the number of cycles on the graph.   We then perturb the Laplacian by a weak magnetic field and view the $n$-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 $s$ of the $n$-th eigenfunction of the unperturbed graph. 
Gregory Berkolaiko
math-ph math.MP