Homological invariants of Cameron–Walker Graphs

Abstract:

Let $G$ be a finite simple connected graph on $[n]$ and \[ R = K[x_1, \ldots , x_n]\] 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 $x_ix_j$ for which $\{i, j\}$ is an edge of $G$. In the present paper, the possible tuples \[ (n, depth(R/I(G)), reg(R/I(G)), \dim R/I(G), \deg h(R/I(G))),\] where $\deg h(R/I(G))$ is the degree of the $h$-polynomial of $R/I(G)$, arising from Cameron–Walker graphs on $[n]$ will be completely determined.