scholarly journals On The Binomial Edge Ideal of a Pair of Graphs

10.37236/2987 ◽  
2013 ◽  
Vol 20 (1) ◽  
Author(s):  
Sara Saeedi Madani ◽  
Dariush Kiani
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Betti Numbers ◽  
Edge Ideal ◽  
Closed Graph ◽  
Linear Relations ◽  
Graded Betti Numbers ◽  
Linear Resolution ◽  
Generic Matrix ◽  
The Ideal

We characterize all pairs of graphs $(G_1,G_2)$, for which the binomial edge ideal $J_{G_1,G_2}$ has linear relations. We show that $J_{G_1,G_2}$ has a linear resolution if and only if $G_1$ and $G_2$ are complete and one of them is just an edge. We also compute some of the graded Betti numbers of the binomial edge ideal of a pair of graphs with respect to some graphical terms. In particular, we show that for every pair of graphs $(G_1,G_2)$ with girth (i.e. the length of a shortest cycle in the graph) greater than 3, $\beta_{i,i+2}(J_{G_1,G_2})=0$, for all $i$. Moreover, we give a lower bound for the Castelnuovo-Mumford regularity of any binomial edge ideal $J_{G_1,G_2}$ and hence the ideal of adjacent $2$-minors of a generic matrix. We also obtain an upper bound for the regularity of $J_{G_1,G_2}$, if $G_1$ is complete and $G_2$ is a closed graph.

scholarly journals Componentwise linear ideals

1999 ◽  
Vol 153 ◽  
pp. 141-153 ◽  
Author(s):  
Jürgen Herzog ◽  
Takayuki Hibi
Keyword(s):  
Simplicial Complex ◽  
Polynomial Ring ◽  
Betti Numbers ◽  
Monomial Ideals ◽  
Homogeneous Polynomials ◽  
Graded Betti Numbers ◽  
Linear Resolution ◽  
Alexander Dual ◽  
Componentwise Linear Ideals ◽  
The Ideal

AbstractA componentwise linear ideal is a graded ideal I of a polynomial ring such that, for each degree q, the ideal generated by all homogeneous polynomials of degree q belonging to I has a linear resolution. Examples of componentwise linear ideals include stable monomial ideals and Gotzmann ideals. The graded Betti numbers of a componentwise linear ideal can be determined by the graded Betti numbers of its components. Combinatorics on squarefree componentwise linear ideals will be especially studied. It turns out that the Stanley-Reisner ideal IΔ arising from a simplicial complex Δ is componentwise linear if and only if the Alexander dual of Δ is sequentially Cohen-Macaulay. This result generalizes the theorem by Eagon and Reiner which says that the Stanley-Reisner ideal of a simplicial complex has a linear resolution if and only if its Alexander dual is Cohen-Macaulay.

scholarly journals Optimal Betti Numbers of Forest Ideals

10.37236/125 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Michael Goff
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Betti Numbers ◽  
Monomial Ideals ◽  
Graded Betti Numbers

We prove a tight lower bound on the algebraic Betti numbers of tree and forest ideals and an upper bound on certain graded Betti numbers of squarefree monomial ideals.

scholarly journals Binomial Edge Ideals of Graphs

10.37236/2349 ◽  
2012 ◽  
Vol 19 (2) ◽  
Author(s):  
Dariush Kiani ◽  
Sara Saeedi
Keyword(s):  
Upper Bound ◽  
Betti Number ◽  
Complete Graphs ◽  
Edge Ideal ◽  
Closed Graph ◽  
Edge Ideals ◽  
Linear Resolution

We characterize all graphs whose binomial edge ideals have a linear resolution. Indeed, we show that complete graphs are the only graphs with this property. We also compute some graded components of the first Betti number of the binomial edge ideal of a graph with respect to the graphical terms. Finally, we give an upper bound for the Castelnuovo-Mumford regularity of the binomial edge ideal of a closed graph.

scholarly journals Rigidity of Linear Strands and Generic Initial Ideals

2008 ◽  
Vol 190 ◽  
pp. 35-61 ◽  
Author(s):  
Satoshi Murai ◽  
Pooja Singla
Keyword(s):  
Polynomial Ring ◽  
Betti Numbers ◽  
Exterior Algebra ◽  
Graded Betti Numbers ◽  
Linear Resolution ◽  
Initial Ideals ◽  
Finite Set ◽  
Generic Initial Ideals ◽  
The Ideal

