An upper bound for the number of independent sets in regular graphs

Write I(G) for the set of independent sets of a graph G and i(G) for |I(G)|. It has been conjectured (by Alon and Kahn) that for an N-vertex, d-regular graph G, i(G)@?(2^d^+^1-1)^N^/^2^d. If true, this bound would be tight, being achieved by the disjoint union of N/2d copies of K"d","d. Kahn established the bound for bipartite G, and later gave an argument that established i(G)@?2^N^2^(^1^+^2^d^) for G not necessarily bipartite. In this note, we improve this to i(G)@?2^N^2^(^1^+^1^+^o^(^1^)^d^) where o(1)->0 as d->~, which matches the conjectured upper bound in the first two terms of the exponent. We obtain this bound as a corollary of a new upper bound on the independent set polynomial P(@l,G)[email protected]?"I"@?"I"("G")@l^|^I^| of an N-vertex, d-regular graph G, namely P(@l,G)@?([email protected])^N^22^N^(^1^+^o^(^1^)^)^2^d valid for all @l>0. This also allows us to improve the bounds obtained recently by Carroll, Galvin and Tetali on the number of independent sets of a fixed size in a regular graph.