  Quantum degeneracy in error correction is a feature unique to quantum error
correcting codes, unlike their classical counterpart. It allows a quantum error
correcting code to correct errors even when they can not uniquely pinpoint the
error. The diagonal distance of a quantum code is an important parameter that
characterizes if the quantum code is degenerate or not. If code has a distance
more than the diagonal distance, then it is degenerate, whereas if it is below
the diagonal distance, then it is nondegenerate. We show that most of the CWS
codes without a cycle of length four attain the upper bound of diagonal
distance d+1, where d is the minimum vertex degree of the associated graph.
Addressing the question of degeneracy, we give necessary conditions on CWS
codes to be degenerate. We show that any degenerate CWS code with graph $G$ and
classical code C will either have a short cycle in the graph $G$ or will be
such that the classical code C has one of the coordinates trivially zero for
all codewords.
