Journal ArticleDOI

# An upper bound for the regularity of binomial edge ideals of trees

17 Jul 2019-Journal of Algebra and Its Applications (World Scientific Publishing Company)-Vol. 18, Iss: 09, pp 1950170

TL;DR: In this article, an improved upper bound for the regularity of binomial edge ideals of trees was obtained, which was later extended to the case of trees with binomial edges.

AbstractIn this article, we obtain an improved upper bound for the regularity of binomial edge ideals of trees.

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TL;DR: A lower bound for the Castelnuovo-Mumford regularity of binomial edge ideals of block graphs is given by computing the two distinguished extremal Betti numbers of a new family of block graph, called flower graphs.
Abstract: We give a lower bound for the Castelnuovo-Mumford regularity of binomial edge ideals of block graphs by computing the two distinguished extremal Betti numbers of a new family of block graphs, calle...

17 citations

### Cites methods from "An upper bound for the regularity o..."

• ...Also using the upper bound proved in [10], we get reg S/JG ≤ 6....

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Journal ArticleDOI
TL;DR: In this paper, an upper bound for the Castelnuovo-Mumford regularity of powers of an ideal generated by a homogeneous quadratic sequence in a polynomial ring in terms of its related ideals and degrees of its generators was obtained.
Abstract: In this article, we obtain an upper bound for the Castelnuovo-Mumford regularity of powers of an ideal generated by a homogeneous quadratic sequence in a polynomial ring in terms of the regularity of its related ideals and degrees of its generators. As a consequence, we compute upper bounds for the regularity of powers of several binomial ideals. We generalize a result of Matsuda and Murai to show that the regularity of J G s is bounded below by 2 s + l ( G ) − 1 for all s ≥ 1 , where J G denotes the binomial edge ideal of a graph G and l ( G ) is the length of a longest induced path in G. We compute the regularity of powers of binomial edge ideals of cycle graphs, star graphs, and balloon graphs explicitly. Also, we give sharp bounds for the regularity of powers of almost complete intersection binomial edge ideals and parity binomial edge ideals.

11 citations

### Cites background from "An upper bound for the regularity o..."

• ...This bound, in general, is a weak one and there are improved bounds for several classes, (see for example [14, 21, 25, 29, 30, 31, 40])....

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Journal ArticleDOI
TL;DR: In this paper, a Hochster type formula for the local cohomology modules of binomial edge ideals is given for the case of Cartwright-Sturmfels ideals.
Abstract: We provide a Hochster type formula for the local cohomology modules of binomial edge ideals. As a consequence we obtain a simple criterion for the Cohen–Macaulayness and Buchsbaumness of these ideals and we describe their Castelnuovo–Mumford regularity and their Hilbert series. Conca and Varbaro (Square-free Groebner degenerations, 2018) have recently proved a conjecture of Conca, De Negri and Gorla (J Comb Algebra 2:231–257, 2018) relating the graded components of the local cohomology modules of Cartwright–Sturmfels ideals and their generic initial ideals. We provide an alternative proof for the case of binomial edge ideals.

11 citations

Journal ArticleDOI
TL;DR: In this article, a lower bound for the regularity of parity binomial edge ideals of graphs was obtained. But the lower bound was not satisfied for all graphs, and it was not shown that all graphs with parity binometric edge ideals have regularity.
Abstract: Let $G$ be a simple graph on $n$ vertices and $\mathcal{I}_G$ denotes parity binomial edge ideal of $G$ in the polynomial ring $S = \mathbb{K}[x_1,\ldots, x_n, y_1, \ldots, y_n].$ We obtain a lower bound for the regularity of parity binomial edge ideals of graphs. We then classify all graphs whose parity binomial edge ideals have regularity $3$. We classify graphs whose parity binomial edge ideals have pure resolution.

2 citations

### Cites background from "An upper bound for the regularity o..."

• ...obtained improved bounds for the regularity of binomial edge ideals of trees, [7, 8]....

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Posted Content
TL;DR: In this paper, a lower bound for the Castelnuovo-Mumford regularity of binomial edge ideals of block graphs was given by computing the two distinguished extremal Betti numbers of a new family of block graph called flower graphs.
Abstract: We give a lower bound for the Castelnuovo-Mumford regularity of binomial edge ideals of block graphs by computing the two distinguished extremal Betti numbers of a new family of block graphs, called flower graphs. Moreover, we present a linear time algorithm to compute the Castelnuovo-Mumford regularity and Krull dimension of binomial edge ideals of block graphs.

