On the roots of Wiener polynomials of graphs

TL;DR

This paper investigates the roots of Wiener polynomials of graphs and trees, revealing their distribution, bounds, and how their roots can have arbitrarily large real and imaginary parts.

Contribution

It characterizes the location and bounds of Wiener polynomial roots for graphs and trees, including the growth of maximum modulus and the real root intervals.

Findings

Maximum root modulus for graphs of order n is (n,2)-1

Maximum root modulus for trees grows linearly with n

Real roots of all graphs are in (-, 0], for trees in (-, -1]

Abstract

The Wiener polynomial of a connected graph $G$ is defined as $W(G;x)=\sum x^{d(u,v)}$, where $d(u,v)$ denotes the distance between $u$ and $v$, and the sum is taken over all unordered pairs of distinct vertices of $G$. We examine the nature and location of the roots of Wiener polynomials of graphs, and in particular trees. We show that while the maximum modulus among all roots of Wiener polynomials of graphs of order $n$ is $\binom{n}{2}-1$, the maximum modulus among all roots of Wiener polynomials of trees of order $n$ grows linearly in $n$. We prove that the closure of the collection of real roots of Wiener polynomials of all graphs is precisely $(-\infty, 0]$, while in the case of trees, it contains $(-\infty, -1]$. Finally, we demonstrate that the imaginary parts and (positive) real parts of roots of Wiener polynomials can be arbitrarily large.