Nodal count for a random signing of a graph with disjoint cycles

A recent conjecture, inspired by quantum chaos and the Bogomolny–Schmit conjecture, suggests that the nodal count of operators on signed graphs exhibits a universal Gaussian-like behavior. We establish this result for the family of graphs composed of disjoint cycles, which serves as a natural starting point by analogy with quantum graphs. Let G be a simple, connected graph with disjoint cycles , and let h be a real symmetric matrix supported on G (e.g., a discrete Schrödinger operator) that satisfies a certain generic condition . The nodal count \nu(h, k) is defined as the number of edges (i, j) where the k -th eigenvector \phi changes sign with respect to h , i.e., h_{ij}\phi_{i} \phi_{j} > 0 . We consider the distribution of nodal counts \nu(h',k) over random signings h' of h , obtained by changing the sign of some off-diagonal elements. We prove, for each k that \sigma(h',k) = \nu(h',k) - (k-1) has a binomial distribution \mathrm{Bin}\bigl(\beta, \frac{1}{2}\bigr) , where \beta is the first Betti number of G . Consequently, the conjecture is validated for graphs with disjoint cycles.