The generalized star product and the factorization of scattering matrices on graphs

In this article we continue our analysis of Schrodinger operators on arbitr ary graphs given as certain Laplace operators. In the present article we gi ve the proof of the composition rule for the scattering matrices. This comp osition rule gives the scattering matrix of a graph as a generalized star p roduct of the scattering matrices corresponding to its subgraphs. We perfor m a detailed analysis of the generalized star product for arbitrary unitary matrices. The relation to the theory of transfer matrices is also discusse d. (C) 2001 American Institute of Physics.