scholarly journals Dynamics of Boundary Graphs

2013 ◽  
Vol 5 (3) ◽  
pp. 447-455
Author(s):  
G. Mariumuthu ◽  
M. S. Saraswathy
Keyword(s):  
Simple Graph ◽  
The Other ◽  
Boundary Vertex ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Periodic Graph ◽  
Vertex Set ◽  
Necessary And Sufficient ◽  
Positive Integers ◽  
Boundary Graph

In a graph G, the distance d(u,v) between a pair of vertices u and v is the length of a shortest path joining them. A vertex v is a boundary vertex of a vertex u if for all The boundary graph B(G) based on a connected graph G is a simple graph which has the vertex set as in G. Two vertices u and v are adjacent in B(G) if either u is a boundary of v or v is a boundary of u. If G is disconnected, then each vertex in a component is adjacent to all other vertices in the other components and is adjacent to all of its boundary vertices within the component. Given a positive integer m, the mth iterated boundary graph of G is defined as A graph G is periodic if for some m. A graph G is said to be an eventually periodic graph if there exist positive integers m and k >0 such that We give the necessary and sufficient condition for a graph to be eventually periodic.  Keywords: Boundary graph; Periodic graph. © 2013 JSR Publications. ISSN: 2070-0237 (Print); 2070-0245 (Online). All rights reserved. doi: http://dx.doi.org/10.3329/jsr.v5i3.14866 J. Sci. Res. 5 (3), xxx-xxx (2013) 

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 Equipartitioning and balancing points of polygons

Pythagoras ◽  
2010 ◽  
Vol 0 (71) ◽  
Author(s):  
Shunmugam Pillay ◽  
Poobhalan Pillay
Keyword(s):  
General Theorem ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
The Other ◽  
Sufficient Condition ◽  
Centre Of Mass ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient

The centre of mass G of a triangle has the property that the rays to the vertices from G sweep out triangles having equal areas. We show that such points, termed equipartitioning points in this paper, need not exist in other polygons. A necessary and sufficient condition for a quadrilateral to have an equipartitioning point is that one of its diagonals bisects the other. The general theorem, namely, necessary and sufficient conditions for equipartitioning points for arbitrary polygons to exist, is also stated and proved. When this happens, they are in general, distinct from the centre of mass. In parallelograms, and only in them, do the two points coincide.

Orthogonal Fractional Factorial Plans

1980 ◽  
Vol 29 (3-4) ◽  
pp. 143-160 ◽  
Author(s):  
Rahul Mukerjee
Keyword(s):  
Broad Class ◽  
Existence Results ◽  
Fractional Factorial ◽  
The Other ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Equal Frequency ◽  
Other Hand ◽  
Necessary And Sufficient ◽  
Generally True

This paper shows that the criterion of proportional frequency for (unblocked) orthogonal fractional factorial plans, as suggested by some previous authors, is not generally true. On the other hand, the criterion of equal frequency has been established as a necessary and sufficient condition in the general case. Some other properties of orthogonal fractional factorial plans have been investigated. A necessary and sufficient condition for designs involving two or more blocks has also been presented. A broad class of non-existence results follow.

Uncertainty Regarding Interpretation of the “Negligence Rule” and Its Implications for the Efficiency of Outcomes

2016 ◽  
Vol 7 (2) ◽  
Author(s):  
Satish K. Jain
Keyword(s):  
Subjective Probability ◽  
The Other ◽  
Sufficient Condition ◽  
Subjective Probabilities ◽  
Necessary And Sufficient Condition ◽  
Liability Rule ◽  
Important Conclusion ◽  
Standard Version ◽  
Necessary And Sufficient ◽  
Negligence Rule

AbstractThere are two ways that the negligence rule is interpreted. Under one interpretation a negligent injurer is liable for the entire harm to the victim; and under the other interpretation a negligent injurer is liable only for that part of the harm which can be ascribed to his negligence. Both these versions are efficient. However, if there is uncertainty regarding whether the court will be employing the full liability version or the incremental liability version for determining the liability of a negligent injurer, notwithstanding the fact that both the versions are efficient, inefficiency is possible. It is shown in the paper that a necessary and sufficient condition for efficiency in all cases is that the subjective probability with which the injurer expects the standard version to be employed must be greater than or equal to the subjective probability with which the victim expects the standard version to be employed. For the subset of applications without complementarities in the cares of the two parties and which are such that the total social costs are minimized at a unique care-configuration, it is shown that efficiency obtains regardless of the subjective probabilities with which the parties expect the two versions. One very important conclusion that emerges from the analysis of this paper is that when courts employ more than one liability rule, even if all the employed rules are efficient, the efficiency of all outcomes cannot be taken for granted merely on the ground of the efficiency of the employed rules.

Generalized unit and unitary Cayley graphs of finite rings

2019 ◽  
Vol 18 (01) ◽  
pp. 1950006 ◽  
Author(s):  
T. Tamizh Chelvam ◽  
S. Anukumar Kathirvel
Keyword(s):  
Commutative Ring ◽  
Undirected Graph ◽  
Unit Element ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Finite Rings ◽  
Vertex Set ◽  
Finite Commutative Ring ◽  
Necessary And Sufficient ◽  
Nonzero Identity

Let [Formula: see text] be a finite commutative ring with nonzero identity and [Formula: see text] be the set of all units of [Formula: see text] The graph [Formula: see text] is the simple undirected graph with vertex set [Formula: see text] in which two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if there exists a unit element [Formula: see text] in [Formula: see text] such that [Formula: see text] is a unit in [Formula: see text] In this paper, we obtain degree of all vertices in [Formula: see text] and in turn provide a necessary and sufficient condition for [Formula: see text] to be Eulerian. Also, we give a necessary and sufficient condition for the complement [Formula: see text] to be Eulerian, Hamiltonian and planar.

