Abstract
On a finite network X, Delta denotes the Laplace operator and for any real-valued function q(x) on X, the operator Delta(q) u(x) = Delta u(x)-q(x)u(x) represents a perturbation of Delta. Assuming that the conductance in X is not necessarily symmetric (non-reversible case) and that the function q(x) is arbitrary (so that it is not anymore necessary the matrix associated to Delta(q), to be positive semi-definite), some results are proved using matrix methods which help solving the Poisson problem of finding a solution u(x) to the equation Delta(q) u(x) = f(x) on X for a given real-valued function f (x). Consequently, Dirichlet-Poisson and Neumann Poisson equations on proper subsets of X are solved.