  We present a study of quantum contextuality of three-dimensional mixed states
for the Klyachko-Can-Binicio\u{g}lu-Shumovsky (KCBS) and the
Kurzy\'{n}ski-Kaszlikowski (KK) noncontextuality inequalities. For any class of
states whose eigenvalues are arranged in decreasing order, a universal set of
measurements always exists for the KK inequality, whereas none does for the
KCBS inequality. This difference can be reflected from the spectral
distribution of the overall measurement matrix. Our results would facilitate
the error analysis for experimental setups, and our spectral method in the
paper combined with graph theory could be useful in future studies on quantum
contextuality.
