Integral Closure of Powers of Generalized Edge Ideals

Abstract

This article studies a new class of monomial ideals associated with a simple graph 𝐺, called generalized edge ideal, denoted by 𝐼 𝑔 (𝐺). Assuming that all the vertices 𝑥 have an exponent greater than 1 in 𝐼 𝑔 (𝐺), we completely characterize the graph 𝐺 for which 𝐼 𝑔 (𝐺) is integrally closed, and show that this is equivalent to 𝐼 𝑔 (𝐺) being normal i.e., all integral powers of 𝐼 𝑔 (𝐺) are integrally clased. We also give a necessary and sufficient condition for when 𝐺 is the star-shaped graph. Finally, we give a necessary and sufficient condition when the generalized edge ideal of a complete graph is integrally closed.