Gaussianity of the nodal count distribution for large graphs

Establish that for finite connected graphs G with large size, when H is a real symmetric matrix strictly supported on G and ψ ranges over eigenvectors corresponding to simple eigenvalues with nonzero entries, the distribution of the nodal count (# of directed edges (r→s) with ψr Hrs ψs > 0) is Gaussian in the large-graph regime.

Background

The weighted cycle intersection form C^T Φ−1 C reflects cycle-level contributions to the nodal count. For graphs with disjoint cycles the form is diagonal and the nodal count has a binomial distribution. For graphs comprising many small blocks, the block structure suggests approximate Gaussian behavior via aggregation of many contributions.

The authors note numerical evidence supporting full Gaussianity of the nodal count for large graphs and refer to works where precise formulations are given, highlighting a conjectural picture that remains to be rigorously established in general.