The talk is concerned with uniqueness, in weighted lp spaces, of solutions to the Schrödinger equation with a potential V , posed on an infite graph. We distinguish the cases 1 ≤ p < 2 and p ≥ 2. Moreover, we discuss uniqueness of bounded solutions, under relaxed assumptions on V . Such results have been recently obtained jointly with S. Biagi (Politecnico di Milano) and G. Meglioli (Bielefeld University).