Our intuition regarding waves suggests that the k-th eigenvector of a graph Laplacian L (or any Discrete Schrodinger operator) should exhibit greater fluctuations as k increases. In this context, the "nodal count" is the number of edges on which the eigenvector changes sign. The works of Fiedler (1975) and Berkolaiko (2007) show that the nodal count is bounded between k-1 and k-1+b, where b is the first Betti number of the graph. We establish that these bounds hold for signed graphs as well when considering sign changes accordingly. The “nodal surplus”, the deviation from k-1, is expected to concentrate around b/2. Numerical observations indicate that the distribution of the nodal surplus, across all eigenvectors and different signings of the graph, resembles a Gaussian distribution centered at b/2, regardless of the graph's characteristics. We prove that it is precisely binomial with mean b/2 in the case of operators on complete graphs with sufficiently high potential.

This outcome, among others, stems from a noteworthy relationship. The magnetic perturbations of L are achieved by multiplying the off-diagonal entries of L by phases (in a Hermitian manner), modulo gauge invariance. The eigenvalues of L extend to piece-wise analytic functions of the phases. At non-degenerate critical points, the Morse index is equal to the associated nodal surplus. If time permits, I will draw the line connecting this work to spectral gaps of periodic operators.
This talk is based on joint works with Mark Goresky and John Urschel