A magnetic interpretation of the nodal count on graphs

The study of nodal sets, i.e. zero sets of eigenfunctions, on geometric objects can be traced back to De Vinci, Galileo, Hook, and Chladni. Today it is a central subject of spectral geometry. Sturm (1836) showed that the n-th eigenfunction of the Laplacian on a finite interval has n-1 zeros, i.e. nodal points. Fiedler (1975) proved a similar result for the discrete Laplacian of finite tree graphs. However, if a graph is not a tree, the nodal count may deviate from n-1. We call this deviation the nodal surplus. Berkolaiko (2007) showed that the nodal surplus is bounded between 0 and the first Betti number of the graph. In 2013 Berkolaiko provided a magnetic interpretation to the nodal surplus.  The nodal surplus of an eigenvector is equal to the Morse index of its eigenvalue under magnetic (or twisting) small perturbations of the Laplacian matrix. In this talk I will describe a different proof for this result, that was given by Colin de Verdiere (2013), and hold for any real symmetric matrix with non-positive off-diagonal entries. I will also explain how this naturally extends to any real symmetric matrix, by modifying the nodal count. In the last part of the talk, I will present some numerical experiments suggesting that the nodal surplus of large random graphs should have universal Gaussian distribution.  The talk should be approachable without any needed background.