Non-locally-finite parahyperbolic networks

In the classification theory of Riemann surfaces the research for positive or bounded solutions of Δu = Pu, where P ＞__- 0 is a C^1－function, has played an important role in establishing the similarities between the solutions of this differential equation and the classical harmonic functions. In the discrete potential theory, the Schrödinger operators are to some extent like the equation Δu = Pu. In this note, we develop on an infinite graph, a theory of functions to reflect the properties of the above solutions, without the use of derivatives. This can be used to study discrete Schrödinger and Helmholtz equations in non-locally-finite networks.