Some Combinatorial Characterizations of Gorenstein Graphs with Independence Number Less than Four

Let $\alpha=\alpha(G)$ be the independence number of a simple graph $G$ with $n$ vertices and $I(G)$ be its edge ideal in $S=K[x_1,\ldots, x_n]$. If $S/I(G)$ is Gorenstein, the graph $G$ is called Gorenstein over $K$ and if $G$ is Gorenstein over every field, then we simply say that $G$ is Gorenstein. In this article, first we state a condition equivalent to $G$ being Gorenstein and using this we give a characterization of Gorenstein graphs with $\alpha=2$. Then we present some properties of Gorenstein graphs with $\alpha=3$ and as an application of these results we characterize triangle-free Gorenstein graphs with $\alpha=3$.