Cameron-Walker图的正则性和$h$ -多项式
The regularity and $h$-polynomial of Cameron-Walker graphs
摘要：
固定一个整数$n \geq 1$，并考虑$n \geq 1$顶点上的所有连通有限简单图的集合。对于该集合中的每个$G$，令$I(G)$表示多项式环$R = K[x_1,\ldots,x_n]$中的$G$的边理想。我们开始研究的集合$\mathcal{RD}(n) \subseteq \mathbb{N}^2$组成的所有对$(r,d)$在$r = {\rm reg}(R/I(G))$，Castelnuovo-Mumford正则性，和$d = {\rm deg} h_{R/I(G)}(t)$，$h$ -多项式的程度，因为我们改变了所有的$n$顶点上的连接图。特别地，我们识别集合$A(n)$和$B(n)$，使得$A(n) \subseteq \mathcal{RD}(n) \subseteq B(n)$。当我们限制在$n$顶点上的Cameron-Walker图族时，我们可以完全刻画所有可能的$(r,d)$ .
英文摘要：
Fix an integer $n \geq 1$, and consider the set of all connected finite simple graphs on $n$ vertices. For each $G$ in this set, let $I(G)$ denote the edge ideal of $G$ in the polynomial ring $R = K[x_1,\ldots,x_n]$. We initiate a study of the set $\mathcal{RD}(n) \subseteq \mathbb{N}^2$ consisting of all the pairs $(r,d)$ where $r = {\rm reg}(R/I(G))$, the Castelnuovo-Mumford regularity, and $d = {\rm deg} h_{R/I(G)}(t)$, the degree of the $h$-polynomial, as we vary over all the connected graphs on $n$ vertices. In particular, we identify sets $A(n)$ and $B(n)$ such that $A(n) \subseteq \mathcal{RD}(n) \subseteq B(n)$. When we restrict to the family of Cameron-Walker graphs on $n$ vertices, we can completely characterize all the possible $(r,d)$.
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