Growth of the Number of Spanning Trees of the Erd\"os-R\'enyi Giant Component

The number of spanning trees in the giant component of the random graph $\text{G}(n,c\mathrm{/}n)$ ($c>1$) grows like $\exp \left\{m\right(f(c)+o(1)\left)\right\}$ as $n\to \mathrm{\infty }$, where $m$ is the number of vertices in the giant component. The function $f$ is not known explicitly, but we show that it is strictly increasing and infinitely differentiable. Moreover, we give an explicit lower bound on $f^{\mathrm{\prime }}(c)$. A key lemma is the following. Let $\text{PGW}(\lambda )$ denote a Galton-Watson tree having Poisson offspring distribution with parameter $\lambda$. Suppose that ${\lambda }^{\ast }>\lambda >1$. We show that $\text{PGW}({\lambda }^{\ast })$ conditioned to survive forever stochastically dominates $\text{PGW}(\lambda )$ conditioned to survive forever.