On roots of Wiener polynomials of trees

TL;DR

This paper studies the roots of Wiener polynomials of trees, showing their roots are dense in the complex plane and real roots can be found for trees with arbitrarily large diameter, with specific bounds on maximum modulus.

Contribution

It proves the density of Wiener roots in the complex plane and real line, determines the tree with maximum Wiener root modulus for large n, and constructs trees with all real roots and large diameter.

Findings

Real Wiener roots are dense in (-∞, 0]

Complex Wiener roots are dense in ℂ

Maximum Wiener root modulus for large n is between 2n-15 and 2n-16

Abstract

The \emph{Wiener polynomial} of a connected graph $G$ is the polynomial $W(G;x) = \sum_{i=1}^{D(G)} d_i(G)x^i$ where $D(G)$ is the diameter of $G$, and $d_i(G)$ is the number of pairs of vertices at distance $i$ from each other. We examine the roots of Wiener polynomials of trees. We prove that the collection of real Wiener roots of trees is dense in $(-\infty, 0]$, and the collection of complex Wiener roots of trees is dense in $\mathbb C$. We also prove that the maximum modulus among all Wiener roots of trees of order $n \ge 31$ is between $2n-15$ and $2n-16$, and we determine the unique tree that achieves the maximum for $n \ge 31$. Finally, we find trees of arbitrarily large diameter whose Wiener roots are all real.