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

TL;DR

This paper investigates the growth rate of the number of spanning trees in the giant component of Erdős-Rényi random graphs, establishing properties of an unknown function related to this growth and providing bounds and stochastic dominance results.

Contribution

It introduces the function describing the growth rate of spanning trees in the giant component and proves its properties, including strict monotonicity, differentiability, and a lower bound on its derivative.

Findings

The number of spanning trees grows exponentially with the size of the giant component.

The function governing growth is strictly increasing and infinitely differentiable.

A stochastic dominance relation for Galton-Watson trees with different parameters is established.

Abstract

The number of spanning trees in the giant component of the random graph $\G(n, c/n)$ ($c>1$) grows like $\exp\big\{m\big(f(c)+o(1)\big)\big\}$ as $n\to\infty$, 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 $f'(c)$. A key lemma is the following. Let $\PGW(\lambda)$ denote a Galton-Watson tree having Poisson offspring distribution with parameter $\lambda$. Suppose that $\lambda^*>\lambda>1$. We show that $\PGW(\lambda^*)$ conditioned to survive forever stochastically dominates $\PGW(\lambda)$ conditioned to survive forever.