On the probability that a random subtree is spanning

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 $e^{-1\mathrm{/}(e{p}_{\mathrm{\infty }})}$ if $p\to p_{\mathrm{\infty }}>0$ and to $0$ if $p\to 0$.