Graph states: two equivalent definitions

Question

I know two definitions of graph states:

• $|G\rangle = K^{a} |G\rangle = \sigma_x^{a} \otimes \sigma_z^{b} |G\rangle$ for all $a \in V$ vertices and for all $b\in Na$ connected to $a$ through an edge.
• $|G\rangle = \Pi_{a,b} Cz^{(a,b)} |+\rangle$ where $a,b$ are the same as before, Cz is a controlled $\sigma_z$, and $|+\rangle = |0\rangle + |1\rangle$ (ignoring normalization)

I would like to prove their equivalence. It's easy to show that the second implies the first one, but I have no idea on how to prove the other arrow. Thank you all!

Answer 1

check this page, based on this paper.