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Integral Closure of Powers of Edge Ideals of Weighted Oriented Graphs
Arindam Banerjee
,
Kanoy Kumar Das
,
Sirajul Haque
Abstract
In this article, we study monomial ideals associated with a simple graph, namely edge ideals of weighted oriented graphs. Let $D$ be a weighted oriented graph. Assuming that all the vertices of $D$ have weights greater than 1, we completely characterize weighted oriented graphs $D$ for which $I(D)$ is integrally closed, and show that this is equivalent to $I(D)$ being normal. We also give an equivalent condition for $(\overline{I(D)}=I(D))$ when the underlying simple graph of $D$ is a complete graph. Finally, we give a necessary and sufficient condition when the edge ideal of a uniform whiskered graph is integrally closed.