Closed neighborhood ideal of a graph

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.

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.

2010

The algebra of basic covers of a graph G, denoted byĀ(G), was introduced by Jürgen Herzog as a suitable quotient of the vertex cover algebra. In this paper we show that if the graph is bipartite thenĀ(G) is a homogeneous algebra with straightening laws and thus is Koszul. Furthermore, we compute the Krull dimension ofĀ(G) in terms of the combinatorics of G. As a consequence we get new upper bounds on the arithmetical rank of monomial ideals of pure codimension 2. Finally, we characterize the Cohen-Macaulay property and the Castelnuovo-Mumford regularity of the edge ideal of a certain class of graphs.

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).

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.

Proceedings of the American Mathematical Society, 2014

In this paper, we explain the regularity, projective dimension and depth of the edge ideal of some classes of graphs in terms of invariants of graphs. We show that for a C 5-free vertex decomposable graph G, reg(R/I(G)) = c G , where c G is the maximum number of 3-disjoint edges in G. Moreover, for this class of graphs we characterize pd(R/I(G)) and depth(R/I(G)). As a corollary we describe these invariants in forests and sequentially Cohen-Macaulay bipartite graphs.

arXiv (Cornell University), 2016

By generalizing the notion of the path ideal of a graph, we study some algebraic properties of some path ideals associated to a line graph. We show that the quotient ring of these ideals are always sequentially Cohen-Macaulay and also provide some exact formulas for the projective dimension and the regularity of these ideals. As some consequences, we give some exact formulas for the depth of these ideals.

2013

In this paper firstly, we generalize the concept of codismantlable graphs to hypergraphs and show that some special vertex decomposable hypergraphs are codismantlable. Then we generalize the concept of bouquet in graphs to hypergraphs to extend some combinatorial invariants of graphs about disjointness of a set of bouquets. We use these invariants to characterize the projective dimension of Stanley-Reisner ring of special hypergraphs in some sense.

Mathematical Communications, 2017

For path ideals of the square of the line graph we compute the Krull dimension, we characterize the linear resolution property in combinatorial terms. We bound the Castelnuovo-Mumford regularity and the projective dimension in terms of the corresponding invariants of two sub-hypergraph. We present some open questions.

Mathematical Problems in Engineering

Elimination ideals are regarded as a special type of Borel type ideals, obtained from degree sequence of a graph, introduced by Anwar and Khalid. In this paper, we compute graphical degree stabilities of K n ∨ C m and K n ∗ C m by using the DVE method. We further compute sharp upper bound for Castelnuovo–Mumford regularity of elimination ideals associated to these families of graphs.