> "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 <A HREF="http://www.norbertwiener.umd.edu/Research/lectures/2014/MBegue_Prelim.pdf">Fourier Analysis on Graphs</A>; another example, to genetic profiling for cancer subtype classification, is discussed in <A HREF="http://www.mit.edu/~segarra/tutorials/GSP_Tutorial_ICASSP.pdf">Graph SP: Fundamentals and Applications.</A>  
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 <A HREF="https://epfl-lts2.github.io/gspbox-html/doc/operators/gsp_gft_code.html">GSP_GFT</A> in the <A HREF="https://epfl-lts2.github.io/gspbox-html/">Graph Signal Processing Toolbox</A>.