Upper bound for the number of spanning forests of regular graphs

We show that if $G$ is a $d$–regular graph on $n$ vertices, then the number of spanning forests $F(G)$ satisfies $F(G)\le d^n$. The previous best bound due to Kahale and Schulman gave $(d+1\mathrm{/}2+O(1\mathrm{/}d){)}^n$. We also have the more precise conjecture that $$F(G{)}^{1\mathrm{/}n}\le \frac{(d-1{)}^{d-1}}{(d^2-2d-1{)}^{d\mathrm{/}2-1}}\mathrm{.}$$ If this conjecture is true, then the expression on the right hand side is the best possible.