  The entanglement properties of a multiparty pure state are invariant under
local unitary transformations. The stabilizer dimension of a multiparty pure
state characterizes how many types of such local unitary transformations
existing for the state. We find that the stabilizer dimension of an $n$-qubit
($n\ge 2$) graph state is associated with three specific configurations in its
graph. We further show that the stabilizer dimension of an $n$-qubit ($n\ge 3$)
graph state is equal to the degree of irreducible two-qubit correlations in the
state.
