Gorenstein homogeneous subrings of graphs

Let $G=(V,E)$ be a connected simple graph, with $n$ vertices such that $S$ is its homogeneous monomial subring. We prove that if $S$ is normal and Gorenstein, then $G$ is unmixed with cover number $\lceil\frac{n}{2}\rceil$ and $G$ has a strong $\lceil\frac{n}{2}\rceil$-$\tau$-reduction. Furthermore, if $n$ is even, then we show that $G$ is bipartite. Finally, if $S$ is normal and $G$ is unmixed whose cover number is $\lceil\frac{n}{2}\rceil$, we give sufficient conditions for $S$ to be Gorenstein.