To answer Hsin-Hao's second question.

The original formulation of the segmentation problem is discrete in the sense, that the indicator vector which tells us whether a pixel belongs to which of the two segments, can only be composed to values. In case of normalized cut it was 1 and -b (where -b was a particular constant related to the volume of the graph).

By allowing the vectors to be continuous, we are let these component values range over the entire set of reals, i.e. they can take any real value.

The key insight to understanding how spectral methods in this case are the following three oberservations

1. if the indicator vector is indeed an eigen vector then the value of the normalized cut is equal to the value of the associated eigenvalue.

2. The constraint yD1=0 is an orthogonality constraint wrt to the first eigenvector of that matrix.

3. The theorem about rayleigh constant which relates the minimization of the rayleigh expression (or the Ncut expression) to its eigen vectors.

hope this helps
sameer