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

Network models play an important role in studying complex systems in many scientific disciplines. Graph signal processing is receiving growing interest as to design novel tools to combine the analysis of topology and signals. The graph Fourier transform, defined as the eigendecomposition of the graph Laplacian, allows extending conventional signal-processing operations to graphs. One main feature is to let emerge global organization from local interactions; i.e., the Fiedler vector has the smallest non-zero eigenvalue and is key for Laplacian embedding and graph clustering. Here, we introduce the design of Slepian graph signals, by maximizing energy concentration in a predefined subgraph for a given spectral bandlimit. We also establish a link with classical Laplacian embedding and graph clustering, for which the graph Slepian design can serve as a generalization.