  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 $\ell^2$. While convergence to the Neumann semigroup
always holds, convergence to the Dirichlet semigroup in $\ell^2$ turns out to
be equivalent to the coincidence of the Dirichlet and Neumann semigroups while
convergence in $\ell^1$ 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.
