  Recently, a novel GHZ/W graphical calculus has been established to study and
reason more intuitively about interacting quantum systems. The compositional
structure of this calculus was shown to be well-equipped to sufficiently
express arbitrary mutlipartite quantum states equivalent under stochastic local
operations and classical communication (SLOCC). However, it is still not clear
how to explicitly identify which graphical properties lead to what states. This
can be achieved if we have well-behaved normal forms for arbitrary graphs
within this calculus. This article lays down a first attempt at realizing such
normal forms for a restricted class of such graphs, namely simple and regular
graphs. These results should pave the way for the most general cases as part of
future work.
