Graceful Labeling of Bow Graphs and Shell-Flower Graphs
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A graceful labeling of a graph G with 'q' edges and vertex set V is an injection f: V(G)-> {0,1,2,....q} with the property that the resulting edge labels are also distinct, where an edge incident with vertices u and v is assigned the label |f(u) - f(v)| . A graph which admits a graceful labeling is called a graceful graph. A Shell graph is defined as a cycle Cn with (n -3) chords sharing a common end point called the apex . Shell graphs are denoted as C(n, n- 3). A multiple shell is defined to be a collection of edge disjoint shells that have their apex in common. Hence a double shell consists of two edge disjoint shells with a common apex. A bow graph is defined to be a double shell in which each shell has any order. In this paper we prove that the bow graph with shell orders 'm'and '2m' is graceful. Further in this paper we define a shell- flower graph as k copies of [C(n, n-3) U K2] and we prove that all shell - flower graphs are graceful for n = 4.
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![International Journal of Computing Algorithm, Vol 2(1), June 2013
ISSN(Print):2278-2397
Website: www.ijcoa.com
Graceful Labeling of Bow Graphs and Shell-Flower
Graphs
J. Jeba Jesintha1
, Ezhilarasi Hilda Stanley2
1
PG Department of Mathematics, Women’s Christian College, Chennai
2
Department of Mathematics, Ethiraj College, Chennai
E-mail: jjesintha_75@yahoo.com
Abstract
A graceful labeling of a graph G with ‘q’ edges and vertex set V is an injection f:
V(G) → {0,1,2,….q} with the property that the resulting edge labels are also distinct, where
an edge incident with vertices u and v is assigned the label |f(u) – f(v)| . A graph
which admits a graceful labeling is called a graceful graph. A Shell graph is defined as a
cycle Cn with (n -3) chords sharing a common end point called the apex. Shell graphs are
denoted as C (n, n- 3). A multiple shell is defined to be a collection of edge disjoint shells
that have their apex in common. Hence a double shell consists of two edge disjoint shells
with a common apex. A bow graph is defined to be a double shell in which each shell has
any order. In this paper we prove that the bow graph with shell orders ‘m’ and ‘2m’ is
graceful. Further in this paper we define a shell – flower graph as k copies of [C(n, n-3) U
K2] and we prove that all shell - flower graphs are graceful for n = 4.](https://image.slidesharecdn.com/v2i12013paper17-151029050846-lva1-app6892/85/Graceful-Labeling-of-Bow-Graphs-and-Shell-Flower-Graphs-1-320.jpg)
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1. A subset S of vertices is a strong triple connected dominating set if S is a strong dominating set and the induced subgraph <S> is triple connected.
2. The strong triple connected domination number stc(G) is the minimum cardinality of a strong triple connected dominating set.
3. Some standard graphs for which the exact stc value is determined include paths, cycles, complete graphs, wheels, and more.
4. Bounds on stc(G) are established, such as 3 ≤ stc(G) ≤ p-1
Graphs-LeX2-given
This document contains lecture notes on graph theory topics for a third year computer science course taught by Dr. Ahmed Ashry. The notes cover definitions and examples of bipartite graphs, complete bipartite graphs, graph substructures like subgraphs and components, graph connectivity concepts like walks, trails, paths and connectedness. It also provides examples of these concepts using graphs like the Petersen graph and assignments for students to practice graph representations and testing bipartiteness.
50120130406004 2-3
This document summarizes an article from the International Journal of Computer Engineering and Technology. The article discusses decomposing complete graphs into circulant graphs. It presents two theorems: 1) A complete graph K2m+1 can be decomposed into m edge-disjoint circulant graphs C2m+1(j) for m ≥ 2. 2) K2m+1 can be decomposed into edge-disjoint circulant graphs when 2m+1 is prime, equals 3⋅3n for n ≥ 1, or equals 3s for s ≥ 5 prime. The article also discusses an algorithm for decomposing the complete graph and applying it to the traveling salesman problem.
IJCER (www.ijceronline.com) International Journal of computational Engineerin...
This document defines and discusses the concept of paired triple connected domination number of a graph. It begins by reviewing existing concepts like domination number, connected domination number, and triple connected domination number. It then introduces the new concept of a paired triple connected dominating set as a triple connected dominating set where the induced subgraph also has a perfect matching. The paired triple connected domination number is defined as the minimum cardinality of such a set. The document explores properties of this number and its relationship to other graph parameters. Examples are provided to illustrate the definitions.
Prime Labelling Of Some Special Graphs
1) The document investigates prime labelling of certain graphs such as the flower graph F, splitting graph of a star, bistar graph, friendship graph, and graph SF(n,1).
2) It proves that these graphs admit prime labelling by providing explicit bijective labellings of the vertices with positive integers such that the greatest common divisor of adjacent vertices is 1.
3) Examples of prime labellings are provided for specific instances of the graphs to illustrate the labellings.
Some New Families on H-Cordial Graphs
The document summarizes some new families of graphs that admit H-cordial and H2-cordial labelings. Specifically:
1) The graphs obtained by duplicating vertices or edges in the cycle Cn for n≥3 admit H2-cordial labelings.
2) The joint sum of two copies of the cycle Cn of even order produces an H2-cordial labeling graph.
3) The split graph of the cycle Cn for even n is H-cordial.
4) The shadow graph D2(Pn) of the path Pn is H-cordial for even n.
