We study heat kernel convergence of induced subgraphs with Neumann boundary conditions. We first establish convergence of the resulting semigroups to the Neumann semigroup in ℓ2. While convergence to the Neumann semigroup always holds, convergence to the Dirichlet semigroup in ℓ2 turns out to be equivalent to the coincidence of the Dirichlet and Neumann semigroups while convergence in ℓ1 is equivalent to stochastic completeness. We then investigate the Feller property for the Neumann semigroup via generalized solutions and give applications to graphs satisfying a condition on the edges as well as birth-death chains.
