The nodal-domain count conjecture for eigenvectors of random signed graphs

Let $G(n,p,q)$ be a randomly sampled signed graph, and let $N(\bm{\varphi})$ denote the number of nodal domains of an eigenvector $\bm{\varphi}$. Fix constants $0<\epsilon,p,q<1$, and let $i\in[\epsilon n,(1-\epsilon)n]$ index an eigenvector of the associated symmetric matrix.

Nodal-domain count conjecture. There are constants $c_1(\epsilon,p,q)$ and $c_2(\epsilon,p,q)$ such that, with probability $1-o_N(1)$, the $i$th eigenvector $\bm{\varphi}$ satisfies

$$c_1\frac{n}{\log n}\leq N(\bm{\varphi})\leq c_2\frac{n}{\log n}.$$

This conjecture predicts that the number of nodal domains of bulk eigenvectors has the same order as the chromatic number of a dense Erdős–Rényi graph. The surrounding discussion explains that the available quantum-ergodicity results only handle fixed-degree polynomial observables and do not provide a sufficiently strong probability estimate for a simultaneous union bound; the conjectured $\Theta(n/\log n)$ behavior is therefore left open.