The document provides definitions, theorems with proofs, and examples
Paper id 71201961
The document discusses claw decompositions of product graphs. It provides necessary and sufficient conditions for decomposing the Cartesian product of standard graphs into claws. Some key results are:
1. If G1 and G2 are claw decomposable, then their Cartesian product G1 x G2 is also claw decomposable.
2. The Cartesian product of complete graphs or complete bipartite graphs is claw decomposable if certain conditions on their vertex sets are met.
3. Sufficient conditions are given for the lexicographic product of a graph G with Km, Kn, or K2 x Kn to be claw decomposable when m and n satisfy certain properties.
So in summary, the document
Graph theory presentation
In mathematics, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects.Graph theory is also important in real life.
Algorithmic Aspects of Vertex Geo-dominating Sets and Geonumber in Graphs
In this paper we study about x-geodominating set, geodetic set, geo-set, geo-number of a graph G. We study the
binary operation, link vectors and some required results to develop algorithms. First we design two algorithms
to check whether given set is an x-geodominating set and to find the minimum x-geodominating set of a graph.
Finally we present another two algorithms to check whether a given vertex is geo-vertex or not and to find the
geo-number of a graph.
Algorithmic Aspects of Vertex Geo-dominating Sets and Geonumber in Graphs
In this paper we study about x-geodominating set, geodetic set, geo-set, geo-number of a graph G. We study the
binary operation, link vectors and some required results to develop algorithms. First we design two algorithms
to check whether given set is an x-geodominating set and to find the minimum x-geodominating set of a graph.
Finally we present another two algorithms to check whether a given vertex is geo-vertex or not and to find the
geo-number of a graph.
Wild knots in higher dimensions as limit sets of kleinian groups
This document summarizes a research paper that constructs wild knots in dimensions 3 through 7 as limit sets of Kleinian groups. The paper introduces the concepts of tangles and knots in higher dimensions, provides background on Kleinian groups, and describes a method of constructing orthogonal ball coverings of Euclidean spaces. Using this method, the paper constructs explicit orthogonal ball coverings for dimensions 1 through 5. It then shows how to construct knots as limit sets of Kleinian groups for these dimensions, and proves properties of the resulting knots, including that they are wildly embedded.
International Journal of Engineering Research and Development
Electrical, Electronics and Computer Engineering,
Information Engineering and Technology,
Mechanical, Industrial and Manufacturing Engineering,
Automation and Mechatronics Engineering,
Material and Chemical Engineering,
Civil and Architecture Engineering,
Biotechnology and Bio Engineering,
Environmental Engineering,
Petroleum and Mining Engineering,
Marine and Agriculture engineering,
Aerospace Engineering.
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Group Magic Labeling of Cycles With a Common Vertex
Group Magic Labeling of Cycles With a Common Vertex
CYCLIC RESOLVING NUMBER OF GRID AND AUGMENTED GRID GRAPHS
CYCLIC RESOLVING NUMBER OF GRID AND AUGMENTED GRID GRAPHS
Even Harmonious Labeling of the Graph H (2n, 2t+1)
Even Harmonious Labeling of the Graph H (2n, 2t+1)
Graph Theory,Graph Terminologies,Planar Graph & Graph Colouring
Graph Theory,Graph Terminologies,Planar Graph & Graph Colouring
IJCER (www.ijceronline.com) International Journal of computational Engineerin...
IJCER (www.ijceronline.com) International Journal of computational Engineerin...
Algorithmic Aspects of Vertex Geo-dominating Sets and Geonumber in Graphs
Algorithmic Aspects of Vertex Geo-dominating Sets and Geonumber in Graphs
Algorithmic Aspects of Vertex Geo-dominating Sets and Geonumber in Graphs
Algorithmic Aspects of Vertex Geo-dominating Sets and Geonumber in Graphs
Wild knots in higher dimensions as limit sets of kleinian groups
Wild knots in higher dimensions as limit sets of kleinian groups
International Journal of Engineering Research and Development
International Journal of Engineering Research and Development
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‘last major merger.’ Hypotheses for the origin of this component include Gaia-Sausage/Enceladus (GSE), where the progenitor
collided with the MW proto-disc 8–11 Gyr ago, and the Virgo Radial Merger (VRM), where the progenitor collided with the
MW disc within the last 3 Gyr. These two scenarios make different predictions about observable structure in local phase space,
because the morphology of debris depends on how long it has had to phase mix. The recently identified phase-space folds in Gaia
DR3 have positive caustic velocities, making them fundamentally different than the phase-mixed chevrons found in simulations
at late times. Roughly 20 per cent of the stars in the prograde local stellar halo are associated with the observed caustics. Based
on a simple phase-mixing model, the observed number of caustics are consistent with a merger that occurred 1–2 Gyr ago.
We also compare the observed phase-space distribution to FIRE-2 Latte simulations of GSE-like mergers, using a quantitative
measurement of phase mixing (2D causticality). The observed local phase-space distribution best matches the simulated data
1–2 Gyr after collision, and certainly not later than 3 Gyr. This is further evidence that the progenitor of the ‘last major merger’
did not collide with the MW proto-disc at early times, as is thought for the GSE, but instead collided with the MW disc within
the last few Gyr, consistent with the body of work surrounding the VRM.
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Unlocking the mysteries of reproduction: Exploring fecundity and gonadosomati...
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