  The $n$-qubit stabilizer states are those left invariant by a $2^n$-element
subset of the Pauli group. The Clifford group is the group of unitaries which
take stabilizer states to stabilizer states; a physically--motivated generating
set, the Hadamard, phase, and CNOT gates which comprise the Clifford gates,
imposes a graph structure on the set of stabilizers. We explicitly construct
these structures, the "reachability graphs," at $n\le5$. When we consider only
a subset of the Clifford gates, the reachability graphs separate into multiple,
often complicated, connected components. Seeking an understanding of the
entropic structure of the stabilizer states, which is ultimately built up by
CNOT gate applications on two qubits, we are motivated to consider the
restricted subgraphs built from the Hadamard and CNOT gates acting on only two
of the $n$ qubits. We show how the two subgraphs already present at two qubits
are embedded into more complicated subgraphs at three and four qubits. We argue
that no additional types of subgraph appear beyond four qubits, but that the
entropic structures within the subgraphs can grow progressively more
complicated as the qubit number increases. Starting at four qubits, some of the
stabilizer states have entropy vectors which are not allowed by holographic
entropy inequalities. We comment on the nature of the transition between
holographic and non-holographic states within the stabilizer reachability
graphs.
