On Gorenstein Graphs with Independence Number at Most Three

Mohammad Reza Oboudi, Ashkan Nikseresht

Published 2019-11-26Version 1

Suppose that $G$ is a simple graph on $n$ vertices and $\alpha=\alpha(G)$ is the independence number of $G$. Let $I(G)$ be the edge ideal of $G$ in $S=K[x_1, \ldots, x_n]$. We say that $G$ is Gorenstein when $S/I(G)$ is so. Here, first we present a condition on $G$ equivalent to $G$ being Gorenstein and use this to get a full characterization of Gorenstein graphs with $\alpha=2$. Then we focus on Gorenstein graphs with $\alpha=3$ and find the number of edges and the independence polynomial of such graphs. Finally, we present a full characterization of triangle-free Gorenstein graphs with $\alpha=3$.