We consider Laplace eigenfunctions of a metric graph satisfying Neumann-Kirchhoff conditions on every vertex. The nodal count of a given eigenfunction is the number of points at which it vanishes. The nodal count of the n-th eigenfunction was shown to be bounded between n-1 and n-1+\beta, where \beta if the first Betti number of the graph. The difference between the nodal count and n-1 is called the nodal surplus. Berkolaiko et al. showed that the n-th nodal surplus equals to a magnetic stability index of the n-th eigenvalue. We present recent results on the statistics of the nodal surplus and conjecture a universal behavior for large graphs.
This talk is based on joint works with Ram Band (Technion) and Gregory Berkolaiko (Texas A&M).