2024/03/01 by Alon, Lior, Goresky, Mark · 2 citations
#05C50 #58J50 #81Q10 #81Q35 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Spectral Theory (math.SP)
paper · doi:10.48550/arxiv.2403.01033
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 Schrödinger 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Part of the proof follows ideas developed by the first author together with Ram Band and Gregory Berkolaiko in a joint unpublished project studying a similar question on quantum graphs.