# Jump Sequences of Edge Ideals

@article{Whieldon2010JumpSO, title={Jump Sequences of Edge Ideals}, author={Gwyneth R. Whieldon}, journal={arXiv: Commutative Algebra}, year={2010} }

Given an edge ideal of graph G, we show that if the first nonlinear strand in the resolution of $I_G$ is zero until homological stage $a_1$, then the next nonlinear strand in the resolution is zero until homological stage $2a_1$. Additionally, we define a sequence, called a \emph{jump sequence}, characterizing the highest degrees of the free resolution of the edge ideal of G via the lower edge of the Betti diagrams of $I_G$. These sequences strongly characterize topological properties of the…

## 10 Citations

On vanishing patterns in j-strands of edge ideals

- Mathematics
- 2015

We consider two problems regarding vanishing patterns in the Betti table of edge ideals I over any fixed field. First, we show that the j-strand is connected if $$j=3$$j=3 (for $$j=2$$j=2 this is…

Algebraic Properties of Product of Graphs

- Mathematics
- 2011

Let G and H be two simple graphs and let G*H denotes the graph theoretical product of G by H. In this article, we provide graded Betti numbers, Castelnuovo–Mumford regularity, projective dimension,…

Edge ideals: algebraic and combinatorial properties

- Mathematics
- 2010

Let C be a clutter and let I(C) be its edge ideal. This is a survey paper on the algebraic and combinatorial properties of R/I(C) and C, respectively. We give a criterion to estimate the regularity…

Bounds on the regularity and projective dimension of ideals associated to graphs

- Mathematics
- 2011

In this paper, we give new upper bounds on the regularity of edge ideals whose resolutions are k-step linear; surprisingly, the bounds are logarithmic in the number of variables. We also give various…

A Beginner’s Guide to Edge and Cover Ideals

- Mathematics
- 2013

Monomial ideals, although intrinsically interesting, play an important role in studying the connections between commutative algebra and combinatorics. Broadly speaking, problems in combinatorics are…

Matchings, coverings, and Castelnuovo-Mumford regularity

- Mathematics
- 2014

We show that the co-chordal cover number of a graph G gives an upper bound for the Castelnuovo-Mumford regularity of the associated edge ideal. Several known combinatorial upper bounds of regularity…

Cover Product and Betti Polynomial of Graphs

- MathematicsCanadian Mathematical Bulletin
- 2015

Abstract The cover product of disjoint graphs $G$ and $H$ with fixed vertex covers $C\left( G \right)$ and $C\left( H \right)$ , is the graph $G\circledast H$ with vertex set $V\left( G \right)\cup…

Castelnuovo-Mumford Regularity of Graphs

- Mathematics, Computer ScienceComb.
- 2018

New combinatorial insights are presented into the calculation of (Castelnuovo-Mumford) regularity of graphs.

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