The Edge Ideals of the Join of Some Vertex Weighted Oriented Graphs

This paper describes the primary decomposition of the edge ideal of the join of weighted oriented graphs in terms of the information from each of those graphs. Using this decomposition and induction, we derive formulas for the depth and regularity of ordinary powers of the edge ideal of the join of two graphs consisting of isolated vertices. Furthermore, we provide upper bounds for the regularity of symbolic powers of such an edge ideal. For the edge ideal of the join of two graphs with at least one oriented edge for each graph, we provide exact formulas for their depth and regularity, as well as upper bounds for the regularity of ordinary powers of such an edge ideal. Some examples demonstrate that these upper bounds are sometimes attainable and sometimes strict.