The reconstruction conjecture and edge ideals

Dalili, Kia; Faridi, Sara; Traves, Will

doi:10.1016/j.disc.2007.04.044

Cited by 4 publications

(5 citation statements)

References 8 publications

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“…Of particular interest is to be able to express reg(R/I(C)) in terms of the combinatorics of C, at least for some special families of clutters. Several authors have studied the regularity of edge ideals of graphs and clutters [18,20,53,74,75,77,81,90,103,106,114]. The main results are general bounds for the regularity and combinatorial formulas for the regularity of special families of clutters.…”

Section: Invariants Of Edge Ideals: Regularity Projective Dimension D...mentioning

confidence: 99%

Edge Ideals: Algebraic and Combinatorial Properties

Morey¹,

Villarreal²

2012

Progress in Commutative Algebra 1

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Let C be a clutter and let I(C) ⊂ R be its edge ideal. This is a survey paper on the algebraic and combinatorial properties of R/I(C) and C, respectively. We give a criterion to estimate the regularity of R/I(C) and apply this criterion to give new proofs of some formulas for the regularity. If R/I(C) is sequentially Cohen-Macaulay, we present a formula for the regularity of the ideal of vertex covers of C and give a formula for the projective dimension of R/I(C). We also examine the associated primes of powers of edge ideals, and show that for a graph with a leaf, these sets form an ascending chain.We are interested in determining which families of clutters have the property that ∆ C is pure, Cohen-Macaulay, or shellable. These properties have been extensively studied, see [10,89,92,93,94,100,102,110,111] and the references there.The above definition of shellable is due to Björner and Wachs [6] and is usually referred to as nonpure shellable, although here we will drop the adjective "nonpure". Originally, the definition of shellable also required that the simplicial complex be pure, that is, all facets have the same dimension. We will say ∆ is pure shellable if it also satisfies this hypothesis. These properties are related to other important properties [10,102,110]:If a shellable complex is not pure, an implication similar to that above holds when Cohen-Macaulay is replaced by sequentially Cohen-Macaulay. Definition 2.2. Let R = K[x 1 , . . . , x n ]. A graded R-module M is called sequentially Cohen-Macaulay (over K) if there exists a finite filtration of graded R-modules (0) = M 0 ⊂ M 1 ⊂ · · · ⊂ M r = M

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Section: Invariants Of Edge Ideals: Regularity Projective Dimension D...mentioning

confidence: 99%

Edge Ideals: Algebraic and Combinatorial Properties

Morey¹,

Villarreal²

2012

Progress in Commutative Algebra 1

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“…, x n ] be a polynomial ring over a field k in n variables. The edge ideal I = I(G) of the graph G is the ideal generated by all monomials of the form x i x j such that {x i , x j } is an edge of G. Edge ideals of graphs have been studied by various authors (see for example [19], [17], [5], [1], and [6]). The focus of this work is to determine the depths of the powers of an edge ideal of a tree.…”

Section: Introductionmentioning

confidence: 99%

Depths of Powers of the Edge Ideal of a Tree

Morey

2010

Communications in Algebra

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Lower bounds are given for the depths of R/I t for t ≥ 1 when I is the edge ideal of a tree or forest. The bounds are given in terms of the diameter of the tree, or in case of a forest, the largest diameter of a connected component and the number of connected components. These lower bounds provide a lower bound on the power for which the depths stabilize.2000 Mathematics Subject Classification. 13A17, 13F55, 05C65, 90C27.

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“…In section 4, generalizing Borzacchini's results [3,4], we prove that both polynomials are reconstructible for hypergraphs. We also prove the reconstruction problems of some algebraic invariants of the independent complex of H, where their graph counter part is proven by Dalili, Faridi and Traves in [6]. That is, we consider reconstructibility of the Hilbert series, the f -vector, the (multi-)graded Betti numbers and some graded Betti tables of the independent complex of H.…”

Section: Introductionmentioning

confidence: 89%

“…The authors in [6] studied reconstructibility of some algebraic invariants of the edge ideal of a graph G such as the Krull dimension, the Hilbert series, and the graded Betti numbers b i,j , where j < n. We extend these results to hypergraphs.…”

Section: Hilbert Series and F -Vectormentioning

confidence: 96%

“…This equation shows it is difficult to determine each b ij only from the data {B l } n l=1 since anti-diagonals of B might contain more than one non-zero entry. We thus have the following which gives a partial answer to [6,Question 5.6]. In fact, in this case, we can compute the non-zero entries from the coefficients of S H (x, y).…”

Section: Multi-graded Betti Numbersmentioning

confidence: 99%

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Using Edge-Induced and Vertex-Induced Subhypergraph Polynomials

Tadesse

2015

MATH. SCAND.

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For a hypergraph H, we consider the edge-induced and vertex-induced subhypergraph polynomials and study their relation. We use this relation to prove that both polynomials are reconstructible, and to prove a theorem relating the Hilbert series of the Stanley-Reisner ring of the independent complex of H and the edge-induced subhypergraph polynomial. We also consider reconstruction of some algebraic invariants of H.

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The reconstruction conjecture and edge ideals

Cited by 4 publications

References 8 publications

Edge Ideals: Algebraic and Combinatorial Properties

Edge Ideals: Algebraic and Combinatorial Properties

Depths of Powers of the Edge Ideal of a Tree

Using Edge-Induced and Vertex-Induced Subhypergraph Polynomials

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