Levinson's theorem for graphs

We prove an analog of Levinson's theorem for scattering on a weighted (m + 1)-vertex graph with a semi-infinite path attached to one of its vertices. In particular, we show that the number of bound states in such a scattering problem is equal to m minus half the winding number of the phase of the reflection coefficient (where each so-called half-bound state is counted as half a bound state).