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

Abstract

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 $\varphi$ changes sign with respect to $h$, i.e., $h_{ij}{\varphi }_i{\varphi }_j>0$. We consider the distribution of nodal counts $\nu (h^{\prime },k)$ over random signings $h^{\prime }$ of $h$, obtained by changing the sign of some off-diagonal elements. We prove, for each $k$ that $\sigma (h^{\prime },k)=\nu (h^{\prime },k)-(k-1)$ has a binomial distribution $\mathrm{Bin}(\beta ,\frac{1}{2})$, where $\beta$ is the first Betti number of $G$. Consequently, the conjecture is validated for graphs with disjoint cycles.