交错多项式：Enumeration、单峰和与代码的连接
Interlace Polynomials: Enumeration, Unimodality, and Connections to Codes
摘要：
交错多项式q由Arratia、Bollobas和Sorkin引入。它在边局部互补（ELC）下编码了图的轨道的许多性质。由Aigner和van der Holst引入的交错多项式Q，同样包含关于局部互补（LC）下图的轨道的信息。我们以前分类LC和ELC轨道，现在给一个枚举的相应的交错多项式的所有图的顺序高达12。最后给出了阶数不超过12的所有圈图的一个计数.我们发现，存在图的所有阶数大于大于9交错多项式q的系数序列是非单峰的，从而反驳了Arratia等人的猜想。我们已经验证了，对于图的阶数高达12，所有的多项式Q有单峰系数。已经证明图的LC和ELC轨道对应于某些纠错码和量子态的等价类。我们表明，这些代码和量子态的属性相关的相关交织多项式的属性。
英文摘要：
The interlace polynomial q was introduced by Arratia, Bollobas, and Sorkin. It encodes many properties of the orbit of a graph under edge local complementation (ELC). The interlace polynomial Q, introduced by Aigner and van der Holst, similarly contains information about the orbit of a graph under local complementation (LC). We have previously classified LC and ELC orbits, and now give an enumeration of the corresponding interlace polynomials of all graphs of order up to 12. An enumeration of all circle graphs of order up to 12 is also given. We show that there exist graphs of all orders greater than 9 with interlace polynomials q whose coefficient sequences are non-unimodal, thereby disproving a conjecture by Arratia et al. We have verified that for graphs of order up to 12, all polynomials Q have unimodal coefficients. It has been shown that LC and ELC orbits of graphs correspond to equivalence classes of certain error-correcting codes and quantum states. We show that the properties of these codes and quantum states are related to properties of the associated interlace polynomials.
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