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 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,...,r − 1} n , generalizing some useful results from the {0, 1} n case. 1 Introduction Consider the following combinatorial problem: Assume that at a given road junction there are n three-position switches that control the red- yellow-green position of the traffic light. You are told that whenever you change the position of all the switches then the color of the light changes. Prove that in fact the light is controlled by only one of the switches. The above problem is a special case of the problem we wish to tackle in this paper, characterizing the optimal colorings and maximal independent sets of products of regular graphs. The configuration space of the switches described above can be modeled by the n-fold product of K 3 . Let us begin by defining the weak graph product of two graphs. The weak product of G and H , denoted by G × H is defined as follows: the vertex set of G × H is the Cartesian product of the vertex sets of G and H . Two vertices (g 1 ,h 1 ) and (g 2 ,h 2 ) are adjacent in G × H if g 1 g 2 is an edge of G and h 1 h 2 is an edge of H . The “times” symbol, ×, is supposed to be reminiscent of the weak product of two edges: |×− = ×. In this paper “graph product” will always mean the weak product. ∗ Schools of Mathematics and Computer Science, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, Israel. Email: noga@math.tau.ac.il. Research supported in part by a USA-Israeli BSF grant, by the Israel Science Foundation and by the Hermann Minkowski Minerva Center for Geometry at Tel Aviv University. † NEC Research, Princeton, NJ 08540, USA. Email: iritd@nec-labs.com. ‡ Institute of Mathematics, Hebrew University, Jerusalem, Israel. Email: ehudf@math.huji.ac.il. Research supported in part by the Israel Science Foundation. Part of this research was done while visiting Microsoft Research and the Institute for Advanced Study in Princeton. § Department of Mathematics, Princeton University, Princeton, NJ 08540, USA and Institute for Advanced Study, Princeton, NJ 08540, USA. Email address: bsudakov@math.princeton.edu. Research supported in part by NSF grants DMS-0106589, CCR-9987845 and by the State of New Jersey. 1