  Write ${\cal I}(G)$ for the set of independent sets of a graph $G$ and $i(G)$
for $|{\cal I}(G)|$. It has been conjectured (by Alon and Kahn) that for an
$N$-vertex, $d$-regular graph $G$, $$ i(G) \leq \left(2^{d+1}-1\right)^{N/2d}.
$$ 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 $$ i(G)\leq
2^{\frac{N}{2}\left(1+\frac{2}{d}\right)} $$ for $G$ not necessarily bipartite.
In this note, we improve this to $$ i(G)\leq
2^{\frac{N}{2}\left(1+\frac{1+o(1)}{d}\right)} $$ where $o(1) \rightarrow 0$ as
$d \rightarrow \infty$, 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 $P(\lambda,G)=\sum_{I \in {\cal I}(G)} \lambda^{|I|}$ of an
$N$-vertex, $d$-regular graph $G$, namely $$ P(\gl,G) \leq
(1+\gl)^{\frac{N}{2}} 2^{\frac{N(1+o(1))}{2d}} $$ valid for all $\gl > 0$. 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.
