This paper extends traditional discrete signal processing (DSP) to signals on graphs, extending its basic tenets such as filters, convolution, z-transform, impulse response, spectral representation, Fourier transform, and frequency response. The authors introduce the concept of graph signals, which are data indexed by nodes of a graph, and develop a linear DSP framework for these signals. They focus on linear, shift-invariant filters and show that all such filters are given by polynomials in the graph shift operator. The paper also discusses the graph Fourier transform, which generalizes the classical Fourier transform to graphs, and its applications in classification, compression, and linear prediction for datasets from various sources, including blogs, weather stations, and mobile service providers. The authors emphasize the contrast between their deterministic framework and statistical approaches like graphical models, and highlight the importance of the Jordan normal form in defining the graph Fourier transform for directed graphs and graphs with negative weights.This paper extends traditional discrete signal processing (DSP) to signals on graphs, extending its basic tenets such as filters, convolution, z-transform, impulse response, spectral representation, Fourier transform, and frequency response. The authors introduce the concept of graph signals, which are data indexed by nodes of a graph, and develop a linear DSP framework for these signals. They focus on linear, shift-invariant filters and show that all such filters are given by polynomials in the graph shift operator. The paper also discusses the graph Fourier transform, which generalizes the classical Fourier transform to graphs, and its applications in classification, compression, and linear prediction for datasets from various sources, including blogs, weather stations, and mobile service providers. The authors emphasize the contrast between their deterministic framework and statistical approaches like graphical models, and highlight the importance of the Jordan normal form in defining the graph Fourier transform for directed graphs and graphs with negative weights.