蜘蛛图的独立多项式的稳定性
Stability of independence polynomials of spiders
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- Taiyuan University of Technology（太原理工大学）
- Beijing Technology and Business University（北京工商大学）
机构由 AI 辅助整理，请以论文原文为准。
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中文总结 AI 辅助
本文证明由星图经任意非均匀边细分得到的蜘蛛图的独立根均在开左半平面内，故所有蜘蛛图都是稳定的，拓展了Brown和Cameron关于稳定图的相关结果。
中文摘要 AI 辅助
对于图$G$，令$i_k(G)$表示基数为$k$的独立集的数量，$I(G,z)=\textstyle\bigoplus_{k\boldsymbol{\u2265}0} i_k(G)z^k$为其独立多项式。依据Brown和Cameron（2018）的定义，若一个图的所有独立多项式零点都位于闭左半平面内，则称该图是稳定的。他们证明了每一个星图都是稳定的，但也构造出了非稳定的树，随后提出了稳定树的刻画问题。在本文中，我们拓展并强化了他们的结果，证明了所有蜘蛛图（由星图经任意、可能非均匀的边细分得到）的独立根都位于开左半平面内，因此每一个蜘蛛图都是稳定的。
英文摘要
For a graph $G$, let $i_k(G)$ denote the number of independent sets of cardinality $k$, and let \[ I(G,z)=\sum_{k\ge0} i_k(G)z^k \] be its independence polynomial. Following Brown and Cameron \cite{BrownCameron2018}, a graph is called stable if all zeros of its independence polynomial lie in the closed left half-plane. They proved that every star is stable, but also constructed nonstable trees. They then asked for a characterization of stable trees. In this paper, we extend and strengthen their result by proving that every spider, obtained from a star by arbitrary and possibly nonuniform subdivisions of its edges, has all its independence roots in the open left half-plane. Hence, every spider is stable.
