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The edge ideals of the join of some vertex weighted oriented graphs

In this paper, we describe primary decomposition of the edge ideal of the join of some graphs in terms of that information of the edge ideal of every weighted oriented graph. Meanwhile, we also study depth and regularity of symbolic powers and ordinary powers of such an edge ideal. We explicitly compute depth and regularity of ordinary powers of the edge ideal of the join of two graphs consisting of isolated vertices, and also provide upper bounds of regularity of symbolic powers of such an edge ideal. For the edge ideal of the join of two graphs with at least an oriented edge for per graph, we give the exact formulas for their depth and regularity, and also provide the upper bounds of regularity of ordinary powers of such an edge ideal. Some examples show that these upper bounds can be obtained, but may be strict.