Related papers: On the linear strand of an edge ideal

Let $G$ be a finite simple graph on the vertex set $V(G)$ and let $r$ be a positive integer. We consider the hypergraph $\mathrm{Con}_r(G)$ whose vertices are the vertices of $G$ and the (hyper)edges are all $A\subseteq V(G)$ such that…

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 $2a_1$.…

We introduce and study the concept which we call the splitting of a graph and compare algebraic properties of the edge ideals of graphs and those of their splitting graphs.

We compute one of the distinguished extremal Betti number of the binomial edge ideal of a block graph, and classify all block graphs admitting precisely one extremal Betti number.

Let $R=\mathbb{K}[x_1,\dots,x_n]$, a graded algebra $S=R/I$ satisfies $N_{k,p}$ if $I$ is generated in degree $k$, and the graded minimal resolution is linear the first $p$ steps, and the $k$-index of $S$ is the largest $p$ such that $S$…

Let $R = \mathbb{K}[x_1, \ldots, x_n]$ and $I \subset R$ be a homogeneous ideal. In this article, we first obtain certain sufficient conditions for the subadditivity of $R/I$. As a consequence, we prove that if $I$ is generated by…

Let $G$ be a simple graph on $n$ vertices and $\mathcal{I}_G$ denotes parity binomial edge ideal of $G$ in the polynomial ring $S = \mathbb{K}[x_1,\ldots, x_n, y_1, \ldots, y_n].$ We obtain a lower bound for the regularity of parity…

We characterize all graphs whose binomial edge ideals have pure resolutions. Moreover, we introduce a new switching of graphs which does not change some algebraic invariants of graphs, and using this, we study the linear strand of the…

We study the equality of the extremal Betti numbers of the binomial edge ideal $J_G$ and those of its initial ideal ${\rm in}(J_G)$ of a closed graph $G$. We prove that in some cases there is an unique extremal Betti number for ${\rm…

We associate a sequence of positive integers, termed the type sequence, with a cochordal graph. Using this type sequence, we compute all graded Betti numbers of its edge ideal. We then classify all positive integer $n$ such that the zero…

We study the Betti tables of reducible algebraic curves with a focus on connected line arrangements and provide a general formula for computing the quadratic strand of the Betti table for line arrangements that satisfy certain hypotheses.…

We study the Betti numbers of binomial edge ideal associated to some classes of graphs with large Castelnuovo-Mumford regularity. As an application we give several lower bounds of the Castelnuovo-Mumford regularity of arbitrary graphs…

We apply some basic notions from combinatorial topology to establish various algebraic properties of edge ideals of graphs and more general Stanley-Reisner rings. In this way we provide new short proofs of some theorems from the literature…

To a matroid M with n edges, we associate the so-called facet ideal F(M) generated by monomials corresponding to bases of M. We show that the Betti numbers related to an N-graded minimal free resolution of F(M) are determined by the Betti…

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…

We determine the arithmetical rank of every edge ideal of a Ferrers graph.

In this article bipartite planar graphs St_r are investigated, r the number of their plane regions. Bounds for the graded Betti numbers and the projective dimension of the quotient ring associated to such graphs are discussed. We prove that…

As a higher analogue of the edge ideal of a graph, we study the $t$-connected ideal $\operatorname{J}_{t}$. This is the monomial ideal generated by the connected subsets of size $t$. For chordal graphs, we show that $\operatorname{J}_{t}$…

Fr\"oberg'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…

Given a finite simple graph one can associate the edge ideal. In this paper we prove that a graded Betti number of the edge ideal does not vanish if the original graph contains a set of complete bipartite subgraphs with some conditions.…