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Independent sets in tensor graph powers Eyal Lubetzky

Summary: Independent sets in tensor graph powers
Noga Alon
Eyal Lubetzky
May 25, 2006
The tensor product of two graphs, G and H, has a vertex set V (G) V (H) and an edge
between (u, v) and (u , v ) iff both uu E(G) and vv E(H). Let A(G) denote the limit
of the independence ratios of tensor powers of G, lim (Gn
)/|V (Gn
)|. This parameter was
introduced in [5], where it was shown that A(G) is lower bounded by the vertex expansion ratio
of independent sets of G. In this note we study the relation between these parameters further,
and ask whether they are in fact equal. We present several families of graphs where equality
holds, and discuss the effect the above question has on various open problems related to tensor
graph products.
1 Introduction
The tensor product (also dubbed as categorical or weak product) of two graphs, GH, is the graph
whose vertex set is V (G)V (H), where two vertices (u, v),(u , v ) are adjacent iff both uu E(G)
and vv E(H), i.e., the vertices are adjacent in each of their coordinates. Clearly, this product is
associative and commutative, thus Gn is well defined to be the tensor product of n copies of G.


Source: Alon, Noga - School of Mathematical Sciences, Tel Aviv University


Collections: Mathematics