How are the two definitions of graph state mathematically equivalent?

Question

There are at least two definitions of Graph State, two of them are shown in Wikipedia. The first definition is via quantum states, while the second one is from the point of view of the stabilizer group. How are the two definitions mathematically equivalent?

Answer 1

The first definition says the nodes start in the $|+\rangle$ state and then you apply a CZ for each edge.

The $|+\rangle$ state's stabilizer is $X$. When you conjugate an $X$ observable by a CZ operation, you end up with $X \otimes Z$. When you conjugate a $Z$ observable by a CZ you just get the $Z$ unchanged. This means propagating the $X$ stabilizer forward through time, through the CZs, kicks out a $Z$ on the other side of each CZ. So you end up with a stabilizer that's $X$ on the node times $Z$ on each neighbor. That's the second definition.