  We propose a systematic procedure for the construction of graphs associated
with binary quantum stabilizer codes. The procedure is characterized by means
of the following three step process. First, the stabilizer code is realized as
a codeword-stabilized (CWS) quantum code. Second, the canonical form of the CWS
code is determined and third, the input vertices are attached to the graphs. In
order to verify the effectiveness of the procedure, we implement the Gottesman
stabilizer code characterized by multi-qubit encoding operators for the
resource-efficient error correction of arbitrary single-qubit errors. Finally,
the error-correcting capabilities of the Gottesman eight-qubit quantum
stabilizer code is verified in graph-theoretic terms as originally advocated by
Schlingemann and Werner.
