scholarly journals Edge δ− Graceful Labeling for Some Cyclic-Related Graphs

2020 ◽  
Vol 2020 ◽  
pp. 1-18
Author(s):  
Mohamed R. Zeen El Deen
Keyword(s):  
Positive Integer ◽  
Graceful Labeling ◽  
Bijective Mapping ◽  
Injective Function ◽  
Wheel Graph ◽  
New Type ◽  
Induced Mapping ◽  
Prism Graph

In this paper, we introduce a new type of labeling of a graph G with p vertices and q edges called edge δ− graceful labeling, for any positive integer δ, as a bijective mapping f of the edge set EG into the set δ,2δ,3δ,⋯,qδ such that the induced mapping f∗:VG→0,δ,2δ,3δ,⋯,qδ−δ, given by f∗u=∑uv∈EGfuvmodδk, where k=maxp,q, is an injective function. We prove the existence of an edge δ− graceful labeling, for any positive integer δ, for some cycle-related graphs like the wheel graph, alternate triangular cycle, double wheel graph Wn,n, the prism graph Πn, the prism of the wheel PWn, the gear graph Gn, the closed helm CHn, the butterfly graph Bn, and the friendship Frn.

scholarly journals Graceful Labeling and Skolem Graceful Labeling on the U-star Graph and It’s Application in Cryptography

2021 ◽  
Vol 3 (2) ◽  
pp. 103-114
Author(s):  
Meliana Pasaribu ◽  
Yundari Yundari ◽  
Muhammad Ilyas
Keyword(s):  
Star Graph ◽  
Graceful Labeling ◽  
Injective Function ◽  
Bijective Function ◽  
Labeling Pattern

Graceful Labeling on graph G=(V, E) is an injective function f from the set of the vertex V(G) to the set of numbers {0,1,2,...,|E(G)|} which induces bijective function f from the set of edges E(G) to the set of numbers {1,2,...,|E(G)|} such that for each edge uv e E(G) with u,v e V(G) in effect f(uv)=|f(u)-f(v)|. Meanwhile, the Skolem graceful labeling is a modification of the Graceful labeling. The graph has graceful labeling or Skolem graceful labeling is called graceful graph or Skolem graceful labeling graph. The graph used in this study is the U-star graph, which is denoted by U(Sn). The purpose of this research is to determine the pattern of the graceful labeling and Skolem graceful labeling on graph U(Sn) apply it to cryptography polyalphabetic cipher. The research begins by forming a graph U(Sn) and they are labeling it with graceful labeling and Skolem graceful labeling. Then, the labeling results are applied to the cryptographic polyalphabetic cipher. In this study, it is found that the U(Sn) graph is a graceful graph and a Skolem graceful graph, and the labeling pattern is obtained. Besides, the labeling results on a graph it U(Sn) can be used to form a table U(Sn) polyalphabetic cipher. The table is used as a key to encrypt messages.

MAXIMUM CUTS IN GRAPHS WITHOUT WHEELS

2018 ◽  
Vol 100 (1) ◽  
pp. 13-26
Author(s):  
JING LIN ◽  
QINGHOU ZENG ◽  
FUYUAN CHEN
Keyword(s):  
Positive Integer ◽  
Lower Bounds ◽  
Bipartite Subgraph ◽  
Wheel Graph

For a graph $G$, let $f(G)$ denote the maximum number of edges in a bipartite subgraph of $G$. Given a fixed graph $H$ and a positive integer $m$, let $f(m,H)$ denote the minimum possible cardinality of $f(G)$, as $G$ ranges over all graphs on $m$ edges that contain no copy of $H$. Alon et al. [‘Maximum cuts and judicious partitions in graphs without short cycles’, J. Combin. Theory Ser. B 88 (2003), 329–346] conjectured that, for any fixed graph $H$, there exists an $\unicode[STIX]{x1D716}(H)>0$ such that $f(m,H)\geq m/2+\unicode[STIX]{x1D6FA}(m^{3/4+\unicode[STIX]{x1D716}})$. We show that, for any wheel graph $W_{2k}$ of $2k$ spokes, there exists $c(k)>0$ such that $f(m,W_{2k})\geq m/2+c(k)m^{(2k-1)/(3k-1)}\log m$. In particular, we confirm the conjecture asymptotically for $W_{4}$ and give general lower bounds for $W_{2k+1}$.

Characterizations for the n-strong Drazin invertibility in a ring

2020 ◽  
pp. 2150141
Author(s):  
Honglin Zou ◽  
Jianlong Chen ◽  
Huihui Zhu ◽  
Yujie Wei
Keyword(s):  
Positive Integer ◽  
Generalized Inverse ◽  
Product Formula ◽  
Drazin Inverse ◽  
Equivalent Definition ◽  
New Type ◽  
Definition Of ◽  
Existence Criteria

Recently, a new type of generalized inverse called the [Formula: see text]-strong Drazin inverse was introduced by Mosić in the setting of rings. Namely, let [Formula: see text] be a ring and [Formula: see text] be a positive integer, an element [Formula: see text] is called the [Formula: see text]-strong Drazin inverse of [Formula: see text] if it satisfies [Formula: see text], [Formula: see text] and [Formula: see text]. The main aim of this paper is to consider some equivalent characterizations for the [Formula: see text]-strong Drazin invertibility in a ring. Firstly, we give an equivalent definition of the [Formula: see text]-strong Drazin inverse, that is, [Formula: see text] is the [Formula: see text]-strong Drazin inverse of [Formula: see text] if and only if [Formula: see text], [Formula: see text] and [Formula: see text]. Also, we obtain some existence criteria for this inverse by means of idempotents. In particular, the [Formula: see text]-strong Drazin invertibility of the product [Formula: see text] are investigated, where [Formula: see text] is regular and [Formula: see text] are arbitrary elements in a ring.

