The Schr\"odinger operator on graphs and topology
  Symplectic vector-valued scalar product is constructed on the spaces of solutions of the real discrete Shrodinger equation with fixed value of the spectral parameter on graphs. It takes values in the first homology group of the graph. This scalar product plays fundamental role in the Scattering theory for the graphs with tails. In particular, all unitarity properties of scattering follow from elementary symplectic geometry 
S.P. Novikov