Mathematical Conversations
A magnetic interpretation of the nodal count on graphs
The study of nodal sets, i.e. zero sets of eigenfunctions, on geometric objects can be traced back to De Vinci, Galileo, Hook, and Chladni. Today it is a central subject of spectral geometry. Sturm (1836) showed that in 1D, the $n$-th eigenfunction of the Laplacian has $n-1$ nodal points. Fiedler (1975) proved a similar result for the discrete Laplacian for tree graphs. However, if a graph is not a tree then the nodal count may deviate from $n-1$. Berkolaiko (2013) showed that this deviation captures the stability of the spectrum under magnetic perturbations of the operator. In this talk, I will sketch a proof given by Colin de Verideire, which highlights the geometry of the problem.
The talk should be approachable without any needed background.