# Theoretical link between the graph diffusion/heat kernel and spectral clustering

The graph diffusion kernel of a Graph is the exponential of its Laplacian $\exp(-\beta L)$ (or a similar expression depending on how you define the kernel). If you have labels on some vertices, you can get labels on the rest of the vertices by a majority vote of adding the kernel contribution. In spectral clustering, the sign of the eigenvector components of the largest eigenvalue of the Laplacian determines the class assignment of the vertices. Since these techniques seem to be doing similar things based on the graph Laplacian, is there a link between spectral clustering and the diffusion or heat kernel? Is one the generalization of the other under some assumptions?

3. Following "Graph spectral image smoothing using the heat kernel" by Zhan and Hancock, we can observe that if $L = USU^T$, (with $U$ as the eigenvectors and $S$ the diagonal matrix of eigenvalues) and $H=\exp(-\beta L)$, then $H = Uexp(-\beta S)U^T$. So for some $\beta$ we just have to take the smallest eigenvalue to get the approximate $H$.