Independent sets in tensor graph powers

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, limα(G)/|V (G)|. 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.