论文标题
Cameron-Walker图的规律性和$ h $ -Polynomial
The regularity and $h$-polynomial of Cameron-Walker graphs
论文作者
论文摘要
修复整数$ n \ geq 1 $，并考虑$ n $顶点上所有连接的有限简单图的集合。对于此组中的每个$ g $，令$ i（g）$表示$ g $ $ g $在多项式环$ r = k [x_1，\ ldots，x_n] $中的边缘理想。我们开始对集合的研究$ \ MATHCAL {rd}（n）\ subseteq \ mathbb {n}^2 $由所有对$（r，d）组成，其中$ r = {\ rm reg}（rm reg}（r/i（g）） h_ {r/i（g）}（t）$，$ h $ -polynomial的度数，因为我们在$ n $顶点上的所有连接图上都有所不同。特别是，我们确定集合$ a（n）$和$ b（n）$，以便$ a（n）\ subseteq \ mathcal {rd}（n）\ subseteq b（n）$。当我们限制在$ n $顶点上的Cameron-Walker图形家族时，我们可以完全表征所有可能的$（R，d）$。
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 $(r,d)$.
