"I am looking for some simple concrete examples of the ways in which real problems go through graph signal processing and how graph Fourier transforms are obtained."
• A concrete example of a graph Fourier transform, to the Minnesota road network, is presented in Fourier Analysis on Graphs; another example, to genetic profiling for cancer subtype classification, is discussed in Graph SP: Fundamentals and Applications.
The graph Fourier transform allows one to introduce the notion of a "band width" to a graph. By analogy with smooth time signals, which have a narrow frequency band width, a graph that exhibits clustering properties (signals vary little within clusters of highly interconnected nodes) will have a narrow band width in the graph Fourier transform. Such a clustered graph would be sparse in the frequency domain, allowing for a more efficient representation of the data.
• To obtain the graph Fourier transform you could use the Matlab routine GSP_GFT in the Graph Signal Processing Toolbox.