On the probability that a random subtree is spanning

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 Erdős–Rényi random graph model G ( n , p ). It is shown that P ( G ) converges in probability to e − 1 ∕ ( e p ∞ ) if p → p ∞ > 0 and to 0 if p → 0. As side results, we find in the dense case that the total number of subtrees satisfies a log‐normal limit law, and that the number of vertices missing in a random subtree asymptotically follows a Poisson distribution.

Solution to a conjecture on the mean subtree order of graphs under edge addition

“…In addition, they studied the ratio $$P(G)$$ of spanning trees to all subtrees in $$G$$ and suggested two conjectures. In 2021, Wagner [12] solved these two conjectures. For the mean subtree order, Chin et al [3] suggested a conjecture as follows:…”