%0 Journal Article %J Journal of Mathematical Physics %D 2012 %T Levinson's theorem for graphs II %A Andrew M. Childs %A David Gosset %X 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. %B Journal of Mathematical Physics %V 53 %P 102207 %8 2012/11/21 %G eng %U http://arxiv.org/abs/1203.6557v2 %N 10 %! J. Math. Phys. %R 10.1063/1.4757665