On roots of Wiener polynomials of trees

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