  Graph states, and the entanglement they posses, are central to modern quantum
computing and communications architectures. Local complementation---the graph
operation that links all local-Clifford equivalent graph states---allows us to
classify all stabiliser states by their entanglement. Here, we study the
structure of the orbits generated by local complementation, mapping them up to
9 qubits and revealing a rich hidden structure. We provide programs to compute
these orbits, along with our data for each of the 587 orbits up to 9 qubits and
a means to visualise them. We find direct links between the connectivity of
certain orbits with the entanglement properties of their component graph
states. Furthermore, we observe the correlations between graph-theoretical
orbit properties, such as diameter and colourability, with Schmidt measure and
preparation complexity and suggest potential applications. It is well known
that graph theory and quantum entanglement have strong interplay---our
exploration deepens this relationship, providing new tools with which to probe
the nature of entanglement.
