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arXiv 1807.10967math.CO
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关于树的Wiener多项式的根

On roots of Wiener polynomials of trees

Danielle Wang

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摘要：

连通图$G$的\emph{Wiener polynomial}是多项式$W(G;x) = \sum_{i=1}^{D(G)} d_i(G)x^i$，其中$D(G)$是$G$的直径，而$d_i(G)$是彼此相距$i$的顶点对的数目。我们研究了树的维纳多项式的根。证明了树的实Wiener根集在$(-\infty, 0]$中是稠密的，树的复Wiener根集在$\mathbb C$中是稠密的.我们还证明了树的顺序$n \ge 31$的所有维纳根之间的最大模是$2n-15$和$2n-16$之间，我们确定的唯一树，达到最大值$n \ge 31$。最后，我们找到任意大直径的树，其Wiener根都是实数。

英文摘要：

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.

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