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 − 1 )N/2d . If true, this bound would be tight, being achieved by the disjoint union ofN/2d copies ofKd,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 (λ,G) = ∑ I∈I(G) λ |I| of an N -vertex, d-regular graph G, namely P (λ,G) ≤ (1 + λ) N 2 2 N(1+o(1)) 2d valid for all λ > 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. 2000 Mathematics Subject Classification: 05C69 (Primary), 05A16, 82B20 (Secondary)