关于独立多项式的复根
On complex roots of the independence polynomial
摘要：
从Shearer（1985）（以及Scott和Sokal（2005））的工作可知，最大次数至多为$d+1$的图$G$的独立多项式$Z_G(λ)$不为零，只要$\vertλ\vert \leq \frac{d^d}{(d+1)^{d+1}}$。最近Peters和Regts（2019）以及Bencs和Csikvári（arxiv：1807.08963）在$\Re λ\geq 0$的案例中给出了该结果的重要扩展。在本文中，我们的动机是进一步推广这些结果，并找到零自由区时，$\Re λ\leq 0$。我们首先给出了新的几何标准，建立零自由区域，以及进行半严格的数值探索。然后，我们提供了两个例子（严格）使用这些标准，通过建立两个新的零自由区域的左半平面。我们还改进了Bencs和Csikvári（arxiv：1807.08963）的右半平面的结果。通过直接应用Barvinok的插值方法，结合Patel和Regts的推广，这些结果也意味着在新的无零区域中有界度图的独立多项式的确定性多项式时间近似算法.
英文摘要：
It is known from the work of Shearer (1985) (and also Scott and Sokal (2005)) that the independence polynomial $Z_G(λ)$ of a graph $G$ of maximum degree at most $d+1$ does not vanish provided that $\vertλ\vert \leq \frac{d^d}{(d+1)^{d+1}}$. Significant extensions of this result have recently been given in the case $\Re λ\geq 0$ by Peters and Regts (2019) and Bencs and Csikvári (arxiv:1807.08963). In this paper, our motivation is to further extend these results and find zero free regions when $\Re λ\leq 0$. We begin by giving new geometric criteria for establishing zero-free regions as well as for carrying out semi-rigorous numerical explorations. We then provide two examples of the (rigorous) use of these criteria, by establishing two new zero-free regions in the left-half plane. We also improve upon the results of Bencs and Csikvári (arxiv:1807.08963) for the right half-plane using our framework. By a direct application of the interpolation method of Barvinok, combined with extensions due to Patel and Regts, these results also imply deterministic polynomial time approximation algorithms for the independence polynomial of bounded degree graphs in the new zero-free regions.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
