Properties of symbolic powers of edge ideals of weighted oriented graphs

Abstract

Let [Formula: see text] be a weighted oriented graph and [Formula: see text] be its edge ideal. We provide one method to find all the minimal generators of [Formula: see text], where [Formula: see text] is a maximal strong vertex cover of [Formula: see text] and [Formula: see text] is the intersections of irreducible ideals associated to the strong vertex covers contained in [Formula: see text]. If [Formula: see text] is an induced digraph of [Formula: see text], under a certain condition on the strong vertex covers of [Formula: see text] and [Formula: see text], we show that [Formula: see text] for some [Formula: see text] implies [Formula: see text]. We provide the necessary and sufficient condition for the equality of ordinary and symbolic powers of edge ideal of the union of two naturally oriented paths with a common sink vertex. We characterize all the maximal strong vertex covers of [Formula: see text] such that at most one edge is oriented into each of its vertices and [Formula: see text] if [Formula: see text] for all [Formula: see text]. Finally, if [Formula: see text] is a weighted rooted tree with the degree of root is [Formula: see text] and [Formula: see text] when [Formula: see text] for all [Formula: see text], we show that [Formula: see text] for all [Formula: see text].