This problem is about quantifying spread of a graph signal. The spread of a graph signal can be defined in the vertex as well as the spectral domains. In the vertex domain, the spread of a signal f lying on a graph G about a node v i is defined as

Question

Based on the graph and spectral spread definitions, answer the following:




(a) Prove that









where ˆf(?`) is the GFT coefficient at frequency ?




(b) Write an expression for spectral spread of the eigenvectors of the graph Laplacian. What




is the relation between the spread of the eigenvectors?




(c) For the graph shown in Figure 11.1, find graph spreads of the eignevectors of the Lapla-




cian

Answer 1

(a)




Proof.









(b) The spectral spread of the eigenvector u` is









(c) The graph and spectral spreads of the Laplacian eigenvectors are listed in Table 11.1









(d) Graph spread will be zero and the spectral spread will be the degree of the Ith node. Therefore,