I have the following questions regarding the graph Laplacian for spectral clustering:

What is the intuition behind projecting the Laplacian (D-A, where D is the degree matrix and A is the affinity matrix) along the k eigenvectors to determine k clusters in the dataset?

The paper: "A tutorial on spectral clustering" by Ulrike von Luxburg mentions that this projection should be done along the first

*k*eigenvectors (eigenvectors with*k**smallest*eigenvalues), while the papers on "Self tuning spectral clustering" by Lihi Zelnik-Manor and Pietro Perona and the Ng Jordan Weiss algorithm suggests selecting the eigenvectors with the largest eigenvalues. My question is why does one suggest selecting the smallest eigenvectors and the others the largest ones for projecting onto a subspace and what is the significance of each?What is the significance of the second smallest eigenvalue being 0.1645?