Algebraic study on Cameron-Walker graphs

Let $G$ be a finite simple graph on $[n]$ and $I(G) \subset S$ the edge ideal of $G$, where $S = K[x_{1}, \ldots, x_{n}]$ is the polynomial ring over a field $K$. Let $m(G)$ denote the maximum size of matchings of $G$ and $im(G)$ that of induced matchings of $G$. It is known that $im(G) \leq \text{reg}(S/I(G)) \leq m(G)$, where $\text{reg}(S/I(G))$ is the Castelnuovo-Mumford regularity of $S/I(G)$. Cameron and Walker succeeded in classifying the finite connected simple graphs $G$ with $im(G) = m(G)$. We say that a finite connected simple graph $G$ is a Cameron-Walker graph if $im(G) = m(G)$ and if $G$ is neither a star nor a star triangle. In the present paper, we study Cameron-Walker graphs from a viewpoint of commutative algebra. First, we prove that a Cameron-Walker graph $G$ is unmixed if and only if $G$ is Cohen-Macaulay and classify all Cohen-Macaulay Cameron-Walker graphs. Second, we prove that there is no Gorenstein Cameron-Walker graph. Finally, we prove that every Cameron--Walker graph is sequentially Cohen-Macaulay.