Homological invariants of Cameron–Walker Graphs

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 \[ $(n,depth(R/I(G)),reg(R/I(G)),\dim R/I(G),\mathrm{deg}h(R/I(G))),$ \] 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.