On the probability that a random subtree is spanning

Abstract

AbstractWe consider the quantity associated with a graph that is defined as the probability that a randomly chosen subtree of 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 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 . It is shown that converges in probability to if and to 0 if . 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.