Integral Closure of Powers of Edge Ideals of Weighted Oriented Graphs

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 I(D)<overline>=I(D)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{I(D)}=I(D)$$\end{document} 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.