  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.
