Why in spectral clustering number of eigen vectors is same as number of clusters that we want? In spectral clustering we take eigenvector corresponding to K smallest eigenvalues. Then we do K means clustering on these eigenvector to get final clusters. What will happen if we take different number of eigenvectors than number of clusters we want ?
Theoretically, log2(k) components could be enough.

But usually clusters are not that well balanced, that you could get all 2^l combinations of eigenvectors stable, usually one will mask the other.

If you go back to the theory of spectral clustering, you'd have one eigenvector for each connected component. Now the clusters are unfortunately connected, so we don't completely get this ideal situation, but you'd expect one eigenvector for each "almost component", too.