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Register a new userOn the index of powers of edge ideals
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The index of a graded ideal measures the number of linear steps in the graded minimal free resolution of the ideal. In this paper we study the index of powers and squarefree powers of edge ideals. Our results indicate that the index as a function of the power of an edge ideal $I$ is strictly increasing if $I$ has linear relations. Examples show that this need not to be the case for monomial ideals generated in degree greater than two.
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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.
Squarefree powers of edge ideals are intimately related to matchings of the underlying graph. In this paper we give bounds for the regularity of squarefree powers of edge ideals, and we consider the question of when such powers are linearly related or have linear resolution. We also consider the so-called squarefree Ratliff property.
In this article, we study the regularity of integral closure of powers of edge ideals. We obtain a lower bound for the regularity of integral closure of powers of edge ideals in terms of induced matching number of graphs. We prove that the regularity of integral closure of powers of edge ideals of graphs with at most two odd cycles is the same as the regularity of their powers.
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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