Regularity and a-invariant of Cameron–Walker graphs

Let $S$ be the polynomial ring over a field $K$ and $I \subset S$ homogeneous ideal. $h(S/I,\lambda)$ $h$-polynomial of $S/I$ $s = \mathrm{deg} h(S/I,\lambda)$ degree $h(S/I,\lambda)$. It follows that inequality - r \leq d e$, where $r \mathrm{reg} (S/I)$, $d \dim S/I$ $e \mathrm{depth} S/I$, is satisfied and, in addition, equality e$ holds if only has unique extremal Betti number. We are interested finding natural class finite simple graphs $G$ for which $S/I(G)$, $I(G)$ edge ideal $G$, satisfies e$. $a(S/I(G))$ denote $a$-invariant $S/I$, i.e., $a(S/I(G)) s d$. One 0$. In present paper, by showing fundamental fact every Cameron--Walker graph 0$, $S/I(G)$ will exhibited.