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 / 2 d 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. ain 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 ) ) 2 d 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.