In this thesis, we prove heat kernel estimates in two main contexts: (1) manifolds with ends with mixed Dirichlet and Neumann boundary condition and (2) infinite (countable) graphs satisfying certain properties, which we call book-like graphs. In both of these settings, we start with ''sufficiently nice'' pieces (pieces satisfying two-sided Gaussian heat kernel estimates) that are ''glued'' together in some sufficiently nice way. The results in setting (1) extend previous results of Grigor'yan and Saloff-Coste in the case of manifolds with ends with Neumann (or no) boundary condition. In setting (2), we are in the discrete case, where there is not direct prior work. This thesis extends some of the continuous setting results of Grigor'yan and Saloff-Coste mentioned above to the discrete setting, and the results here are also related to results of Grigor'yan and Ishiwata regarding gluing two copies of n-dimensional real space via a surface of revolution. In both settings, the results of this thesis rely heavily on the h-transform technique and understanding particular harmonic functions and hitting probabilities. In the setting of (1), we show the existence of a global harmonic function satisfying particular properties. In the setting of (2), we give estimates on certain hitting probabilities that naturally arise from considering subgraphs of larger graphs. All work in this thesis is joint with Laurent Saloff-Coste.