scholarly journals Squarefree Powers of Edge Ideals of Forests

2021 ◽  
Vol 28 (2) ◽  
Author(s):  
Nursel Erey ◽  
Takayuki Hibi
Keyword(s):  
Positive Integer ◽  
Upper Bound ◽  
Matching Number ◽  
Edge Ideal ◽  
Combinatorial Formula ◽  
Edge Ideals ◽  
Linear Resolution

Let $I(G)^{[k]}$ denote the $k$th squarefree power of the edge ideal of $G$. When $G$ is a forest, we provide a sharp upper bound for the regularity of $I(G)^{[k]}$ in terms of the $k$-admissable matching number of $G$. For any positive integer $k$, we classify all forests $G$ such that $I(G)^{[k]}$ has linear resolution. We also give a combinatorial formula for the regularity of $I(G)^{[2]}$ for any forest $G$.

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 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 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.

Cover Product and Betti Polynomial of Graphs

2015 ◽  
Vol 58 (2) ◽  
pp. 320-333
Author(s):  
Aurora Llamas ◽  
Josá Martínez–Bernal
Keyword(s):  
Positive Integer ◽  
Bipartite Graph ◽  
Betti Numbers ◽  
Recursive Formula ◽  
Bipartite Graphs ◽  
Matching Number ◽  
Edge Ideal ◽  
Graded Betti Numbers ◽  
Vertex Set ◽  
Chordal Bipartite Graphs

AbstractThe cover product of disjoint graphs G and H with fixed vertex covers C(G) and C(H), is the graphwith vertex set V(G) ∪ V(H) and edge setWe describe the graded Betti numbers of GeH in terms of those of. As applications we obtain: (i) For any positive integer k there exists a connected bipartite graph G such that reg R/I(G) = μS(G) + k, where, I(G) denotes the edge ideal of G, reg R/I(G) is the Castelnuovo–Mumford regularity of R/I(G) and μS(G) is the induced or strong matching number of G; (ii)The graded Betti numbers of the complement of a tree depends only upon its number of vertices; (iii)The h-vector of R/I(G e H) is described in terms of the h-vectors of R/I(G) and R/I(H). Furthermore, in a diòerent direction, we give a recursive formula for the graded Betti numbers of chordal bipartite graphs.

scholarly journals Induced Matchings and the v-Number of Graded Ideals

Mathematics ◽  
2021 ◽  
Vol 9 (22) ◽  
pp. 2860
Author(s):  
Gonzalo Grisalde ◽  
Enrique Reyes ◽  
Rafael H. Villarreal
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Matching Number ◽  
Edge Ideal ◽  
Induced Matching ◽  
Simplicial Partition ◽  
The Family ◽  
Induced Matchings ◽  
Edge Ring ◽  
Insight Into

We give a formula for the v-number of a graded ideal that can be used to compute this number. Then, we show that for the edge ideal I(G) of a graph G, the induced matching number of G is an upper bound for the v-number of I(G) when G is very well-covered, or G has a simplicial partition, or G is well-covered connected and contains neither four, nor five cycles. In all these cases, the v-number of I(G) is a lower bound for the regularity of the edge ring of G. We classify when the induced matching number of G is an upper bound for the v-number of I(G) when G is a cycle and classify when all vertices of a graph are shedding vertices to gain insight into the family of W2-graphs.

scholarly journals On The Binomial Edge Ideal of a Pair of Graphs

10.37236/2987 ◽  
2013 ◽  
Vol 20 (1) ◽  
Author(s):  
Sara Saeedi Madani ◽  
Dariush Kiani
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Betti Numbers ◽  
Edge Ideal ◽  
Closed Graph ◽  
Linear Relations ◽  
Graded Betti Numbers ◽  
Linear Resolution ◽  
Generic Matrix ◽  
The Ideal

We characterize all pairs of graphs $(G_1,G_2)$, for which the binomial edge ideal $J_{G_1,G_2}$ has linear relations. We show that $J_{G_1,G_2}$ has a linear resolution if and only if $G_1$ and $G_2$ are complete and one of them is just an edge. We also compute some of the graded Betti numbers of the binomial edge ideal of a pair of graphs with respect to some graphical terms. In particular, we show that for every pair of graphs $(G_1,G_2)$ with girth (i.e. the length of a shortest cycle in the graph) greater than 3, $\beta_{i,i+2}(J_{G_1,G_2})=0$, for all $i$. Moreover, we give a lower bound for the Castelnuovo-Mumford regularity of any binomial edge ideal $J_{G_1,G_2}$ and hence the ideal of adjacent $2$-minors of a generic matrix. We also obtain an upper bound for the regularity of $J_{G_1,G_2}$, if $G_1$ is complete and $G_2$ is a closed graph.

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 Depth and Stanley depth of the edge ideals of the powers of paths and cycles

2019 ◽  
Vol 27 (3) ◽  
pp. 113-135
Author(s):  
Zahid Iqbal ◽  
Muhammad Ishaq
Keyword(s):  
Positive Integer ◽  
Lower Bounds ◽  
Quotient Ring ◽  
Edge Ideal ◽  
Stanley Depth ◽  
Edge Ideals

AbstractLet k be a positive integer. We compute depth and Stanley depth of the quotient ring of the edge ideal associated to the kth power of a path on n vertices. We show that both depth and Stanley depth have the same values and can be given in terms of k and n. If n≣0, k + 1, k + 2, . . . , 2k(mod(2k + 1)), then we give values of depth and Stanley depth of the quotient ring of the edge ideal associated to the kth power of a cycle on n vertices and tight bounds otherwise, in terms of n and k. We also compute lower bounds for the Stanley depth of the edge ideals associated to the kth power of a path and a cycle and prove a conjecture of Herzog for these ideals.

scholarly journals On the Ternary Exponential Diophantine Equation Equating a Perfect Power and Sum of Products of Consecutive Integers

Mathematics ◽  
2021 ◽  
Vol 9 (15) ◽  
pp. 1813
Author(s):  
S. Subburam ◽  
Lewis Nkenyereye ◽  
N. Anbazhagan ◽  
S. Amutha ◽  
M. Kameswari ◽  
...  
Keyword(s):  
Positive Integer ◽  
Diophantine Equation ◽  
Upper Bound ◽  
Exponential Diophantine Equation ◽  
Research Article ◽  
All Solutions ◽  
Sum Of Products ◽  
Consecutive Integers ◽  
Sums Of Products ◽  
Positive Integers

Consider the Diophantine equation yn=x+x(x+1)+⋯+x(x+1)⋯(x+k), where x, y, n, and k are integers. In 2016, a research article, entitled – ’power values of sums of products of consecutive integers’, primarily proved the inequality n= 19,736 to obtain all solutions (x,y,n) of the equation for the fixed positive integers k≤10. In this paper, we improve the bound as n≤ 10,000 for the same case k≤10, and for any fixed general positive integer k, we give an upper bound depending only on k for n.

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.

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