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

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 Schr\"odinger operator). The eigenvalues of h are ordered increasingly, λ₁ ≤ ⋯ ≤ λₙ, and if φ is the eigenvector corresponding to λₖ, the nodal (edge) count ν(h,k) is the number of edges $(rs)$ such that hᵣₛφᵣφₛ>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)~ Bin(β,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Part of the proof follows ideas developed by the first author together with Ram Band and Gregory Berkolaiko in a joint unpublished project studying a similar question on quantum graphs.