Growth of the Number of Spanning Trees of the Erdős–Rényi Giant Component

Abstract

The number of spanning trees in the giant component of the random graph (n, c/n) (c > 1) grows like exp{m(f(c)+o(1))} as n → ∞, 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 f2(c). A key lemma is the following. Let PGW (λ) denote a GaltonWatson tree having Poisson offspring distribution with parameter λ. Suppose that λ*>λ>1. We show that PGW(λ*) conditioned to survive forever stochastically dominates PGW(λ) conditioned to survive forever.