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 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 Invariants of the symbolic powers of edge ideals

2019 ◽  
Vol 19 (10) ◽  
pp. 2050184
Author(s):  
Bidwan Chakraborty ◽  
Mousumi Mandal
Keyword(s):  
Complete Graph ◽  
Explicit Description ◽  
Edge Ideal ◽  
Edge Ideals ◽  
Single Vertex ◽  
Symbolic Powers ◽  
Odd Cycles ◽  
Waldschmidt Constant

Let [Formula: see text] be a graph and [Formula: see text] be its edge ideal. When [Formula: see text] is the clique sum of two different length odd cycles joined at single vertex then we give an explicit description of the symbolic powers of [Formula: see text] and compute the Waldschmidt constant. When [Formula: see text] is complete graph then we describe the generators of the symbolic powers of [Formula: see text] and compute the Waldschmidt constant and the resurgence of [Formula: see text]. Moreover for complete graph we prove that the Castelnuovo–Mumford regularity of the symbolic powers and ordinary powers of the edge ideal coincide.

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 An Upper Bound for the Regularity of Symbolic Powers of Edge Ideals of Chordal Graphs

10.37236/8566 ◽  
2019 ◽  
Vol 26 (2) ◽  
Author(s):  
Seyed Amin Seyed Fakhari
Keyword(s):  
Upper Bound ◽  
Chordal Graph ◽  
Chordal Graphs ◽  
Symbolic Power ◽  
Matching Number ◽  
Edge Ideal ◽  
Edge Ideals ◽  
Symbolic Powers ◽  
Reg I

Assume that $G$ is a chordal graph with edge ideal $I(G)$ and ordered matching number $\nu_{o}(G)$. For every integer $s\geq 1$, we denote the $s$-th symbolic power of $I(G)$ by $I(G)^{(s)}$. It is shown that ${\rm reg}(I(G)^{(s)})\leq 2s+\nu_{o}(G)-1$. As a consequence, we determine the regularity of symbolic powers of edge ideals of chordal Cameron-Walker graphs.

scholarly journals On the k-Buchsbaum property of powers of Stanley–Reisner ideals

2014 ◽  
Vol 213 ◽  
pp. 127-140 ◽  
Author(s):  
Nguyên Công Minh ◽  
Yukio Nakamura
Keyword(s):  
Simplicial Complex ◽  
Polynomial Ring ◽  
Residue Class ◽  
Simple Graph ◽  
Edge Ideal ◽  
Vertex Set

AbstractLet S = K[x1,x2,…,xn] be a polynomial ring over a field K. Let Δ be a simplicial complex whose vertex set is contained in {1, 2,…,n}. For an integer k ≥ 0, we investigate the k-Buchsbaum property of residue class rings S/I(t); and S/It for the Stanley-Reisner ideal I = IΔ. We characterize the k-Buchsbaumness of such rings in terms of the simplicial complex Δ and the power t. We also give a characterization in the case where I is the edge ideal of a simple graph.

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.

Regularity of powers of edge ideals of product of graphs

2018 ◽  
Vol 17 (07) ◽  
pp. 1850128 ◽  
Author(s):  
S. Selvaraja
Keyword(s):  
Upper Bounds ◽  
Edge Ideal ◽  
Edge Ideals ◽  
Wheel Graphs ◽  
Product Of Graphs

In this paper, we study the Castelnuovo–Mumford regularity of powers of edge ideals of product of graphs. We find new upper bounds for [Formula: see text] for various classes of graphs [Formula: see text] and [Formula: see text], where [Formula: see text] denotes the product of [Formula: see text] and [Formula: see text], and [Formula: see text] denote the corresponding edge ideal. Using this result, we explicitly compute [Formula: see text] for several classes of graphs [Formula: see text] and [Formula: see text]. In particular, we also explicitly compute the regularity of powers of wheel graphs.

scholarly journals Weakly Closed Graphs and F-Purity of Binomial Edge Ideals

2018 ◽  
Vol 25 (04) ◽  
pp. 567-578
Author(s):  
Kazunori Matsuda
Keyword(s):  
Polynomial Ring ◽  
Quotient Ring ◽  
Gröbner Bases ◽  
Grobner Bases ◽  
Edge Ideal ◽  
Closed Graph ◽  
Edge Ideals ◽  
Weakly Closed

Herzog, Hibi, Hreindóttir et al. introduced the class of closed graphs, and they proved that the binomial edge ideal JG of a graph G has quadratic Gröbner bases if G is closed. In this paper, we introduce the class of weakly closed graphs as a generalization of the closed graph, and we prove that the quotient ring S/JG of the polynomial ring [Formula: see text] with K a field and [Formula: see text] is F-pure if G is weakly closed. This fact is a generalization of Ohtani’s theorem.

scholarly journals On the k-Buchsbaum property of powers of Stanley–Reisner ideals

2014 ◽  
Vol 213 ◽  
pp. 127-140
Author(s):  
Nguyên Công Minh ◽  
Yukio Nakamura
Keyword(s):  
Simplicial Complex ◽  
Polynomial Ring ◽  
Residue Class ◽  
Simple Graph ◽  
Edge Ideal ◽  
Vertex Set

AbstractLetS=K[x1,x2,…,xn] be a polynomial ring over a fieldK. Let Δ be a simplicial complex whose vertex set is contained in {1, 2,…,n}. For an integerk≥ 0, we investigate thek-Buchsbaum property of residue class ringsS/I(t); andS/Itfor the Stanley-Reisner idealI=IΔ. We characterize thek-Buchsbaumness of such rings in terms of the simplicial complex Δ and the powert. We also give a characterization in the case whereIis the edge ideal of a simple graph.

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