4:00 pm Thursday, March 1, 2018

Gregory Berkolaiko (Texas A&M University)

Abstract: We start by reviewing the notion of quantum graph, its eigenfunctions and the problem of counting the number of their zeros. The nodal surplus of the n-th eigenfunction is defined as the number of its zeros minus (n-1). When the graph is composed of two or more blocks separated by bridges, we propose a way to define a local nodal surplus of a given block. Since the eigenfunction index n has no local meaning, the local nodal surplus has to be defined in an indirect way via the nodal-magnetic theorem of Berkolaiko, Colin de Verdire and Weyand. We will discuss the properties of the local nodal surplus and their consequences. In particular, its symmetry properties allow us to prove the long-standing conjecture that the nodal surplus distribution for graphs with beta disjoint loops is binomial with parameters (beta,1/2). The talk is based on joint work with Lior Alon and Ram Band, arXiv:1709.10413 (accepted to CMP).