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

Abstract

Let $G$ be a simple, connected graph on $n$ vertices, and further assume that $G$ has disjoint cycles. Let $h$ be a real symmetric matrix supported on $G$ (for example, a discrete Schrödinger operator). The eigenvalues of $h$ are ordered increasingly, ${\lambda }_1\le \cdots \le {\lambda }_n$, and if $\varphi$ is the eigenvector corresponding to ${\lambda }_k$, the nodal (edge) count $\nu (h,k)$ is the number of edges $(rs)$ such that $h_{rs}{\varphi }_r{\varphi }_s>0$. The nodal surplus is $\sigma (h,k)=\nu (h,k)-(k-1)$. Let $h^{\prime }$ be a random signing of $h$, that is a real symmetric matrix obtained from $h$ by changing the sign of some of its off-diagonal elements. If $h$ satisfies a certain generic condition, we show for each $k$ that the nodal surplus has a binomial distribution $\sigma (h^{\prime },k)\sim Bin(\beta ,\frac{1}{2})$. Part of the proof follows ideas developed by the first author together with Ram Band and Gregory Berkolaiko in a joint unpublished project studying a similar question on quantum graphs.