The regularity of edge ideals of graphs

2010

In this paper we give upper bounds for the regularity of edge ideal of some classes of graphs in terms of invariants of graph. We introduce two numbers a ′ (G) and n(G) depending on graph G and show that for a vertex decomposable graph G, reg(R/I(G)) ≤ min{a ′ (G), n(G)} and for a shellable graph G, reg(R/I(G)) ≤ n(G). Moreover it is shown that for a graph G, where G c is a d-tree, we have pd(R/I(G)) = max v∈V (G) {deg G (v)}.

Journal of Algebraic Combinatorics, 2012

In this paper we give new upper bounds on the regularity of edge ideals whose resolutions are k-steps linear; surprisingly, the bounds are logarithmic in the number of variables. We also give various bounds for the projective dimension of such ideals, generalizing other recent results. By Alexander duality, our results also apply to unmixed square-free monomial ideals of codimension two. We also discuss and connect these results to more classical topics in commutative algebra.

In this paper we prove the existence of a special order on the set of minimal monomial generators of powers of edge ideals of arbitrary graphs. Using this order we find new upper bounds on the regularity of powers of edge ideals of graphs whose complement does not have any induced four cycle.

Journal of Algebraic Combinatorics, 2016

Fröberg's classical theorem about edge ideals with 2-linear resolution can be regarded as a classification of graphs whose edge ideals have linearity defect zero. Extending his theorem, we classify all graphs whose edge ideals have linearity defect at most 1. Our characterization is independent of the characteristic of the base field: the graphs in question are exactly weakly chordal graphs with induced matching number at most 2. The proof uses the theory of Betti splittings of monomial ideals due to Francisco, Hà, and Van Tuyl and the structure of weakly chordal graphs. Along the way, we compute the linearity defect of edge ideals of cycles and weakly chordal graphs. We are also able to recover and generalize previous results due to Dochtermann-Engström, Kimura and Woodroofe on the projective dimension and Castelnuovo-Mumford regularity of edge ideals.

Rocky Mountain Journal of Mathematics

Let C * (E) be the graph C *-algebra of a row-finite graph E. We give a complete description of the vertex sets of the gauge-invariant regular ideals of C * (E). It is shown that when E satisfies Condition (L) the regular ideals C * (E) are a class of gauge-invariant ideals which preserve Condition (L) under quotients. That is, we show that if E satisfies Condition (L) then a regular ideal J C * (E) is necessarily gauge-invariant. Further, if J C * (E) is a regular ideal, it is shown that C * (E)/J ≃ C * (F) where F satisfies Condition (L).

Journal of Pure and Applied Algebra, 2017

In this paper we study the Castelnuovo-Mumford regularity of the path ideals of finite simple graphs. We find new upper bounds for various path ideals of gap free graphs. In particular we prove that the path ideals of gap free and claw graphs have linear minimal free resolutions.

Let K be a field and S = K[x 1 , . . . , x n ] be the polynomial ring in n variables over K. Let G be a graph with n vertices. Assume that I = I(G) is the edge ideal of G and J = J(G) is its cover ideal. We prove that sdepth(J) ≥ n−ν o (G) and sdepth(S/J) ≥ n − ν o (G) − 1, where ν o (G) is the ordered matching number of G. We also prove the inequalities sdepth(J k ) ≥ depth(J k ) and sdepth(S/J k ) ≥ depth(S/J k ), for every integer k ≫ 0, when G is a bipartite graph. Moreover, we provide an elementary proof for the known inequality reg(S/I) ≤ ν o (G).

Discrete Mathematics, 2008

Given a simple graph G on n vertices, we prove that it is possible to reconstruct several algebraic properties of the edge ideal from the deck of G, that is, from the collection of subgraphs obtained by removing a vertex from G. These properties include the Krull dimension, the Hilbert function, and all the graded Betti numbers i,j where j < n. We also state many further questions that arise from our study.

Bulletin of the Korean Mathematical Society, 2015

Let G be a graph on the vertex set V (G) = {x 1 ,. .. , xn} with the edge set E(G), and let R = K[x 1 ,. .. , xn] be the polynomial ring over a field K. Two monomial ideals are associated to G, the edge ideal I(G) generated by all monomials x i x j with {x i , x j } ∈ E(G), and the vertex cover ideal I G generated by monomials x i ∈C x i for all minimal vertex covers C of G. A minimal vertex cover of G is a subset C ⊂ V (G) such that each edge has at least one vertex in C and no proper subset of C has the same property. Indeed, the vertex cover ideal of G is the Alexander dual of the edge ideal of G. In this paper, for an unmixed bipartite graph G we consider the lattice of vertex covers L G and we explicitly describe the minimal free resolution of the ideal associated to L G which is exactly the vertex cover ideal of G. Then we compute depth, projective dimension, regularity and extremal Betti numbers of R/I(G) in terms of the associated lattice.

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.