The regularity and $h$-polynomial of Cameron-Walker graphs

Abstract

Fix an integer $n\ge 1$, 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 $R=K[x_1,\dots ,x_n]$. We initiate a study of the set $\mathcal{RD}(n)\subseteq {\mathbb{N}}^2$ consisting of all the pairs $(r,d)$ where $r=\mathrm{reg}(R/I(G))$, the Castelnuovo-Mumford regularity, and $d=\mathrm{deg}{h}_{R/I(G)}(t)$, 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 $(r,d)$.

15 pages; comments welcomed