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Register a new userOrdinary and symbolic powers of edge ideals of weighted oriented graphs
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Let $mathcal{D}$ be a weighted oriented graph and $I(mathcal{D})$ be its edge ideal. In this paper, we show that all the symbolic and ordinary powers of $I(mathcal{D})$ coincide when $mathcal{D}$ is a weighted oriented certain class of tree. Finally, we give necessary and sufficient conditions for the equality of ordinary and symbolic powers of naturally oriented lines.
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We compute the Betti numbers for all the powers of initial and final lexsegment edge ideals. For the powers of the edge ideal of an anti-$d-$path, we prove that they have linear quotients and we characterize the normally torsion-free ideals. We determine a class of non-squarefree ideals, arising from some particular graphs, which are normally torsion-free.
In this article, we prove that for several classes of graphs, the Castelnuovo-Mumford regularity of symbolic powers of their edge ideals coincide with that of their ordinary powers.
We characterize unmixed and Cohen-Macaulay edge-weighted edge ideals of very well-covered graphs. We also provide examples of oriented graphs which have unmixed and non-Cohen-Macaulay vertex-weighted edge ideals, while the edge ideal of their underlying graph is Cohen-Macaulay. This disproves a conjecture posed by Pitones, Reyes and Toledo.
In this paper, we compute the regularity and Hilbert series of symbolic powers of the cover ideal of a graph $G$ when $G$ is either a crown graph or a complete multipartite graph. We also compute the multiplicity of symbolic powers of cover ideals in terms of the number of edges.
Let $mathcal{D}$ be a weighted oriented graph and let $I(mathcal{D})$ be its edge ideal. Under a natural condition that the underlying (undirected) graph of $mathcal{D}$ contains a perfect matching consisting of leaves, we provide several equivalent conditions for the Cohen-Macaulayness of $I(mathcal{D})$. We also completely characterize the Cohen-Macaulayness of $I(mathcal{D})$ when the underlying graph of $mathcal{D}$ is a bipartite graph. When $I(mathcal{D})$ fails to be Cohen-Macaulay, we give an instance where $I(mathcal{D})$ is shown to be sequentially Cohen-Macaulay.
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