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, $λ_1 \le \cdots \le λ_n$, and if $ϕ$ is the eigenvector corresponding to $λ_k$, the nodal (edge) count $ν(h,k)$ is the number of edges $(rs)$ such that $ h_{rs}ϕ_{r}ϕ_{s}>0$. The nodal surplus is $σ(h,k)= ν(h,k) - (k-1)$. Let $h'$ 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 $σ(h',k)\sim Bin(β,\frac{1}{2})$. Part of the proof follows ideas developed by the first author together with Ram Band and Gregory Berkolaiko in a joint unpublished project studying a similar question on quantum graphs.