scholarly journals Vertex Isoperimetric Inequalities for a Family of Graphs on $\mathbb{Z}^k$

10.37236/2426 ◽  
2012 ◽  
Vol 19 (2) ◽  
Author(s):  
Ellen Veomett ◽  
Andrew John Radcliffe
Keyword(s):  
Positive Integer ◽  
Isoperimetric Inequality ◽  
Isoperimetric Inequalities ◽  
The Family ◽  
Vertex Set

We consider the family of graphs whose vertex set is $\mathbb{Z}^k$ where two vertices are connected by an edge when their $\ell_\infty$-distance is 1.  We prove the optimal vertex isoperimetric inequality for this family of graphs.  That is, given a positive integer $n$, we find a set $A\subset \mathbb{Z}^k$ of size $n$ such that the number of vertices who share an edge with some vertex in $A$ is minimized.  These sets of minimal boundary  are nested, and the proof uses the technique of compression.We also show a method of calculating the vertex boundary for certain subsets in  this family of graphs.  This calculation and the isoperimetric inequality allow us to indirectly find the sets which minimize the function calculating the boundary.

scholarly journals Bipartite graphs with close domination and k-domination numbers

2020 ◽  
Vol 18 (1) ◽  
pp. 873-885
Author(s):  
Gülnaz Boruzanlı Ekinci ◽  
Csilla Bujtás
Keyword(s):  
Positive Integer ◽  
Dominating Set ◽  
Domination Number ◽  
Bipartite Graphs ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Minimum Cardinality ◽  
Vertex Set ◽  
Necessary And Sufficient ◽  
The Given

Abstract Let k be a positive integer and let G be a graph with vertex set V(G) . A subset D\subseteq V(G) is a k -dominating set if every vertex outside D is adjacent to at least k vertices in D . The k -domination number {\gamma }_{k}(G) is the minimum cardinality of a k -dominating set in G . For any graph G , we know that {\gamma }_{k}(G)\ge \gamma (G)+k-2 where \text{Δ}(G)\ge k\ge 2 and this bound is sharp for every k\ge 2 . In this paper, we characterize bipartite graphs satisfying the equality for k\ge 3 and present a necessary and sufficient condition for a bipartite graph to satisfy the equality hereditarily when k=3 . We also prove that the problem of deciding whether a graph satisfies the given equality is NP-hard in general.

scholarly journals On Total Vertex Irregularity Strength of Hexagonal Cluster Graphs

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Nurdin Hinding ◽  
Hye Kyung Kim ◽  
Nurtiti Sunusi ◽  
Riskawati Mise
Keyword(s):  
Lower Bound ◽  
Positive Integer ◽  
Simple Graph ◽  
Vertex Set ◽  
Cluster Graph ◽  
Cluster Graphs ◽  
Irregularity Strength

For a simple graph G with a vertex set V G and an edge set E G , a labeling f : V G ∪ ​ E G ⟶ 1,2 , ⋯ , k is called a vertex irregular total k − labeling of G if for any two different vertices x and y in V G we have w t x ≠ w t y where w t x = f x + ∑ u ∈ V G f x u . The smallest positive integer k such that G has a vertex irregular total k − labeling is called the total vertex irregularity strength of G , denoted by tvs G . The lower bound of tvs G for any graph G have been found by Baca et. al. In this paper, we determined the exact value of the total vertex irregularity strength of the hexagonal cluster graph on n cluster for n ≥ 2 . Moreover, we show that the total vertex irregularity strength of the hexagonal cluster graph on n cluster is 3 n 2 + 1 / 2 .

MINIMUM WEIGHT FEEDBACK VERTEX SETS IN CIRCLE n-GON GRAPHS AND CIRCLE TRAPEZOID GRAPHS

2011 ◽  
Vol 03 (03) ◽  
pp. 323-336 ◽  
Author(s):  
FANICA GAVRIL
Keyword(s):  
Minimum Weight ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Circle Graphs ◽  
Intersection Model ◽  
The Family ◽  
Positive Weights ◽  
Vertex Set ◽  
Vertex Sets ◽  
Trapezoid Graphs

A circle n-gon is the region between n or fewer non-crossing chords of a circle, no chord connecting the arcs between two other chords; the sides of a circle n-gon are either chords or arcs of the circle. A circle n-gon graph is the intersection graph of a family of circle n-gons in a circle. The family of circle trapezoid graphs is exactly the family of circle 2-gon graphs and the family of circle graphs is exactly the family of circle 1-gon graphs. The family of circle n-gon graphs contains the polygon-circle graphs which have an intersection representation by circle polygons, each polygon with at most n chords. We describe a polynomial time algorithm to find a minimum weight feedback vertex set, or equivalently, a maximum weight induced forest, in a circle n-gon graph with positive weights, when its intersection model by n-gon-interval-filaments is given.

