Homological invariants of Cameron--Walker graphs

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 $(n, {rm depth} (R/I(G)), {rm reg} (R/I(G)), dim R/I(G), {rm deg} h(R/I(G)))$, where ${rm deg} h(R/I(G))$ is the degree of the $h$-polynomial of $R/I(G)$, arising from Cameron--Walker graphs on $[n]$ will be completely determined.