4 problems

GOE nodal-count central limit conjecture. For every $k\in[n]$,

Let $G$ be a graph, let $A$ be a generic real symmetric matrix supported on $G$, and let $A_{\mathrm{sgn}}$ be obtained by independently and uniformly randomly changing the signs o…

Let a quantum graph have suitable mild assumptions on its graph structure and edge lengths, and let the zeros of its $k$-th eigenfunction be counted. Smilansky–Gnutzmann–Weber conj…

An isospectral pair of graphs is a pair with the same spectrum, and a graph's nodal properties record the nodal counts of its eigenfunctions. Nodal-distinguishability conjecture. E…