Matrix Transformations Based on Dirichlet Convolution

1997 ◽  
Vol 40 (4) ◽  
pp. 498-508
Author(s):  
Chikkanna Selvaraj ◽  
Suguna Selvaraj
Keyword(s):  
Bounded Variation ◽  
Matrix Transformations ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Summability Methods ◽  
The Matrix ◽  
Dirichlet Convolution ◽  
Necessary And Sufficient ◽  
Positive Integers

AbstractThis paper is a study of summability methods that are based on Dirichlet convolution. If f(n) is a function on positive integers and x is a sequence such that then x is said to be Af-summable to L. The necessary and sufficient condition for the matrix Af to preserve bounded variation of sequences is established. Also, the matrix Af is investigated as ℓ − ℓ and G − G mappings. The strength of the Af-matrix is also discussed.

A Criterion for (Non-)Planarity of The Block-Transformation Graph Gαβγ when αβγ = 101

2014 ◽  
Vol 10 ◽  
pp. 38-47 ◽  
Author(s):  
B. Basavanagoud ◽  
Jaishri B. Veeragoudar
Keyword(s):  
General Concept ◽  
Crossing Number ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Vertex Set ◽  
Necessary And Sufficient

The general concept of the block-transformation graph Gαβγ was introduced in [1]. The vertices and blocks of a graph are its members. The block-transformation graph G101 of a graph G is the graph, whose vertex set is the union of vertices and blocks of G, in which two vertices are adjacent whenever the corresponding vertices of G are adjacent or the corresponding blocks of G are nonadjacent or the corresponding members of G are incident. In this paper, we present characterizations of graphs whose block-transformation graphs G101 are planar, outerplanar or minimally nonouterplanar. Further we establish a necessary and sufficient condition for the block-transformation graph G101 to have crossing number one.

scholarly journals COARSE AMENABILITY AND DISCRETENESS

2015 ◽  
Vol 100 (1) ◽  
pp. 65-77 ◽  
Author(s):  
JERZY DYDAK
Keyword(s):  
The Other ◽  
Asymptotic Dimension ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Property A ◽  
Main Concept ◽  
Other Hand ◽  
Necessary And Sufficient ◽  
Decomposition Complexity

This paper is devoted to dualization of paracompactness to the coarse category via the concept of $R$-disjointness. Property A of Yu can be seen as a coarse variant of amenability via partitions of unity and leads to a dualization of paracompactness via partitions of unity. On the other hand, finite decomposition complexity of Guentner, Tessera, and Yu and straight finite decomposition complexity of Dranishnikov and Zarichnyi employ $R$-disjointness as the main concept. We generalize both concepts to that of countable asymptotic dimension and our main result shows that it is a subclass of spaces with Property A. In addition, it gives a necessary and sufficient condition for spaces of countable asymptotic dimension to be of finite asymptotic dimension.

Properties of bisect-diagonal quadrilaterals

2017 ◽  
Vol 101 (551) ◽  
pp. 214-226 ◽  
Author(s):  
Martin Josefsson
Keyword(s):  
General Class ◽  
The Other ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Basic Properties ◽  
Necessary And Sufficient

The general class of quadrilaterals where one diagonal is bisected by the other diagonal has appeared very rarely in the geometrical literature, but they have been named several times in connection with quadrilateral classifications. Günter Graumann strangely gave these objects two different names in [1, pp. 192, 194]: sloping-kite and sliding-kite. A. Ramachandran called them slant kites in [2, p. 54] and Michael de Villiers called them bisecting quadrilaterals in [3, pp. 19, 206]. The latter is a pretty good name, although a bit confusing: what exactly is bisected?We have found no papers and only two books where any theorems on such quadrilaterals are studied. In each of the books, one necessary and sufficient condition for such quadrilaterals is proved (see Theorem 1 and 2 in the next section). The purpose of this paper is to investigate basic properties ofconvexbisecting quadrilaterals, but we have chosen to give them a slightly different name. Let us first remind the reader that a quadrilateral whose diagonals have equal lengths is called an equidiagonal quadrilateral and one whose diagonals are perpendicular is called an orthodiagonal quadrilateral.

scholarly journals Limits of unbounded sequences of continued fractions

1990 ◽  
Vol 41 (3) ◽  
pp. 509-512
Author(s):  
Jingcheng Tong
Keyword(s):  
Continued Fractions ◽  
Continued Fraction ◽  
Rational Numbers ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Partial Quotients ◽  
Limit Points ◽  
Necessary And Sufficient ◽  
Positive Integers

Let X = {xk}k≥1 be a sequence of positive integers. Let Qk = [O;xk,xk−1,…,x1] be the finite continued fraction with partial quotients xi(1 ≤ i ≤ k). Denote the set of the limit points of the sequence {Qk}k≥1 by Λ(X). In this note a necessary and sufficient condition is given for Λ(X) to contain no rational numbers other than zero.

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