We present a direct and simple method for the computation of the total
scattering matrix of an arbitrary finite noncompact connected quantum graph
given its metric structure and local scattering data at each vertex. The method
is inspired by the formalism of Reflection-Transmission algebras and quantum
field theory on graphs though the results hold independently of this formalism.
It yields a simple and direct algebraic derivation of the formula for the total
scattering and has a number of advantages compared to existing recursive
methods. The case of loops (or tadpoles) is easily incorporated in our method.
This provides an extension of recent similar results obtained in a completely
different way in the context of abstract graph theory. It also allows us to
discuss briefly the inverse scattering problem in the presence of loops using
an explicit example to show that the solution is not unique in general. On top
of being conceptually very easy, the computational advantage of the method is
illustrated on two examples of "three-dimensional" graphs (tetrahedron and
cube) for which other methods are rather heavy or even impractical.Comment: 20 pages, 4 figure
