When Slepian Meets Fiedler: Putting a Focus on the Graph Spectrum

The study of complex systems greatly benefits from graph models and their analysis. In particular, the eigendecomposition of the graph Laplacian lets emerge properties of global organization from local interactions; e.g., the Fiedler vector has the smallest nonzero eigenvalue and plays a key role for graph clustering. Graph signal processing focuses on the analysis of signals that are attributed to the graph nodes. Again, the eigendecomposition of the graph Laplacian is important to define the graph Fourier transform and extend conventional signal-processing operations to graphs. Here, we introduce the design of Slepian graph signals by maximizing energy concentration in a predefined subgraph given a graph spectral bandlimit. We establish a novel link with classical Laplacian embedding and graph clustering, which provides a meaning to localized graph frequencies.