 
Summary: Graph Products, Fourier Analysis and Spectral Techniques
Noga Alon
Irit Dinur
Ehud Friedgut
Benny Sudakov §
October 21, 2003
Abstract
We consider powers of regular graphs defined by the weak graph product and give a character
ization of maximumsize 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 maximumsize
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, . . . , r  1}n
,
generalizing some useful results from the {0, 1}n
case.
1 Introduction
