  Discrete versions of the Laplace and Dirac operators haven been studied in
the context of combinatorial models of statistical mechanics and quantum field
theory. In this paper we introduce several variations of the Laplace and Dirac
operators on graphs, and we investigate graph-theoretic versions of the
Schr\"odinger and Dirac equation. We provide a combinatorial interpretation for
solutions of the equations and we prove gluing identities for the Dirac
operator on lattice graphs, as well as for graph Clifford algebras.
