Homological invariants of Cameron--Walker graphs

Abstract

Let $G$ be a finite simple connected graph on $[n]$ and $R=K[x_1,\dots ,x_n]$ the polynomial ring in $n$ variables over a field $K$. The edge ideal of $G$ is the ideal $I(G)$ of $R$ which is generated by those monomials $x_i{x}_j$ for which $\{i,j\}$ is an edge of $G$. In the present paper, the possible tuples $(n,\mathrm{depth}(R/I(G)),\mathrm{reg}(R/I(G)),\dim R/I(G),\mathrm{deg}h(R/I(G)))$, where $\mathrm{deg}h(R/I(G))$ is the degree of the $h$-polynomial of $R/I(G)$, arising from Cameron--Walker graphs on $[n]$ will be completely determined.

26 pages, 9 figures; revised and corrected