论文标题
在内部和外部多项式的系数上
On coefficients of the interior and exterior polynomials
论文作者
论文摘要
内部多项式和外部多项式是对$（1/ξ，1）$和$（1,1/η）$的估值的概括，分别是多项式$ t_g（x，y）的图表。由连接的两分图引起的一对超图是抽象的二元，被证明具有相同的内部多项式，但可能具有不同的外部多项式。特殊交替链路的Homfly多项式的顶部与该链接的Seifert图引起的一对超图的内部多项式相吻合。令$ g =（v \ cup e，\ varepsilon）$为连接的两部分图。在本文中，我们主要研究内部和外部多项式的系数。我们证明，连接的两分图的内部多项式正在插值。我们在连接的两分图的内部多项式方面加强了已知的结果，并以$ v $或$ e $削减了2个vertex。我们证明，平衡双分配图家族的内部多项式是一致的，并且任何连接的双分子图的内部多项式都可以写为连接平衡的两极图的内部多项式的线性组合。事实证明，超图的外部多项式也被插值。众所周知，内部多项式的线性项的系数是两分图图的无效，我们会在外部多项式的线性项的系数上获得“双重”结果：如果$ g-e $连接到e $中的每个$ e \ e $的每个$ e \ exter linear of exter of exter us $ iS $ is $ iS $ vnomial in $。计算一些两分图家族的内部和外部多项式。
The interior polynomial and the exterior polynomial are generalizations of valuations on $(1/ξ,1)$ and $(1,1/η)$ of the Tutte polynomial $T_G(x,y)$ of graphs to hypergraphs, respectively. The pair of hypergraphs induced by a connected bipartite graph are abstract duals and are proved to have the same interior polynomial, but may have different exterior polynomials. The top of the HOMFLY polynomial of a special alternating link coincides with the interior polynomial of the pair of hypergraphs induced by the Seifert graph of the link. Let $G=(V\cup E, \varepsilon)$ be a connected bipartite graph. In this paper, we mainly study the coefficients of the interior and exterior polynomials. We prove that the interior polynomial of a connected bipartite graph is interpolating. We strengthen the known result on the degree of the interior polynomial for connected bipartite graphs with 2-vertex cuts in $V$ or $E$. We prove that interior polynomials for a family of balanced bipartite graphs are monic and the interior polynomial of any connected bipartite graph can be written as a linear combination of interior polynomials of connected balanced bipartite graphs. The exterior polynomial of a hypergraph is also proved to be interpolating. It is known that the coefficient of the linear term of the interior polynomial is the nullity of the bipartite graph, we obtain a `dual' result on the coefficient of the linear term of the exterior polynomial: if $G-e$ is connected for each $e\in E$, then the coefficient of the linear term of the exterior polynomial is $|V|-1$. Interior and exterior polynomials for some families of bipartite graphs are computed.