1 citations

##### References
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TL;DR: It follows that all binomial edge ideals are radical ideals, and the results apply for the class of conditional independence ideals where a fixed binary variable is independent of a collection of other variables, given the remaining ones.
Abstract: We introduce binomial edge ideals attached to a simple graph G and study their algebraic properties. We characterize those graphs for which the quadratic generators form a Grobner basis in a lexicographic order induced by a vertex labeling. Such graphs are chordal and claw-free. We give a reduced squarefree Grobner basis for general G. It follows that all binomial edge ideals are radical ideals. Their minimal primes can be characterized by particular subsets of the vertices of G. We provide sufficient conditions for Cohen-Macaulayness for closed and nonclosed graphs. Binomial edge ideals arise naturally in the study of conditional independence ideals. Our results apply for the class of conditional independence ideals where a fixed binary variable is independent of a collection of other variables, given the remaining ones. In this case the primary decomposition has a natural statistical interpretation.

202 citations

• ..., [4] and independently by Ohtani [11]....

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Journal ArticleDOI
TL;DR: In this article, the Grobner basis of I G of S generated by 2-minors [i, j] of X which correspond to edges of G was constructed.
Abstract: Let G be a finite graph on [n] = {1, 2,…, n}, X a 2 × n matrix of indeterminates over a field K, and S = K[X] a polynomial ring over K. In this article, we study about ideals I G of S generated by 2-minors [i, j] of X which correspond to edges {i, j} of G. In particular, we construct a Grobner basis of I G as a set of paths of G and compute a primary decomposition.

138 citations

• ..., [4] and independently by Ohtani [11]....

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• ...This notion was introduced by Herzog et al., [4] and independently by Ohtani [11]....

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Posted Content

TL;DR: In this article, the authors introduced binomial edge ideals attached to a simple graph and studied their algebraic properties, and provided sufficient conditions for Cohen-Macaulayness for closed and non-closed graphs.
Abstract: We introduce binomial edge ideals attached to a simple graph $G$ and study their algebraic properties. We characterize those graphs for which the quadratic generators form a Gr\"obner basis in a lexicographic order induced by a vertex labeling. Such graphs are chordal and claw-free. We give a reduced squarefree Gr\"obner basis for general $G$. It follows that all binomial edge ideals are radical ideals. Their minimal primes can be characterized by particular subsets of the vertices of $G$. We provide sufficient conditions for Cohen--Macaulayness for closed and nonclosed graphs. Binomial edge ideals arise naturally in the study of conditional independence ideals. Our results apply for the class of conditional independence ideals where a fixed binary variable is independent of a collection of other variables, given the remaining ones. In this case the primary decomposition has a natural statistical interpretation

132 citations

Journal ArticleDOI
TL;DR: In this article, the depth of classes of binomial edge ideals and classifications of closed graphs with Cohen-Macaulay edge ideal were studied and the binomial-edge ideal is defined.
Abstract: We study the depth of classes of binomial edge ideals and classify all closed graphs whose binomial edge ideal is Cohen-Macaulay.

108 citations

Journal ArticleDOI
TL;DR: In this article, it was shown that the Castelnuovo-Mumford regularity of the binomial edge ideal of a graph is bounded by the length of its longest induced path and bounded above by the number of its vertices.
Abstract: We show that the Castelnuovo-Mumford regularity of the binomial edge ideal of a graph is bounded below by the length of its longest induced path and bounded above by the number of its vertices.

92 citations

### "An upper bound for the regularity o..." refers background in this paper

• ...rity of the binomial edge ideals using combinatorial invariants. It is known that ℓ≤ reg(S/J G) ≤ n− 1, where n is the number of vertices in G and ℓ denotes the length of a longest induced path in G, [10]. Further, in the same article, Matsuda and Murai conjectured that reg(S/J G) = n− 1 if and only if Gis a path. This conjecture was settled in the aﬃrmative by Kiani and Saeedi Madani, [8]. A vertex v...

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