If we can label the nodes in a graph such that the Laplacian matrix of the graph is circulant, then such a graph is known as a shift invariant graph [16]. The eigenvectors of the Laplacian matrix of a shift invariant graph are the same as the columns of the DFT matrix. One example of a shift invariant graph is the ring graph. Draw two more shift invariant graphs and verify that the eigenvectors of the corresponding Laplacian matrices are the same as the columns of the DFT matrix. Also comment on the eigenvalues of the Laplacians of the shift invariant graphs.
Figure 8.4 shows two graphs that are shift-invariant: an infinite 2-D lattice and a complete
graph.
The eigenvalues of the Laplacian matrix of a shift-invariant graph is
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