  We aim to give more insights on adiabatic evolution concerning the occurrence
of anti-crossings and their link to the spectral minimum gap $\Delta_{min}$. We
study in detail adiabatic quantum computation applied to a specific
combinatorial problem called weighted max $k$-clique. A clear intuition of the
parametrization introduced by V. Choi is given which explains why the
characterization isn't general enough. We show that the instantaneous vectors
involved in the anti-crossing vary brutally through it making the instantaneous
ground-state hard to follow during the evolution. This result leads to a
relaxation of the parametrization to be more general.
