  Let ${\rm ind}(G)$ be the number of independent sets in a graph $G$. We show
that if $G$ has maximum degree at most $5$ then $$ {\rm ind}(G) \leq 2^{{\rm
iso}(G)} \prod_{uv \in E(G)} {\rm ind}(K_{d(u),d(v)})^{\frac{1}{d(u)d(v)}} $$
(where $d(\cdot)$ is vertex degree, ${\rm iso}(G)$ is the number of isolated
vertices in $G$ and $K_{a,b}$ is the complete bipartite graph with $a$ vertices
in one partition class and $b$ in the other), with equality if and only if each
connected component of $G$ is either a complete bipartite graph or a single
vertex. This bound (for all $G$) was conjectured by Kahn.
  A corollary of our result is that if $G$ is $d$-regular with $1 \leq d \leq
5$ then $$ {\rm ind}(G) \leq \left(2^{d+1}-1\right)^\frac{|V(G)|}{2d}, $$ with
equality if and only if $G$ is a disjoint union of $V(G)/2d$ copies of
$K_{d,d}$. This bound (for all $d$) was conjectured by Alon and Kahn and
recently proved for all $d$ by the second author, without the characterization
of the extreme cases.
  Our proof involves a reduction to a finite search. For graphs with maximum
degree at most $3$ the search could be done by hand, but for the case of
maximum degree $4$ or $5$, a computer is needed.
