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

TL;DR

This paper studies the roots of the subtree polynomial in dense graphs, showing they are close to zero and that the number of missing vertices in a random subtree follows a Poisson distribution.

Contribution

It establishes the asymptotic distribution of missing vertices and the proximity of subtree polynomial roots to zero in dense graphs.

Findings

Number of missing vertices in a random subtree is asymptotically Poisson-distributed.

All roots of the subtree polynomial are close to zero in dense graphs.

Results apply to graphs with minimum degree linear in the number of vertices.

Abstract

For a graph $G$ with $n$ vertices and a positive integer $k \leq n$, 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 $0$.