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 \(\alpha =3\).

令\（\ alpha = \ alpha（G）\）是具有n个顶点的简单图G的独立性，而I（G）是\（S = K [x_1，\ ldots，x_n] \）中的边缘理想。如果S / I（G）是Gorenstein，则图G在K上称为Gorenstein ，如果G在每个域上都是Gorenstein，则我们简单地说G是Gorenstein。在本文中，首先我们陈述一个等于G为Gorenstein的条件，并使用它给出\（\ alpha = 2 \）的Gorenstein图的特征。。然后，我们展示具有\（\ alpha = 3 \）的Gorenstein图的一些性质，并作为这些结果的应用，我们描述具有\（\ alpha = 3 \）的无三角形Gorenstein图。