Monomial and toric ideals associated to Ferrers graphs

Journal of Algebraic Combinatorics, 2008

We introduce a specialization technique in order to study monomial ideals that are generated in degree two by using our earlier results about Ferrers ideals. It allows us to describe explicitly a cellular minimal free resolution of various ideals including any strongly stable and any squarefree strongly stable ideal whose minimal generators have degree two. In particular, this shows that threshold graphs can be obtained as specializations of Ferrers graphs, which explains their similar properties.

Analele Universitatii "Ovidius" Constanta - Seria Matematica, 2014

Let J G denote the binomial edge ideal of a connected undirected graph on n vertices. This is the ideal generated by the binomials x i y j − x j y i , 1 ≤ i < j ≤ n, in the polynomial ring S = K[x 1 ,. .. , x n , y 1 ,. .. , y n ] where {i, j} is an edge of G. We study the arithmetic properties of S/J G for G, the complete bipartite graph. In particular we compute dimensions, depths, Castelnuovo-Mumford regularities, Hilbert functions and multiplicities of them. As main results we give an explicit description of the modules of deficiencies, the duals of local cohomology modules, and prove the purity of the minimal free resolution of S/J G .

Transactions of the American Mathematical …, 2004

Abstract. For a graph G, we construct two algebras whose dimensions are both equal to the number of spanning trees of G. One of these algebras is the quotient of the polynomial ring modulo certain monomial ideal, while the other is the quotient of the polynomial ring ...

Journal of Algebraic Combinatorics, 2007

We use the correspondence between hypergraphs and their associated edge ideals to study the minimal graded free resolution of squarefree monomial ideals. The theme of this paper is to understand how the combinatorial structure of a hypergraph H appears within the resolution of its edge ideal I(H). We discuss when recursive formulas to compute the graded Betti numbers of I(H) in terms of its subhypergraphs can be obtained; these results generalize our previous work (Hà, H.T., Van Tuyl, A. in J. Algebra 309:405-425, 2007) on the edge ideals of simple graphs. We introduce a class of hypergraphs, which we call properly-connected, that naturally generalizes simple graphs from the point of view that distances between intersecting edges are "well behaved." For such a hypergraph H (and thus, for any simple graph), we give a lower bound for the regularity of I(H) via combinatorial information describing H and an upper bound for the regularity when H = G is a simple graph. We also introduce triangulated hypergraphs that are properly-connected hypergraphs generalizing chordal graphs. When H is a triangulated hypergraph, we explicitly compute the regularity of I(H) and show that the graded Betti numbers of I(H) are independent of the ground field. As a consequence, many known results about the graded Betti numbers of forests can now be extended to chordal graphs. Keywords Hypergraphs • Chordal graphs • Monomial ideals • Graded resolutions • Regularity Dedicated to Anthony V. Geramita on the occasion of his 65th birthday.

Journal of Pure and Applied Algebra, 2004

For positive integers n; b1 6 b2 6 • • • 6 bn and t 6 n, let It be the transversal monomial ideal generated by square-free monomials yi 1 j 1 yi 2 j 2 • • • yi t jt ; 1 6 i1 ¡ i2 ¡ • • • ¡ it 6 n; 1 6 j k 6 bi k ; k = 1; : : : ; t; (*) where yij's are distinct indeterminates. It is observed that the simplicial complex associated to this ideal is pure shellable if and only if b1=• • •=bn=1, but its Alexander dual is always pure and shellable. The simplicial complex admits some weaker shelling which leads to the computation of its Hilbert series. The main result is the construction of the minimal free resolution for the quotient ring of It. This class of monomial ideals includes the ideals of t-minors of generic pluri-circulant matrices under a change of coordinates. The last family of ideals arise from some specializations of the deÿning ideals of generic singularities of algebraic varieties.

2018

For a graph G, we define G-parking functions and show that their number is equal to the number of spanning trees of G. We construct a certain monomial ideal and a certain ideal generated by powers of linear forms. The dimension of the quotient of the polynomial ring modulo either of these two ideals equals the number of spanning trees of G. The monomials corresponding to G-parking functions form linear bases in each of these two algebras. Then we investigate the general class of monotone monomial ideals and their deformations. We show that the Hilbert series of a monotone monomial ideal is always bounded by the Hilbert series of its deformation. We prove several formulas for the Hilbert series.

Journal of Pure and Applied Algebra, 2005

Let R = K[x 1 , . . . , x n ] be a polynomial ring over a field K and let I be an ideal of R generated by a set x 1 , . . . , x q of square-free monomials of degree two such that the graph G defined by those monomials is bipartite. We study the Rees algebra R(I ) of I, by studying both the Rees cone R + A generated by the set A = {e 1 , . . . , e n , , . . . , ( q , 1)} and the matrix C whose columns are the vectors in A . It is shown that C is totally unimodular. We determine the irreducible representation of the Rees cone in terms of the minimal vertex covers of G. Then we compute the a-invariant of R(I ).

Communications in Algebra, 2019

In this article, we shall investigate the numerical invariants encoded in the minimal graded free resolution of edge ideal of k-partite n-crown graph(or multipartite crown graph) C ðkÞ n : We deduce the structure of minimal graded free resolution and obtain the lower bounds on graded Betti numbers of edge ideal IðC ðkÞ n Þ: In particular, we obtain the graded Betti numbers of edge ideal of C ð2Þ n , already studied, and the initial graded Betti numbers b i, iþ1 ðC ð3Þ n Þ: Also, the extremal graded Betti number b knÀ2, kn ðC ðkÞ n Þ, projective dimension and Castelnuovo-Mumford regularity of these ideals are obtained.

Journal of Algebra, 2007

We provide a new combinatorial approach to study the minimal free resolutions of edge ideals, that is, quadratic square-free monomial ideals. With this method we can recover most of the known results on resolutions of edge ideals with fuller generality, and at the same time, obtain new results. Past investigations on the resolutions of edge ideals usually reduced the problem to computing the dimensions of reduced homology or Koszul homology groups. Our approach circumvents the highly nontrivial problem of computing the dimensions of these groups and turns the problem into combinatorial questions about the associated simple graph. We also show that our technique extends successfully to the study of graded Betti numbers of arbitrary square-free monomial ideals viewed as facet ideals of simplicial complexes.

2013

In a 2008 paper, the first author and Van Tuyl proved that the regularity of the edge ideal of a graph G is at most one greater than the matching number of G. In this note, we provide a generalization of this result to any square-free monomial ideal. We define a 2-collage in a simple hypergraph to be a collection of edges with the property that for any edge E of the hypergraph, there exists an edge F in the 2-collage such that |E \ F | ≤ 1. The Castelnuovo-Mumford regularity of the edge ideal of a simple hypergraph is bounded above by a multiple of the minimum size of a 2-collage. We also give a recursive formula to compute the regularity of a vertex-decomposable hypergraph. Finally, we show that regularity in the graph case is bounded by a certain statistic based on maximal packings of nondegenerate star subgraphs.