Nodal Statistics on Quantum Graphs Academic Article uri icon


  • 2018, Springer-Verlag GmbH Germany, part of Springer Nature. It has been suggested that the distribution of the suitably normalized number of zeros of Laplacian eigenfunctions contains information about the geometry of the underlying domain. We study this distribution (more precisely, the distribution of the nodal surplus) for Laplacian eigenfunctions of a metric graph. The existence of the distribution is established, along with its symmetry. One consequence of the symmetry is that the graphs first Betti number can be recovered as twice the average nodal surplus of its eigenfunctions. Furthermore, for graphs with disjoint cycles it is proven that the distribution has a universal formit is binomial over the allowed range of values of the surplus. To prove the latter result, we introduce the notion of a local nodal surplus and study its symmetry and dependence properties, establishing that the local nodal surpluses of disjoint cycles behave like independent Bernoulli variables.

published proceedings

  • Communications in Mathematical Physics

altmetric score

  • 1

author list (cited authors)

  • Alon, L., Band, R., & Berkolaiko, G.

citation count

  • 10

complete list of authors

  • Alon, Lior||Band, Ram||Berkolaiko, Gregory

publication date

  • September 2018