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 field $K$. The edge ideal of is $I(G)$ $R$ which generated by those monomials $x_ix_j$ for $\{i, j\}$ an $G$. In present paper, possible tuples $(n, {\rm depth} (R/I(G)), reg} \dim R/I(G), deg} \ h(R/I(G)))$, where ${\rm h(R/I(G))$ degree $h$-polynomial $R/I(G)$, arising from Cameron--Walker graphs will completely determined.