Levinson's theorem for graphs II

We prove Levinson's theorem for scattering on an (m+n)-vertex graph with n
semi-infinite paths each attached to a different vertex, generalizing a
previous result for the case n=1. This theorem counts the number of bound
states in terms of the winding of the determinant of the S-matrix. We also
provide a proof that the bound states and incoming scattering states of the
Hamiltonian together form a complete basis for the Hilbert space, generalizing
another result for the case n=1.