  We investigate the properties of different levels of entanglement in graph
states which correspond to connected graphs. Combining the operational
definition of graph states and the postulates of entanglement measures, we
prove that in connected graph states of N>3 qubits there is no genuine
three-qubit entanglement. For certain classes of graph states, all genuine
k-qubit entanglement, 2\leq k\leq N-1, among every k qubits vanishes. These
results about connected graph states naturally lead to the definition of fully
multi-qubit entangled states. We also find that the connected graph states of
four qubits is one but not the only one class of fully four-qubit entangled
states.
