We introduce the discrete p-Schrödinger operator L_p,ω and solve the following p-Schrödinger equation: L_p,ωu = −Δ_p,ωu + q|u|^p−2u = f on networks. To show the uniqueness of solutions of the p-Schrödinger equation, we first solve the eigenvalue problem for the p-Schrödinger operator and obtain some properties of the smallest eigenvalue and its corresponding eigenfunction of the p-Schrödinger operator.