Let K be a field, S a polynomial ring and E an exterior algebra over K, both in a finite set of variables. We study rigidity properties of the graded Betti numbers of graded ideals in S and E when passing to their generic initial ideals. First, we prove that if the graded Betti numbers for some i > 1 and k ≥ 0, then for all q ≥ i, where I ⊂ S is a graded ideal. Second, we show that if for some i > 1 and k ≥ 0, then for all q ≥ 1, where I ⊂ E is a graded ideal. In addition, it will be shown that the graded Betti numbers for all i ≥ 1 if and only if I(k) and I(k+1) have a linear resolution. Here I(d) is the ideal generated by all homogeneous elements in I of degree d, and R can be either the polynomial ring or the exterior algebra.

Weakly Polymatroidal Ideals

2006 ◽  
Vol 13 (04) ◽  
pp. 711-720 ◽  
Author(s):  
Masako Kokubo ◽  
Takayuki Hibi
Keyword(s):  
Betti Numbers ◽  
Partially Ordered Sets ◽  
Finite Sequence ◽  
Ordered Sets ◽  
Partially Ordered ◽  
Graded Betti Numbers ◽  
Linear Resolution ◽  
Fundamental Fact ◽  
Polymatroidal Ideals ◽  
Polymatroidal Ideal

The concept of the weakly polymatroidal ideal, which generalizes both the polymatroidal ideal and the prestable ideal, is introduced. A fundamental fact is that every weakly polymatroidal ideal has a linear resolution. One of the typical examples of weakly polymatroidal ideals arises from finite partially ordered sets. We associate each weakly polymatroidal ideal with a finite sequence, alled the polymatroidal sequence, which will be useful for the computation of graded Betti numbers of weakly polymatroidal ideals as well as for the construction of weakly polymatroidal ideals.

scholarly journals On the Betti numbers and Rees algebras of ideals with linear powers

2021 ◽  
Vol 53 (2) ◽  
pp. 575-592
Author(s):  
Lisa Nicklasson
Keyword(s):  
Betti Numbers ◽  
Free Resolution ◽  
Minimal Free Resolution ◽  
Linear Relations ◽  
Rees Algebras ◽  
Low Dimension ◽  
The Ideal

AbstractAn ideal $$I \subset \mathbb {k}[x_1, \ldots , x_n]$$ I ⊂ k [ x 1 , … , x n ] is said to have linear powers if $$I^k$$ I k has a linear minimal free resolution, for all integers $$k>0$$ k > 0 . In this paper, we study the Betti numbers of $$I^k$$ I k , for ideals I with linear powers. We provide linear relations on the Betti numbers, which holds for all ideals with linear powers. This is especially useful for ideals of low dimension. The Betti numbers are computed explicitly, as polynomials in k, for the ideal generated by all square-free monomials of degree d, for $$d=2, 3$$ d = 2 , 3 or $$n-1$$ n - 1 , and the product of all ideals generated by s variables, for $$s=n-1$$ s = n - 1 or $$n-2$$ n - 2 . We also study the generators of the Rees ideal, for ideals with linear powers. Particularly, we are interested in ideals for which the Rees ideal is generated by quadratic elements. This problem is related to a conjecture on matroids by White.

Cover Product and Betti Polynomial of Graphs

2015 ◽  
Vol 58 (2) ◽  
pp. 320-333
Author(s):  
Aurora Llamas ◽  
Josá Martínez–Bernal
Keyword(s):  
Positive Integer ◽  
Bipartite Graph ◽  
Betti Numbers ◽  
Recursive Formula ◽  
Bipartite Graphs ◽  
Matching Number ◽  
Edge Ideal ◽  
Graded Betti Numbers ◽  
Vertex Set ◽  
Chordal Bipartite Graphs

