The paper defines an Entanglement Matrix and claims maximum entanglement of n-qubit graph states is N^2-N for odd N, with piecewise quadratic formulas for even N and a special enhancement for multiples of 12.
Whereas Secondary midpoints can only contribute to the diagonal entries
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Entanglement Classification in the Graph States: The generalization to $n$-Qubits States using the Entanglement Matrix
The paper defines an Entanglement Matrix and claims maximum entanglement of n-qubit graph states is N^2-N for odd N, with piecewise quadratic formulas for even N and a special enhancement for multiples of 12.