Basically, spectral clustering is an application of spectral graph theory, which utilizes the eigenvalues and eigenvectors of a Laplacian matrix or adjacency matrix to disclose the connected components of a graph.

Let's talk about the Laplacian matrix. Ideally, for a graph with a few connected components, the values of the eigenvector corresponding to the 2nd smallest eigenvalue is a piecewise linear vector that perfectly corresponds to the components. As showed in the following figure:

The figure shows five connected components (clusters). However, I wonder if there are any actual meaning for the magnitudes of these values?

For example, the 4th component has the largest values while 5th component has the smallest, what does that tell us about these two components?

Basically, spectral clustering is an application of spectral graph theory, which utilizes the eigenvalues and eigenvectors of a Laplacian matrix or adjacency matrix to disclose the connected components of a graph.

Let's talk about the Laplacian matrix. Ideally, for a graph with a few connected components, the values of the eigenvector corresponding to the 2nd smallest eigenvalue is a piecewise linear vector that perfectly corresponds to the components. As showed in the following figure:

The figure shows five connected components (clusters). However, I wonder if there are any actual meaning for the magnitudes of these values?

For example, the 4th component has the smallest absolute values while 5th component has the largest, what does that tell us about these two components?