论文标题
图复杂性和链接着色
Graph Complexity and Link Colorings
论文作者
论文摘要
有限签名的图的（扭转）复杂性被定义为其Laplacian矩阵提出的Abelian组的扭转亚组的顺序。当$ g $是$ d $ - periodic（即，$ g $具有免费的$ {\ mathbb z}^d $ - 通过图形自动形态具有有限商的图形）Mahler衡量其Laplacian多项式的MAHLER量度是$ g $ $ g $的有限标准的复杂性的增长率。任何1个周期的平面图$ g $都用未打结的组件$ c $确定链接$ \ ell \ ell \ cup c $。在这种情况下，$ g $的拉普拉斯多项式与链接的亚历山大多项式有关。 Lehmer的问题是关于Monic积分多项式根源的一个开放问题，相当于关于不一定嵌入的签名的1个周期图的复杂性生长的问题。
The (torsion) complexity of a finite signed graph is defined to be the order of the torsion subgroup of the abelian group presented by its Laplacian matrix. When $G$ is $d$-periodic (i.e., $G$ has a free ${\mathbb Z}^d$-action by graph automorphisms with finite quotient) the Mahler measure of its Laplacian polynomial is the growth rate of the complexity of finite quotients of $G$. Any 1-periodic plane graph $G$ determines a link $\ell \cup C$ with unknotted component $C$. In this case the Laplacian polynomial of $G$ is related to the Alexander polynomial of the link. Lehmer's question, an open question about the roots of monic integral polynomials, is equivalent to a question about the complexity growth of signed 1-periodic graphs that are not necessarily embedded.
