Edge Ideals with Regularity No More than Three

2020 ◽  
Vol 27 (04) ◽  
pp. 761-766
Author(s):  
Aming Liu ◽  
Tongsuo Wu
Keyword(s):  
Simple Graph ◽  
Edge Ideal ◽  
Free Graph ◽  
Edge Ideals

We prove that if G is a gap-free and chair-free simple graph, then the regularity of the edge ideal of G is no more than 3. If G is a gap-free and P4-free graph, then it is a chair-free graph; furthermore, the complement of G is chordal, and thus the regularity of G is 2.

scholarly journals An upper bound for the regularity of powers of edge ideals

2020 ◽  
Vol 126 (2) ◽  
pp. 165-169
Author(s):  
Jürgen Herzog ◽  
Takayuki Hibi
Keyword(s):  
Upper Bound ◽  
Simple Graph ◽  
Edge Ideal ◽  
Edge Ideals

For a finite simple graph $G$ we give an upper bound for the regularity of the powers of the edge ideal $I(G)$.

scholarly journals Upper bounds for the regularity of symbolic powers of certain classes of edge ideals

2021 ◽  
Author(s):  
Arvind Kumar ◽  
S. Selvaraja
Keyword(s):  
Polynomial Ring ◽  
Upper Bounds ◽  
Simple Graph ◽  
Edge Ideal ◽  
Edge Ideals ◽  
Symbolic Powers

Let [Formula: see text] be a finite simple graph and [Formula: see text] denote the corresponding edge ideal in a polynomial ring over a field [Formula: see text]. In this paper, we obtain upper bounds for the Castelnuovo–Mumford regularity of symbolic powers of certain classes of edge ideals. We also prove that for several classes of graphs, the regularity of symbolic powers of their edge ideals coincides with that of their ordinary powers.

scholarly journals Certain algebraic invariants of edge ideals of join of graphs

2020 ◽  
pp. 2150099
Author(s):  
Arvind Kumar ◽  
Rajiv Kumar ◽  
Rajib Sarkar
Keyword(s):  
Simple Graph ◽  
Chordal Graphs ◽  
Edge Ideal ◽  
Edge Ideals ◽  
Symbolic Powers ◽  
Multipartite Graphs ◽  
Wheel Graphs ◽  
Join Of Graphs ◽  
Depth Dimension ◽  
Complete Multipartite

Let [Formula: see text] be a simple graph and [Formula: see text] be its edge ideal. In this paper, we study the Castelnuovo–Mumford regularity of symbolic powers of edge ideals of join of graphs. As a consequence, we prove Minh’s conjecture for wheel graphs, complete multipartite graphs, and a subclass of co-chordal graphs. We obtain a class of graphs whose edge ideals have regularity three. By constructing graphs, we prove that the multiplicity of edge ideals of graphs is independent from the depth, dimension, regularity, and degree of [Formula: see text]-polynomial. Also, we demonstrate that the depth of edge ideals of graphs is independent from the regularity and degree of [Formula: see text]-polynomial by constructing graphs.

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.

scholarly journals Comparing powers of edge ideals

2019 ◽  
Vol 18 (10) ◽  
pp. 1950184 ◽  
Author(s):  
Mike Janssen ◽  
Thomas Kamp ◽  
Jason Vander Woude
Keyword(s):  
Symbolic Power ◽  
Edge Ideal ◽  
Homogeneous Ideal ◽  
Edge Ideals ◽  
Ideal Formula ◽  
Number Of Generators ◽  
Recent Interest ◽  
Odd Cycle ◽  
Ordinary Power

