Olga Aryasova (Inst. of Geophysics, Nat. Acad. of Sciences of Ukraine / Friedrich–Schiller–Univ. Jena)
Prof. Dr. Ram Band (Israel Institute of Technology - Technion)
We discuss the number of zeros of Laplacian eigenfunctions on a metric (quantum) graph.
The n-th eigenfunction has at least n-1 zeros and at most n-1+\beta zeros, where \beta is the number of graph cycles (graph's first Betti number).
The nodal surplus of an eigenfunction is defined as the number of its zeros minus (n-1).
For a given graph, one might study the distribution of the nodal surplus of its eigenfuncitons.
This distribution is interestingly connected to the graph's topology and we show some recent results in this direction.
Furthermore, numerical studies suggest that the nodal surplus distribution has a universal form: it converges to a normal distribution as the number of cycles grows.
We state this conjecture and discuss our recent progress in proving it.
The talk is based on joint works with Lior Alon and Gregory Berkolaiko.