Morse theory for discrete magnetic operators and nodal count distribution for graphs

Given a discrete Schrodinger operator h on a finite connected graph G of n vertices, the nodal count 0(h, k) denotes the number of edges on which the k-th eigenvector changes sign. A signing h0 of h is any real symmetric matrix constructed by changing the sign of some off-diagonal entries of h, and its nodal count is defined according to the signing. The set of signings of h lie in a naturally defined torus Th of "magnetic perturbations" of h. G. Berkolaiko [Anal. PDE 6 (2013), 1213-1233] discovered that every signing h0 of h is a critical point of every eigenvalue Ak:Th R, with Morse index equal to the nodal surplus. We add further Morse theoretic information to this result. We show if h E Th is a critical point of Ak and the eigenvector vanishes at a single vertex v of degree d, then the critical point lies in a nondegenerate critical submanifold of dimension d + n - 4, closely related to the configuration space of a planar linkage. We compute its Morse index in terms of spectral data. The average nodal surplus distribution is the distribution of values of 0(h0, k) - (k - 1), averaged over all signings h0 of h. If all critical points correspond to simple eigenvalues with nowhere-vanishing eigenvectors, then the average nodal surplus distribution is binomial. In general, we conjecture that the nodal surplus distribution converges to a Gaussian in a CLT fashion as the first Betti number of G goes to infinity.