On the lack of bound states for certain NLS equations on metric graphs
  The purpose of this paper is to prove some results on the absence of bound states for certain nonlinear Schr\"odinger equations on noncompact metric graphs with localized nonlinearity. In particular, we show how the topological and metric properties of graphs affect the existence/nonexistence of bound states. This work completes the discussion initiated in [17, 18]. 
Enrico Serra, Lorenzo Tentarelli
math.AP