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

Abstract

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 $\alpha =3$.