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Graph products, Fourier analysis and spectral techniques. (English) Zbl 1056.05104
This is a deserving paper with an informative abstract: We consider powers of regular graphs defined by the weak graph product and give a characterization of maximum-size independent sets for a wide family of base graphs which includes, among others, complete graphs, line graphs of regular graphs which contain a perfect matching and Kneser graphs. In many cases this also characterizes the optimal colorings of these products.
We show that the independent sets induced by the base graph are the only maximum-size independent sets. Furthermore we give a qualitative stability statement: any independent set of size close to the maximum is close to some independent set of maximum size.
Our approach is based on Fourier analysis on abelian groups and on spectral techniques. To this end we develop some basic lemmas regarding the Fourier transform of functions on $$\{0,\ldots,r-1\}^n$$, generalizing some useful results from the $$\{0,1\}^n$$ case.

##### MSC:
 05C69 Vertex subsets with special properties (dominating sets, independent sets, cliques, etc.) 42A16 Fourier coefficients, Fourier series of functions with special properties, special Fourier series
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