I will explain a combinatorial toy model for a Quantum Field Theory (QFT), where the space-time is a graph. The focus will lie on two aspects of this toy model: First, one can define a ``cobordism category'' of graphs and show that the theory defines a functor out of that category. I will also discuss an extended (higher categorical) version of this. Second, in the ``first quantization formalism'', all objects in the QFT have combinatorial expressions in terms of sums over paths in the graph.