Regularity and $a$-invariant of Cameron--Walker graphs

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