From the Outside In

2015 ◽  
Author(s):  
Derek Smith
Keyword(s):  
Positive Integer ◽  
Infinite Family ◽  
Open Problems ◽  
The Family

This chapter discusses Slothouber–Graatsma–Conway puzzle, which asks one to assemble six 1 × 2 × 2 pieces and three 1 × 1 × 1 pieces into the shape of a 3 × 3 × 3 cube. The puzzle has been generalized to larger cubes, and there is an infinite family of such puzzles. The chapter's primary argument is that, for any odd positive integer n = 2k + 1, there is exactly one way, up to symmetry, to make an n × n × n cube out of n tiny 1 × 1 × 1 cubes and six of each of a set of rectangular blocks. The chapter describes a way to solve each puzzle in the family and explains why there are no other solutions. It then presents several related open problems.

Total k-rainbow reinforcement number in graphs

2020 ◽  
pp. 2050101
Author(s):  
L. Shahbazi ◽  
H. Abdollahzadeh Ahangar ◽  
R. Khoeilar ◽  
S. M. Sheikholeslami
Keyword(s):  
Open Neighborhood ◽  
Minimum Weight ◽  
Domination Number ◽  
The Family ◽  
Rainbow Domination ◽  
Minimum Number ◽  
Vertex Set ◽  
Isolated Vertex ◽  
Complete Bipartite Graphs ◽  
Dominating Function

Let [Formula: see text] be an integer, and let [Formula: see text] be a graph. A k-rainbow dominating function (or [Formula: see text]RDF) of [Formula: see text] is a function [Formula: see text] from the vertex set [Formula: see text] to the family of all subsets of [Formula: see text] such that for very [Formula: see text] with [Formula: see text], the condition [Formula: see text] is fulfilled, where [Formula: see text] is the open neighborhood of [Formula: see text]. The weight of a [Formula: see text]RDF [Formula: see text] of [Formula: see text] is the value [Formula: see text]. A k-rainbow dominating function [Formula: see text] in a graph with no isolated vertex is called a total k-rainbow dominating function if the subgraph of [Formula: see text] induced by the set [Formula: see text] has no isolated vertices. The total k-rainbow domination number of [Formula: see text], denoted by [Formula: see text], is the minimum weight of the total [Formula: see text]-rainbow dominating function on [Formula: see text]. The total k-rainbow reinforcement number of [Formula: see text], denoted by [Formula: see text], is the minimum number of edges that must be added to [Formula: see text] in order to decrease the total k-rainbow domination number. In this paper, we investigate the properties of total [Formula: see text]-rainbow reinforcement number in graphs. In particular, we present some sharp bounds for [Formula: see text] and we determine the total [Formula: see text]-rainbow reinforcement number of some classes of graphs including paths, cycles and complete bipartite graphs.

scholarly journals Monophonic wirelength on Circulant Networks

2018 ◽  
Vol 7 (3) ◽  
pp. 1154
Author(s):  
K Uma Samundesvari ◽  
Sindhuja G. Michael
Keyword(s):  
The Family ◽  
Vertex Set ◽  
Sum Of Distances ◽  
Circulant Networks

In this paper, we define the monophonic embedding of graph G into graph H and we present an algorithm for finding the monophonic wirelength of circulant networks into the family of grids M(n×2), n≥2.The monophonic embedding of a graph G into a graph H is an embedding denoted by fm is a bijective map from the vertex set of G into the vertex set of H and fm is a one-one mapping from the edge set (x, y) of G into Pm(H) where Pm(H) is the set of monophonic paths between fm(x) and fm(y) for every fm(x), fm(y) H. The monophonic wirelength of fm of G into H is the sum of distances of monophonic paths between two vertices fm(x) and fm(y) in H such that (x, y) E(G). This paper presents a monophonic algorithm to find the monophonic wirelength of circulant networks G(2n, ±S) , where S {1,2,3,…,n} into the family of grids M[n×2],n≥2. We also derived a Lemma to get the monophonic edge congestion MEC(G,H).  

scholarly journals ON THE FAMILY OF DIOPHANTINE TRIPLES {k− 1,k+ 1, 16k3− 4k}

2007 ◽  
Vol 49 (2) ◽  
pp. 333-344 ◽  
Author(s):  
YANN BUGEAUD ◽  
ANDREJ DUJELLA ◽  
MAURICE MIGNOTTE
Keyword(s):  
Positive Integer ◽  
Recent Result ◽  
The Family ◽  
Diophantine Triples ◽  
Image Position ◽  
Diophantine Quadruples