Given a nontrivial homogeneous ideal [Formula: see text], a problem of great recent interest has been the comparison of the [Formula: see text]th ordinary power of [Formula: see text] and the [Formula: see text]th symbolic power [Formula: see text]. This comparison has been undertaken directly via an exploration of which exponents [Formula: see text] and [Formula: see text] guarantee the subset containment [Formula: see text] and asymptotically via a computation of the resurgence [Formula: see text], a number for which any [Formula: see text] guarantees [Formula: see text]. Recently, a third quantity, the symbolic defect, was introduced; as [Formula: see text], the symbolic defect is the minimal number of generators required to add to [Formula: see text] in order to get [Formula: see text]. We consider these various means of comparison when [Formula: see text] is the edge ideal of certain graphs by describing an ideal [Formula: see text] for which [Formula: see text]. When [Formula: see text] is the edge ideal of an odd cycle, our description of the structure of [Formula: see text] yields solutions to both the direct and asymptotic containment questions, as well as a partial computation of the sequence of symbolic defects.

scholarly journals Edge ideals of clique clutters of comparability graphs and the normality of monomial ideals

2010 ◽  
Vol 106 (1) ◽  
pp. 88 ◽  
Author(s):  
Luis A. Dupont ◽  
Rafael H. Villarreal
Keyword(s):  
Monomial Ideal ◽  
Lattice Points ◽  
Edge Ideal ◽  
Comparability Graph ◽  
Decomposition Property ◽  
Edge Ideals ◽  
Finite Poset ◽  
Integer Decomposition Property ◽  
Torsion Free ◽  
Min Cut

The normality of a monomial ideal is expressed in terms of lattice points of blocking polyhedra and the integer decomposition property. For edge ideals of clutters this property characterizes normality. Let $G$ be the comparability graph of a finite poset. If $\mathrm{cl}(G)$ is the clutter of maximal cliques of $G$, we prove that $\mathrm{cl}(G)$ satisfies the max-flow min-cut property and that its edge ideal is normally torsion free. Then we prove that edge ideals of complete admissible uniform clutters are normally torsion free.

scholarly journals On the sum-connectivity index

Filomat ◽  
2011 ◽  
Vol 25 (3) ◽  
pp. 29-42 ◽  
Author(s):  
Shilin Wang ◽  
Zhou Bo ◽  
Nenad Trinajstic
Keyword(s):  
Lower Bound ◽  
Minimum Degree ◽  
Simple Graph ◽  
Connectivity Index ◽  
Extremal Graphs ◽  
Free Graph ◽  
Mathematical Chemistry ◽  
Degree Of Vertex

The sum-connectivity index of a simple graph G is defined in mathematical chemistry as R+(G) = ? uv?E(G)(du+dv)?1/2, where E(G) is the edge set of G and du is the degree of vertex u in G. We give a best possible lower bound for the sum-connectivity index of a graph (a triangle-free graph, respectively) with n vertices and minimum degree at least two and characterize the extremal graphs, where n ? 11.

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 ON THE DEPTH OF SYMBOLIC POWERS OF EDGE IDEALS OF GRAPHS

2020 ◽  
pp. 1-13
Author(s):  
S. A. SEYED FAKHARI
Keyword(s):  
Chordal Graph ◽  
Symbolic Power ◽  
Edge Ideal ◽  
Packing Number ◽  
Edge Ideals ◽  
Symbolic Powers

Abstract Assume that G is a graph with edge ideal $I(G)$ and star packing number $\alpha _2(G)$ . We denote the sth symbolic power of $I(G)$ by $I(G)^{(s)}$ . It is shown that the inequality $ \operatorname {\mathrm {depth}} S/(I(G)^{(s)})\geq \alpha _2(G)-s+1$ is true for every chordal graph G and every integer $s\geq 1$ . Moreover, it is proved that for any graph G, we have $ \operatorname {\mathrm {depth}} S/(I(G)^{(2)})\geq \alpha _2(G)-1$ .

scholarly journals Cohen-Macaulay binomial edge ideals

2011 ◽  
Vol 204 ◽  
pp. 57-68 ◽  
Author(s):  
Viviana Ene ◽  
Jürgen Herzog ◽  
Takayuki Hibi
Keyword(s):  
Edge Ideal ◽  
Edge Ideals

AbstractWe study the depth of classes of binomial edge ideals and classify all closed graphs whose binomial edge ideal is Cohen-Macaulay.

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