On the extremal Betti numbers of the binomial edge ideal of closed graphs

Let G be a simple graph on the vertex set [n] and let JG be the corresponding binomial edge ideal. Let G=v∗H be the cone of v on H. In this article, we compute all the Betti numbers of JG in terms of the Betti numbers of JH and as a consequence, we get the Betti diagram of wheel graph. Also, we study Cohen–Macaulay defect of S/JG in terms of Cohen–Macaulay defect of SH/JH and using this we construct a graph with Cohen–Macaulay defect q for any q≥1. We obtain the depth of binomial edge ideal of join of graphs. Also, we prove that for any pair (r,b) of positive integers with 1≤b

“…Also, if G is a connected graph on n vertices such that

S / J_{G}

is Cohen–Macaulay, then

{prefixdepth}_{S} true(S / J_{G} true) = n + 1

. In [2], de Alba and Hoang studied the depth of some subclass of closed graphs. However not much more is known about the depth of binomial edge ideal.…”

Section: Introductionmentioning

confidence: 99%

“…for trees and unicyclic graphs [12]. Extremal Betti numbers of binomial edge ideals of closed graphs were studied by de Alba and Hoang in [2]. In [7], Herzog and Rinaldo studied extremal Betti number of binomial edge ideal of block graphs.…”

Section: Introductionmentioning

confidence: 99%

Depth and extremal Betti number of binomial edge ideals

Kumar

Sarkar

2020

“…See [14] and [22]. The extremal Betti numbers of the binomial edge ideal of certain graphs were studied in [8] and [15]. In [8], the authors also posed a question, see [8, Question 1].…”

Section: Introductionmentioning

confidence: 99%

“…The extremal Betti numbers of the binomial edge ideal of certain graphs were studied in [8] and [15]. In [8], the authors also posed a question, see [8, Question 1]. Indeed, the authors ask in this question if the initial ideals (with respect to the lexicographic order induced by

x_{1} > \dots > x_{n} > y_{1} > \dots > y_{n}

) of the so‐called closed graphs have the unique extremal Betti number.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Induced matchings in strongly biconvex graphs and some algebraic applications

Madani

Kiani

2021

In this paper, motivated by a question posed by H. de Alba and D. T. Hoang, we introduce strongly biconvex graphs as a subclass of weakly chordal and bipartite graphs. We give a linear time algorithm to find an induced matching for such graphs and we prove that this algorithm indeed gives a maximum induced matching. Applying this algorithm, we provide a strongly biconvex graph whose (monomial) edge ideal does not admit a unique extremal Betti number. Using this constructed graph, we provide an infinite family of the so‐called closed graphs (also known as proper interval graphs) whose binomial edge ideals do not have a unique extremal Betti number. This, in particular, answers the aforementioned question.

Gorenstein binomial edge ideals

González-Martínez

2021