 
Summary: The Shannon capacity of a graph and the independence numbers of
its powers
Noga Alon
Eyal Lubetzky
June 15, 2005
Abstract
The independence numbers of powers of graphs have been long studied, under several def
initions of graph products, and in particular, under the strong graph product. We show that
the series of independence numbers in strong powers of a fixed graph can exhibit a complex
structure, implying that the Shannon Capacity of a graph cannot be approximated (up to a
subpolynomial factor of the number of vertices) by any arbitrarily large, yet fixed, prefix of the
series. This is true even if this prefix shows a significant increase of the independence number
at a given power, after which it stabilizes for a while.
1 Introduction
Given two graphs, G1 and G2, their strong graph product G1 · G2 has a vertex set V (G1) × V (G2),
and two distinct vertices (v1, v2) and (u1, u2) are connected iff they are adjacent or equal in each
coordinate (i.e., for i {1, 2}, either vi = ui or viui E(Gi)). This product is associative and
commutative, and we can thus define Gk as the product of k copies of G. In [11], Shannon introduced
the parameter c(G), the Shannon Capacity of a graph G, which is the limit limk
k
