scholarly journals On the Extremal Betti Numbers of Binomial Edge Ideals of Block Graphs

10.37236/7689 ◽  
2018 ◽  
Vol 25 (1) ◽  
Author(s):  
Jürgen Herzog ◽  
Giancarlo Rinaldo
Keyword(s):  
Betti Number ◽  
Betti Numbers ◽  
Block Graph ◽  
Edge Ideal ◽  
Edge Ideals ◽  
Block Graphs ◽  
Extremal Betti Number

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.

On the Betti numbers of edge ideals of skew Ferrers graphs

2019 ◽  
Vol 30 (01) ◽  
pp. 125-139
Author(s):  
Do Trong Hoang
Keyword(s):  
Explicit Formula ◽  
Betti Number ◽  
Betti Numbers ◽  
Edge Ideal ◽  
Edge Ideals ◽  
Ferrers Graph ◽  
Ferrers Graphs ◽  
Extremal Betti Number

We prove that [Formula: see text] for any staircase skew Ferrers graph [Formula: see text], where [Formula: see text] and [Formula: see text]. As a consequence, Ene et al. conjecture is confirmed to hold true for the Betti numbers in the last column of the Betti table in a particular case. An explicit formula for the unique extremal Betti number of the binomial edge ideal of some closed graphs is also given.

scholarly journals Betti Numbers of Weighted Oriented Graphs

10.37236/9887 ◽  
2021 ◽  
Vol 28 (2) ◽  
Author(s):  
Beata Casiday ◽  
Selvi Kara
Keyword(s):  
Betti Number ◽  
Betti Numbers ◽  
Oriented Graph ◽  
Projective Dimension ◽  
Edge Ideal ◽  
Oriented Graphs ◽  
Edge Ideals ◽  
Mapping Cone ◽  
Recursive Formulas ◽  
Extremal Betti Number

Let $\mathcal{D}$ be a weighted oriented graph and $I(\mathcal{D})$ be its edge ideal. In this paper, we investigate the Betti numbers of $I(\mathcal{D})$ via upper-Koszul simplicial complexes, Betti splittings and the mapping cone construction. In particular, we provide recursive formulas for the Betti numbers of edge ideals of several classes of weighted oriented graphs. We also identify classes of weighted oriented graphs whose edge ideals have a unique extremal Betti number which allows us to compute the regularity and projective dimension for the identified classes. Furthermore, we characterize the structure of a weighted oriented graph $\mathcal{D}$ on $n$ vertices such that $\textrm{pdim } (R/I(\mathcal{D}))=n$ where $R=k[x_1,\ldots, x_n]$.

scholarly journals On the binomial edge ideals of block graphs

2016 ◽  
Vol 24 (2) ◽  
pp. 149-158 ◽  
Author(s):  
Faryal Chaudhry ◽  
Ahmet Dokuyucu ◽  
Rida Irfan
Keyword(s):  
Block Graph ◽  
Edge Ideal ◽  
Edge Ideals ◽  
Initial Ideal ◽  
Block Graphs

Abstract We find a class of block graphs whose binomial edge ideals have minimal regularity. As a consequence, we characterize the trees whose binomial edge ideals have minimal regularity. Also, we show that the binomial edge ideal of a block graph has the same depth as its initial ideal.

scholarly journals Binomial edge ideals of generalized block graphs

2020 ◽  
Vol 30 (08) ◽  
pp. 1537-1554 ◽  
Author(s):  
Arvind Kumar
Keyword(s):  
Upper Bound ◽  
Betti Number ◽  
Edge Ideals ◽  
Minimal Cut Sets ◽  
Minimal Cut ◽  
Block Graphs ◽  
Bounded Below ◽  
Extremal Betti Number

We classify generalized block graphs whose binomial edge ideals admit a unique extremal Betti number. We prove that the Castelnuovo–Mumford regularity of binomial edge ideals of generalized block graphs is bounded below by [Formula: see text], where [Formula: see text] is the number of minimal cut sets of the graph [Formula: see text] and obtain an improved upper bound for the regularity in terms of the number of maximal cliques and pendant vertices of [Formula: see text].

