Consider one dimensional quantum codes $[[ n,k=0]] $. One way to describe them is using stabilizer framework with $n$ independent Pauli matrices. Usually, one considers them in the graph state model as well. If you use the distance definition by treating them as stabilizer codes, then as $N(S)=S$, it would detect all errors. Is there a notion of distance of such graph states? I have seen distance being defined after restriction that they are non-degenerate with a mention that graph states are traditionally treated as non-degenerate. Are there any applications where this distance (or some other notion of distance of graph states) plays an important role?