Discrete Dirichlet integral formula

The (discrete) Dirichlet integral is one of the most important quantities in the discrete potential theory and the network theory. In many situations, the dissipation formula which assures that the Dirichlet integral of a function u is expressed as the sum of -u(x)Δu(x)] seems to play an essential role, where Δu(x) denotes the (discrete) Laplacian of u. This formula can be regarded as a special case of the discrete analogue of Green's Formula. In this paper, we aim to determine the class of functions which satisfy the dissipation formula.