On the probability that a random subtree is spanning

We consider the quantity P ( G ) associated with a graph that is defined as probability randomly chosen subtree of spanning. Motivated by conjectures due to Chin, Gordon, MacPhee and Vincent on behaviour this invariant depending edge density, we establish first bounded below positive constant provided minimum degree linear function in number vertices. Thereafter, focus shifted classical Erd?s–Rényi random model n , p . It shown converges e ? 1 ? ? if ? > 0 0. As side results, find dense case total subtrees satisfies log-normal limit law, vertices missing asymptotically follows Poisson distribution.