In this paper, we deal with the discrete p-Laplacian opera- tors with a potential term having the smallest nonnegative eigenvalue. Such operators are classied as its smallest eigenvalue is positive or zero. We discuss differences between them such as an existence of solutions of p-Laplacian equations on networks and properties of the energy func- tional. Also, we give some examples of Poisson equations which suggest a difference between linear types and nonlinear types. Finally, we study characteristics of the set of a potential those involving operator has the smallest positive eigenvalue.