4 problems

Let $G_j$ be a sequence of graphs with first Betti number $\beta(G_j)$ tending to infinity. For a random signing $h_j'$ and eigenfunction index $k$, let $\Sigma(h_j',k)$ be the nor…

Uniform Gaussian nodal surplus conjecture. One should have

Aomoto–Gorin conjecture. The random variable $\sigma(h',k)$ should have mean $\beta(G)/2$, and, as $\beta(G)\to\infty$, $\Sigma(h',k)$ should converge to the standard normal distri…

Let $G=G([n],E)$ be a simple graph with first Betti number … For a symmetric matrix $h$ supported on $G$, let $h'$ range over its signings, and let $\sigma(h',k)$ denote the nodal…