AbstractThe cover product of disjoint graphs G and H with fixed vertex covers C(G) and C(H), is the graphwith vertex set V(G) ∪ V(H) and edge setWe describe the graded Betti numbers of GeH in terms of those of. As applications we obtain: (i) For any positive integer k there exists a connected bipartite graph G such that reg R/I(G) = μS(G) + k, where, I(G) denotes the edge ideal of G, reg R/I(G) is the Castelnuovo–Mumford regularity of R/I(G) and μS(G) is the induced or strong matching number of G; (ii)The graded Betti numbers of the complement of a tree depends only upon its number of vertices; (iii)The h-vector of R/I(G e H) is described in terms of the h-vectors of R/I(G) and R/I(H). Furthermore, in a diòerent direction, we give a recursive formula for the graded Betti numbers of chordal bipartite graphs.

scholarly journals Some Families of Componentwise Linear Monomial Ideals

2007 ◽  
Vol 187 ◽  
pp. 115-156 ◽  
Author(s):  
Christopher A. Francisco ◽  
Adam Van Tuyl
Keyword(s):  
Polynomial Ring ◽  
Betti Numbers ◽  
Monomial Ideals ◽  
Fat Points ◽  
Graded Betti Numbers ◽  
Alexander Duality ◽  
Componentwise Linear Ideals ◽  
Positive Integers ◽  
The Ideal

AbstractLet R = k[x1,…,xn] be a polynomial ring over a field k. Let J = {j1,…,jt} be a subset of {1,…, n}, and let mJ ⊂ R denote the ideal (xj1,…,xjt). Given subsets J1,…,Js of {1,…, n} and positive integers a1,…,as, we study ideals of the form These ideals arise naturally, for example, in the study of fat points, tetrahedral curves, and Alexander duality of squarefree monomial ideals. Our main focus is determining when ideals of this form are componentwise linear. Using polymatroidality, we prove that I is always componentwise linear when s ≤ 3 or when Ji ∪ Jj = [n] for all i ≠ j. When s ≥ 4, we give examples to show that I may or may not be componentwise linear. We apply these results to ideals of small sets of general fat points in multiprojective space, and we extend work of Fatabbi, Lorenzini, Valla, and the first author by computing the graded Betti numbers in the s = 2 case. Since componentwise linear ideals satisfy the Multiplicity Conjecture of Herzog, Huneke, and Srinivasan when char(k) = 0, our work also yields new cases in which this conjecture holds.

scholarly journals Induced Matchings and the v-Number of Graded Ideals

Mathematics ◽  
2021 ◽  
Vol 9 (22) ◽  
pp. 2860
Author(s):  
Gonzalo Grisalde ◽  
Enrique Reyes ◽  
Rafael H. Villarreal
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Matching Number ◽  
Edge Ideal ◽  
Induced Matching ◽  
Simplicial Partition ◽  
The Family ◽  
Induced Matchings ◽  
Edge Ring ◽  
Insight Into

We give a formula for the v-number of a graded ideal that can be used to compute this number. Then, we show that for the edge ideal I(G) of a graph G, the induced matching number of G is an upper bound for the v-number of I(G) when G is very well-covered, or G has a simplicial partition, or G is well-covered connected and contains neither four, nor five cycles. In all these cases, the v-number of I(G) is a lower bound for the regularity of the edge ring of G. We classify when the induced matching number of G is an upper bound for the v-number of I(G) when G is a cycle and classify when all vertices of a graph are shedding vertices to gain insight into the family of W2-graphs.

scholarly journals Squarefree Powers of Edge Ideals of Forests

2021 ◽  
Vol 28 (2) ◽  
Author(s):  
Nursel Erey ◽  
Takayuki Hibi
Keyword(s):  
Positive Integer ◽  
Upper Bound ◽  
Matching Number ◽  
Edge Ideal ◽  
Combinatorial Formula ◽  
Edge Ideals ◽  
Linear Resolution

Let $I(G)^{[k]}$ denote the $k$th squarefree power of the edge ideal of $G$. When $G$ is a forest, we provide a sharp upper bound for the regularity of $I(G)^{[k]}$ in terms of the $k$-admissable matching number of $G$. For any positive integer $k$, we classify all forests $G$ such that $I(G)^{[k]}$ has linear resolution. We also give a combinatorial formula for the regularity of $I(G)^{[2]}$ for any forest $G$.

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