  In this article we formulate and discuss one particle quantum scattering
theory on an arbitrary finite graph with $n$ open ends and where we define the
Hamiltonian to be (minus) the Laplace operator with general boundary conditions
at the vertices. This results in a scattering theory with $n$ channels. The
corresponding on-shell S-matrix formed by the reflection and transmission
amplitudes for incoming plane waves of energy $E>0$ is explicitly given in
terms of the boundary conditions and the lengths of the internal lines. It is
shown to be unitary, which may be viewed as the quantum version of Kirchhoff's
law. We exhibit covariance and symmetry properties. It is symmetric if the
boundary conditions are real. Also there is a duality transformation on the set
of boundary conditions and the lengths of the internal lines such that the low
energy behaviour of one theory gives the high energy behaviour of the
transformed theory. Finally we provide a composition rule by which the on-shell
S-matrix of a graph is factorizable in terms of the S-matrices of its
subgraphs. All proofs only use known facts from the theory of self-adjoint
extensions, standard linear algebra, complex function theory and elementary
arguments from the theory of Hermitean symplectic forms.
