On the probability that a random subtree is spanning

TL;DR

This paper investigates the probability that a randomly chosen subtree of a graph is spanning, establishing bounds based on minimum degree and analyzing its behavior in Erdős-Rényi random graphs as the edge probability varies.

Contribution

It proves a lower bound for the spanning subtree probability based on minimum degree and characterizes its asymptotic behavior in Erdős-Rényi graphs.

Findings

$P(G)$ is bounded below by a positive constant if minimum degree is linear in vertices.

In $G(n,p)$, $P(G)$ converges to $e^{-1/(ep_{ ext{infty}})}$ for fixed $p_{ ext{infty}}>0$.

$P(G)$ tends to zero as $p$ approaches zero.

Abstract

We consider the quantity $P(G)$ associated with a graph $G$ that is defined as the probability that a randomly chosen subtree of $G$ is spanning. Motivated by conjectures due to Chin, Gordon, MacPhee and Vincent on the behaviour of this graph invariant depending on the edge density, we establish first that $P(G)$ is bounded below by a positive constant provided that the minimum degree is bounded below by a linear function in the number of vertices. Thereafter, the focus is shifted to the classical Erd\H{o}s-R\'enyi random graph model $G(n,p)$. It is shown that $P(G)$ converges in probability to $e^{-1/(ep_{\infty})}$ if $p \to p_{\infty} > 0$ and to $0$ if $p \to 0$.