scholarly journals Graceful labeling on torch graph

2018 ◽  
Vol 2 (1) ◽  
pp. 14
Author(s):  
Jona Martinus Manulang ◽  
Kiki A. Sugeng
Keyword(s):  
Graceful Labeling ◽  
Injective Function ◽  
Vertex Set

Let G be a graph with vertex set V=V(G) and edge set E=E(G). An injective function f:V<span style="font-family: symbol;"> --&gt; </span>{0,1,2,...,|E|} is called graceful labeling if f induces a function f<sup>*</sup>(uv)=|f(u)<span style="font-family: symbol;">-</span>f(v)| which is a bijection from E(G) to the set {1,2,3,...,|E|}. A graph which admits a graceful labeling is called a graceful graph. In this paper, we show that torch graph O<sub>n</sub> is a graceful graph.

scholarly journals On Super (a,d)-edge antimagic total labeling of branched-prism graph

2021 ◽  
Vol 5 (1) ◽  
pp. 11
Author(s):  
Khairannisa Al Azizu ◽  
Lyra Yulianti ◽  
Narwen Narwen ◽  
Syafrizal Sy
Keyword(s):  
Positive Integer ◽  
Negative Integer ◽  
Prism Graph

Let <em>H</em> be a branched-prism graph, denoted by <em>H</em> = (<em>C<sub>m</sub></em> x <em>P</em><sub>2</sub>) ⊙ Ǩ<sub>n</sub> for odd <em>m</em>, <em>m</em> ≥ 3 and <em>n</em> ≥ 1. This paper considers about the existence of the super (<em>a</em>,<em>d</em>)-edge antimagic total labeling of <em>H</em>, for some positive integer <em>a</em> and some non-negative integer <em>d</em>.

scholarly journals Achromatic Colouring of the Central Graph of Some Specialgraphs

2019 ◽  
Vol 8 (11S) ◽  
pp. 169-171
Keyword(s):  
Structural Properties ◽  
Wheel Graph ◽  
Prism Graph

In this research investigation, the achromatic number of central graph of double wheel graph, wind mill graph andn- anti prism graph have been studied. In addition the structural properties of these graphs have also been studied.

scholarly journals Prime Graceful Labeling

2018 ◽  
Vol 7 (4.36) ◽  
pp. 750
Author(s):  
T. Hameed Hassan ◽  
R. Mohammad Abbas
Keyword(s):  
Graceful Labeling ◽  
Injective Function

A graph G with m vertices and n edges, is said to be prime graceful labeling, if there is an injection   from the vertices of G to {1, 2, ..., k} where k = min {2m, 2n} such that  gcd ( ( ),  ( )=1 and the induced injective function   from the edges of G to {1, 2, ..., k − 1} defined by  ( ) = |  ( ) − ( ) | , the resulting edge labels are distinct. In this paper path  , cycle Cn , star K1,n , friendship graph Fn , bistar Bn,n, C4 ∪ Pn , Km,2 and Km,2 ∪ Pn are shown to be Prime Graceful Labeling . 

Odd Graceful Labeling of Some New Type of Graphs

2017 ◽  
Vol 41 (1) ◽  
pp. 9-15
Author(s):  
Sushant Kumar Rout ◽  
Debdas Mishra ◽  
Purna chandra Nayak
Keyword(s):  
Graceful Labeling ◽  
New Type

scholarly journals New classes of graphs with edge $ \; \delta- $ graceful labeling

2022 ◽  
Vol 7 (3) ◽  
pp. 3554-3589
Author(s):  
Mohamed R. Zeen El Deen ◽  
◽  
Ghada Elmahdy ◽  
Keyword(s):  
Positive Integer ◽  
Mathematical Models ◽  
Communication Networks ◽  
Data Security ◽  
Coding Theory ◽  
Graph Labeling ◽  
Graceful Labeling ◽  
Extensive Range ◽  
Fan Graph ◽  
Positive Integers

<abstract><p>Graph labeling is a source of valuable mathematical models for an extensive range of applications in technologies (communication networks, cryptography, astronomy, data security, various coding theory problems). An edge $ \; \delta - $ graceful labeling of a graph $ G $ with $ p\; $ vertices and $ q\; $ edges, for any positive integer $ \; \delta $, is a bijective $ \; f\; $ from the set of edge $ \; E(G)\; $ to the set of positive integers $ \; \{ \delta, \; 2 \delta, \; 3 \delta, \; \cdots\; , \; q\delta\; \} $ such that all the vertex labels $ \; f^{\ast} [V(G)] $, given by: $ f^{\ast}(u) = (\sum\nolimits_{uv \in E(G)} f(uv)\; )\; mod\; (\delta \; k) $, where $ k = max (p, q) $, are pairwise distinct. In this paper, we show the existence of an edge $ \; \delta- $ graceful labeling, for any positive integer $ \; \delta $, for the following graphs: the splitting graphs of the cycle, fan, and crown, the shadow graphs of the path, cycle, and fan graph, the middle graphs and the total graphs of the path, cycle, and crown. Finally, we display the existence of an edge $ \; \delta- $ graceful labeling, for the twig and snail graphs.</p></abstract>

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