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 Matching Numbers and the Regularity of the Rees Algebra of an Edge Ideal

2020 ◽  
Vol 24 (3) ◽  
pp. 577-586
Author(s):  
Jürgen Herzog ◽  
Takayuki Hibi
Keyword(s):  
Lower Bound ◽  
Simple Graph ◽  
Matching Number ◽  
Edge Ideal ◽  
Rees Algebra ◽  
Induced Matching

Abstract The regularity $${\text {reg}}R(I(G))$$ reg R ( I ( G ) ) of the Rees ring R(I(G)) of the edge ideal I(G) of a finite simple graph G is studied. It is shown that, if R(I(G)) is normal, one has $${\text {mat}}(G) \le {\text {reg}}R(I(G)) \le {\text {mat}}(G) + 1$$ mat ( G ) ≤ reg R ( I ( G ) ) ≤ mat ( G ) + 1 , where $${\text {mat}}(G)$$ mat ( G ) is the matching number of G. In general, the induced matching number is a lower bound for the regularity, which can be shown by applying the squarefree divisor complex.

scholarly journals Regularity of bicyclic graphs and their powers

2019 ◽  
Vol 19 (03) ◽  
pp. 2050057 ◽  
Author(s):  
Yairon Cid-Ruiz ◽  
Sepehr Jafari ◽  
Navid Nemati ◽  
Beatrice Picone
Keyword(s):  
Base Case ◽  
Matching Number ◽  
Edge Ideal ◽  
Induced Matching ◽  
Base Graph ◽  
Bicyclic Graph ◽  
Bicyclic Graphs

Let [Formula: see text] be the edge ideal of a bicyclic graph [Formula: see text] with a dumbbell as the base graph. In this paper, we characterize the Castelnuovo–Mumford regularity of [Formula: see text] in terms of the induced matching number of [Formula: see text]. For the base case of this family of graphs, i.e. dumbbell graphs, we explicitly compute the induced matching number. Moreover, we prove that [Formula: see text], for all [Formula: see text], when [Formula: see text] is a dumbbell graph with a connecting path having no more than two vertices.

The Family of Lodato Proximities Compatible With a Given Topological Space

1974 ◽  
Vol 26 (02) ◽  
pp. 388-404 ◽  
Author(s):  
W. J. Thron ◽  
R. H. Warren
Keyword(s):  
Lower Bound ◽  
Topological Space ◽  
Distributive Lattice ◽  
Structural Properties ◽  
Upper Bound ◽  
Set Inclusion ◽  
The Family

Let (X, ) be a topological space. By we denote the family of all Lodato proximities on X which induce . We show that is a complete distributive lattice under set inclusion as ordering. Greatest lower bound and least upper bound are characterized. A number of techniques for constructing elements of are developed. By means of one of these constructions, all covers of any member of can be obtained. Several examples are given which relate to the lattice of all compatible proximities of Čech and the family of all compatible proximities of Efremovič. The paper concludes with a chart which summarizes many of the structural properties of , and .

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 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 A Stop Loss Inequality for Compound Poisson Processes with a Unimodal Claimsize Distribution

1977 ◽  
Vol 9 (1-2) ◽  
pp. 247-256 ◽  
Author(s):  
H. G. Verbeek
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Compound Poisson Process ◽  
Compound Poisson ◽  
Jump Size ◽  
Specific Assumption ◽  
Stop Loss ◽  
Size Variable ◽  
Insight Into ◽  
Underlying Risk

The paper considers the problem of finding an upper bound for the Stop loss premium.We will start with a brief sketch of the practical context in which this problem is relevant.If it is reasonable to assume, that the accumulated claims variable of the underlying risk can be represented by a Compound Poisson Process, the following data are needed for fixing the Stop loss premium:— the claims intensity,— the distribution of the claimsizes (jump-size variable).In practical situations it is usually possible to find a reasonable estimate for the claims intensity (expected number of claims in a given period).Generally speaking, however, it is not so easy to get sufficient data on the claimsize distribution. Ordinarily only its mean is known. This deficiency in information can of course be offset by assuming the unknown distribution to be one of the familiar types, such as Exponential, Gamma or Pareto.Stop loss premiums are however very sensitive to variations in the type of claimsize distribution and consequently it can make a lot of difference in the result what particular choice is made.To gain some insight into the consequences of a specific assumption, it is useful to know within what range the premium can move for varying distributional suppositions. This means establishing an upper bound and a lower bound. The lower bound is trivially obtained if the mass of the claimsize distribution is solely concentrated at its mean. The upper bound on the other hand should correspond to the “worst” possible claimsize distribution. This means, that we have to look for a distribution which maximizes the Stop loss premium.

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.

Neutrosophic Soft Filter Structures Concerning Soft Points

2021 ◽  
pp. 87-94
Author(s):  
Naime Demirtas ◽  
◽  
Abdullah Demirtas ◽  
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Basic Properties ◽  
Soft Topology ◽  
The Family ◽  
Set Up ◽  
Filter Structures

In this paper, the concept of neutrosophic soft filter (NSF) and its basic properties are introduced. Later, we set up a neutrosophic soft topology with the help of an NSF. We also give the notions of the greatest lower bound and the least upper bound of the family of neutrosophic soft filters (NSFs), NSF subbase, NSF base and explore some basic properties of them.

A Review of Phytochemical and Pharmacological Studies of Inula Species

2020 ◽  
Vol 16 (5) ◽  
pp. 557-567
Author(s):  
Aparoop Das ◽  
Anshul Shakya ◽  
Surajit Kumar Ghosh ◽  
Udaya P. Singh ◽  
Hans R. Bhat
Keyword(s):  
Bacterial Infections ◽  
Chemical Constituents ◽  
Valuable Insight ◽  
Important Ingredient ◽  
Pharmacological Activities ◽  
Medicinal Uses ◽  
The Family ◽  
Pharmacological Studies ◽  
Perennial Herbs ◽  
Insight Into

Background: Plants of the genus Inula are perennial herbs of the family Asteraceae. This genus includes more than 100 species, widely distributed throughout Europe, Africa and Asia including India. Many of them are indicated in traditional medicine, e.g., in Ayurveda. This review explores chemical constituents, medicinal uses and pharmacological actions of Inula species. Methods: Major databases and research and review articles retrieved through Scopus, Web of Science, and Medline were consulted to obtain information on the pharmacological activities of the genus Inula published from 1994 to 2017. Results: Inula species are used either alone or as an important ingredient of various formulations to cure dysfunctions of the cardiovascular system, respiratory system, urinary system, central nervous system and digestive system, and for the treatment of asthma, diabetes, cancers, skin disorders, hepatic disease, fungal and bacterial infections. A range of phytochemicals including alkaloids, essential and volatile oils, flavonoids, terpenes, and lactones has been isolated from herbs of the genus Inula, which might possibly explain traditional uses of these plants. Conclusion: The present review is focused on chemical constituents, medicinal uses and pharmacological actions of Inula species and provides valuable insight into its medicinal potential.

scholarly journals Multiple closed orbits for N-body-type problems

1998 ◽  
Vol 58 (1) ◽  
pp. 1-13 ◽  
Author(s):  
Shiqing Zhang
Keyword(s):  
Lower Bound ◽  
Periodic Solutions ◽  
Upper Bound ◽  
Critical Value ◽  
Body Type ◽  
Fixed Energy ◽  
Closed Orbits

Using the equivariant Ljusternik-Schnirelmann theory and the estimate of the upper bound of the critical value and lower bound for the collision solutions, we obtain some new results in the large concerning multiple geometrically distinct periodic solutions of fixed energy for a class of planar N-body type problems.

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