The Distribution Of Subtrees In Dense Graphs And The Roots Of The Subtree Polynomial

For a graph $G$ with $n$ vertices and a positive integer $k\le n$, let $s_k(G)$ be the number of subtrees (subgraphs that are trees, not necessarily induced) of $G$ with $k$ vertices. The subtree polynomial of $G$ is $S(G;x)={\sum }_{k=1}^n{s}_k(G)x^k$. In this paper, we consider dense connected graphs with a minimum degree that is linear in the number of vertices. We prove that the number of missing vertices in a random subtree is asymptotically Poisson-distributed and deduce that all the roots of the subtree polynomial have to be close to $0$.