  We characterize some graphs with a Gorenstein edge ideal. In particular, we
show that if $G$ is a circulant graph with vertex degree at most four or a
circulant graph of the form $C_n(1,\ldots, d)$ for some $d\leq n/2$, then $G$
is Gorenstein if and only if $G\cong tK_2$, $G\cong t\overline{C_n}$ or $G\cong
tC_{13}(1,5)$ for some integers $t$ and $n\geq 4$. Also we prove that if $G$ is
a \mathcal{SQC}\ graph, then $G$ is Gorenstein if and only if each component of
$G$ is either an edge or a 5-cycle.