AbstractIt is proven that ifk≥ 2 is an integer anddis a positive integer such that the product of any two distinct elements of the setincreased by 1 is a perfect square, thend= 4kord= 64k5−48k3+8k. Together with a recent result of Fujita, this shows that all Diophantine quadruples of the form {k− 1,k+ 1,c,d} are regular.

scholarly journals VERY WELL-COVERED GRAPHS OF GIRTH AT LEAST FOUR AND LOCAL MAXIMUM STABLE SET GREEDOIDS

2011 ◽  
Vol 03 (02) ◽  
pp. 245-252 ◽  
Author(s):  
VADIM E. LEVIT ◽  
EUGEN MANDRESCU
Keyword(s):  
Perfect Matching ◽  
Local Maximum ◽  
Stable Set ◽  
Stable Sets ◽  
Maximum Cardinality ◽  
Math Program ◽  
The Family ◽  
Maximum Stable Set ◽  
Vertex Set ◽  
Appl Math

A maximum stable set in a graph G is a stable set of maximum cardinality. S is a local maximum stable set of G, and we write S ∈ Ψ(G), if S is a maximum stable set of the subgraph induced by S ∪ N(S), where N(S) is the neighborhood of S. Nemhauser and Trotter Jr. [Vertex packings: structural properties and algorithms, Math. Program.8 (1975) 232–248], proved that any S ∈ Ψ(G) is a subset of a maximum stable set of G. In [Levit and Mandrescu, A new greedoid: the family of local maximum stable sets of a forest, Discrete Appl. Math.124 (2002) 91–101] we have shown that the family Ψ(T) of a forest T forms a greedoid on its vertex set. The cases where G is bipartite, triangle-free, well-covered, while Ψ(G) is a greedoid, were analyzed in [Levit and Mandrescu, Local maximum stable sets in bipartite graphs with uniquely restricted maximum matchings, Discrete Appl. Math.132 (2004) 163–174], [Levit and Mandrescu, Triangle-free graphs with uniquely restricted maximum matchings and their corresponding greedoids, Discrete Appl. Math.155 (2007) 2414–2425], [Levit and Mandrescu, Well-covered graphs and greedoids, Proc. 14th Computing: The Australasian Theory Symp. (CATS2008), Wollongong, NSW, Conferences in Research and Practice in Information Technology, Vol. 77 (2008) 89–94], respectively. In this paper we demonstrate that if G is a very well-covered graph of girth ≥4, then the family Ψ(G) is a greedoid if and only if G has a unique perfect matching.

THE ITERATION DIGRAPHS OF GROUP RINGS OVER FINITE FIELDS

2014 ◽  
Vol 13 (05) ◽  
pp. 1350162 ◽  
Author(s):  
YANGJIANG WEI ◽  
GAOHUA TANG ◽  
JIZHU NAN
Keyword(s):  
Positive Integer ◽  
Finite Field ◽  
Finite Fields ◽  
Sufficient Conditions ◽  
Group Rings ◽  
Directed Edge ◽  
Cycle Structure ◽  
Vertex Set ◽  
Finite Commutative Ring ◽  
Necessary And Sufficient

For a finite commutative ring R and a positive integer k ≥ 2, we construct an iteration digraph G(R, k) whose vertex set is R and for which there is a directed edge from a ∈ R to b ∈ R if b = ak. In this paper, we investigate the iteration digraphs G(𝔽prCn, k) of 𝔽prCn, the group ring of a cyclic group Cn over a finite field 𝔽pr. We study the cycle structure of G(𝔽prCn, k), and explore the symmetric digraphs. Finally, we obtain necessary and sufficient conditions on 𝔽prCn and k such that G(𝔽prCn, k) is semiregular.

scholarly journals Heat kernel estimates for an operator with a singular drift and isoperimetric inequalities

2018 ◽  
Vol 2018 (736) ◽  
pp. 1-31
Author(s):  
Alexander Grigor’yan ◽  
Shunxiang Ouyang ◽  
Michael Röckner
Keyword(s):  
Heat Kernel ◽  
Isoperimetric Inequality ◽  
Lower Bounds ◽  
Isoperimetric Inequalities ◽  
Upper And Lower Bounds ◽  
Radial Part ◽  
Kernel Estimates ◽  
Heat Kernel Estimates ◽  
Singular Drift

AbstractIn the present paper we prove upper and lower bounds of the heat kernel for the operator{\Delta-\nabla({|x|^{-\alpha}})\cdot\nabla}in{\mathbb{R}^{n}\setminus\{0\}}, where{\alpha>0}. We obtain these bounds from an isoperimetric inequality for a measure{\mathrm{e}^{-{|x|^{-\alpha}}}dx}on{\mathbb{R}^{n}\setminus\{0\}}. The latter amounts to a certain functional isoperimetric inequality for the radial part of this measure.

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