Geometric & Functional Analysis GAFA

, Volume 14, Issue 5, pp 913–940

Graph Products, Fourier Analysis and Spectral Techniques

Original Paper

DOI: 10.1007/s00039-004-0478-3

Cite this article as:
Alon, N., Dinur, I., Friedgut, E. et al. Geom. funct. anal. (2004) 14: 913. doi:10.1007/s00039-004-0478-3


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.

Copyright information

© Birkhäuser Verlag, Basel 2004

Authors and Affiliations

  1. 1.Schools of Mathematics and Computer Science, Sackler Faculty of Exact SciencesTel Aviv UniversityTel AvivIsrael
  2. 2.NEC ResearchPrincetonUSA
  3. 3.Institute of MathematicsHebrew UniversityJerusalemIsrael
  4. 4.Department of MathematicsPrinceton UniversityPrincetonUSA
  5. 5.Institute for Advanced StudyPrincetonUSA