  We discuss Neumann problems for self-adjoint Laplacians on (possibly
infinite) graphs. Under the assumption that the heat semigroup is
ultracontractive we discuss the unique solvability for non-empty subgraphs with
respect to the vertex boundary and provide analytic and probabilistic
representations for Neumann solutions. A second result deals with Neumann
problems on canonically compactifiable graphs with respect to the Royden
boundary and provides conditions for unique solvability and analytic and
probabilistic representations.
