  We consider an algebraic formulation of Quantum Theory and develop a
combinatorial model of the Heisenberg-Weyl algebra structure. It is shown that
by lifting this structure to the richer algebra of graph operator calculus, we
gain a simple interpretation involving, for example, the natural composition of
graphs. This provides a deeper insight into the algebraic structure of Quantum
Theory and sheds light on the intrinsic combinatorial underpinning of its
abstract formalism.