scholarly journals Binomial edge ideals of unicyclic graphs

2021 ◽  
pp. 1-26
Author(s):  
Rajib Sarkar
Keyword(s):  
Betti Number ◽  
Connected Graph ◽  
Betti Numbers ◽  
Unicyclic Graph ◽  
Unicyclic Graphs ◽  
Edge Ideals ◽  
Vertex Set ◽  
Bounded Below ◽  
Extremal Betti Number

Let [Formula: see text] be a connected graph on the vertex set [Formula: see text]. Then [Formula: see text]. In this paper, we prove that if [Formula: see text] is a unicyclic graph, then the depth of [Formula: see text] is bounded below by [Formula: see text]. Also, we characterize [Formula: see text] with [Formula: see text] and [Formula: see text]. We then compute one of the distinguished extremal Betti numbers of [Formula: see text]. If [Formula: see text] is obtained by attaching whiskers at some vertices of the cycle of length [Formula: see text], then we show that [Formula: see text]. Furthermore, we characterize [Formula: see text] with [Formula: see text], [Formula: see text] and [Formula: see text]. In each of these cases, we classify the uniqueness of the extremal Betti number of these graphs.

scholarly journals Binomial Edge Ideals of Graphs

10.37236/2349 ◽  
2012 ◽  
Vol 19 (2) ◽  
Author(s):  
Dariush Kiani ◽  
Sara Saeedi
Keyword(s):  
Upper Bound ◽  
Betti Number ◽  
Complete Graphs ◽  
Edge Ideal ◽  
Closed Graph ◽  
Edge Ideals ◽  
Linear Resolution

We characterize all graphs whose binomial edge ideals have a linear resolution. Indeed, we show that complete graphs are the only graphs with this property. We also compute some graded components of the first Betti number of the binomial edge ideal of a graph with respect to the graphical terms. Finally, we give an upper bound for the Castelnuovo-Mumford regularity of the binomial edge ideal of a closed graph.

scholarly journals Krull dimension and regularity of binomial edge ideals of block graphs

2019 ◽  
Vol 19 (07) ◽  
pp. 2050133 ◽  
Author(s):  
Carla Mascia ◽  
Giancarlo Rinaldo
Keyword(s):  
Lower Bound ◽  
Linear Time ◽  
Betti Numbers ◽  
Krull Dimension ◽  
Edge Ideals ◽  
New Family ◽  
Block Graphs ◽  
Linear Time Algorithms

We give a lower bound for the Castelnuovo-Mumford regularity of binomial edge ideals of block graphs by computing the two distinguished extremal Betti numbers of a new family of block graphs, called flower graphs. Moreover, we present linear time algorithms to compute the Castelnuovo–Mumford regularity and the Krull dimension of binomial edge ideals of block graphs.

scholarly journals Depth and extremal Betti number of binomial edge ideals

2020 ◽  
Vol 293 (9) ◽  
pp. 1746-1761 ◽  
Author(s):  
Arvind Kumar ◽  
Rajib Sarkar
Keyword(s):  
Betti Number ◽  
Edge Ideals ◽  
Extremal Betti Number

scholarly journals On the Betti Numbers of some Classes of Binomial Edge Ideals

10.37236/3689 ◽  
2013 ◽  
Vol 20 (4) ◽  
Author(s):  
Sohail Zafar ◽  
Zohaib Zahid
Keyword(s):  
Lower Bounds ◽  
Betti Numbers ◽  
Edge Ideal ◽  
Induced Subgraphs ◽  
Edge Ideals ◽  
Arbitrary Graphs

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 depending on induced subgraphs.

scholarly journals Stanley depth of edge ideals

2012 ◽  
Vol 49 (4) ◽  
pp. 501-508 ◽  
Author(s):  
Muhammad Ishaq ◽  
Muhammad Qureshi
Keyword(s):  
Complete Graph ◽  
Upper Bound ◽  
Edge Ideal ◽  
Stanley Depth ◽  
Edge Ideals

We give an upper bound for the Stanley depth of the edge ideal I of a k-partite complete graph and show that Stanley’s conjecture holds for I. Also we give an upper bound for the Stanley depth of the edge ideal of a s-uniform complete bipartite hypergraph.

Sign in / Sign up

Export Citation Format

Share Document