We prove a Feynman path integral formula for the unitary group $ \exp(-itL_{v,θ})$, $t\geq 0$, associated with a discrete magnetic Schrödinger operator $L_{v,θ}$ on a large class of weighted infinite graphs. As a consequence, we get a new Kato-Simon estimate $$ |\exp(-itL_{v,θ})(x,y)|\leq \exp(-tL_{-\mathrm{deg},0})(x,y), $$ which controls the unitary group uniformly in the potentials in terms of a Schrödinger semigroup, where the potential $\mathrm{deg}$ is the weighted degree function of the graph.