scholarly journals H-Coverings of Path-Amalgamated Ladders and Fans

2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
Yijun Xiong ◽  
Huajun Wang ◽  
Muhammad Awais Umar ◽  
Yu-Ming Chu ◽  
Basharat Rehman Ali ◽  
...  
Keyword(s):  
Simple Graph ◽  
Antimagic Labeling ◽  
Covering Graph ◽  
Edge Covering

Let G be a connected, simple graph with finite vertices v and edges e . A family G 1 , G 2 , … , G p ⊂ G of subgraphs such that for all e ∈ E , e ∈ G l , for some l ,   l = 1,2 , … , p is an edge-covering of G . If G l ≅ ℍ , ∀ l , then G has an ℍ -covering. Graph G with ℍ -covering is an a d , d - ℍ -antimagic if ψ : V G ∪ E G ⟶ 1,2 , … , v + e a bijection exists and the sum over all vertex-weights and edge-weights of ℍ forms a set a d , a d + d , … , a d + p − 1 d . The labeling ψ is super for ψ V G = 1,2,3 , … , v and graph G is ℍ -supermagic for d = 0 . This manuscript proves results about super ℍ -antimagic labeling of path amalgamation of ladders and fans for several differences.

scholarly journals Pelabelan Selimut Bintang Ajaib Super Pada Graf Bintang

2021 ◽  
Vol 18 (1) ◽  
pp. 95-109
Author(s):  
N Mattiro ◽  
I W Sudarsana
Keyword(s):  
Research Methodology ◽  
Simple Graph ◽  
The Other ◽  
Star Graph ◽  
Literature Study ◽  
Covering Graph ◽  
Edge Covering ◽  
The Given

Let  be a simple graph. An edge covering of  is a family of subgraphs  such that each edge of graph  belongs to at least one of the ,  subgraphs. If each  is isomorphic with the given graph , then it is said that contains a  covering. The graph G contains a covering  and   the bijectif function  is said an the magic labeling of a graph G if for each subgraph  of  is isomorphic to , so that is a constant. It is said that the graph G has a super magic if  in this case, the graph G which can be labeled with  magic is called the covering graph  magic. A star graph with n points is a graph with  points and  sides, where point is  degree and the other  point has degree  denoted by . This study aims to determine the presence of covering labeling for the super-magic star on the  star graph. The research methodology is literature study. The results show that the  star graph for   has   magic covering labeling with magic constants for all covering is  and the super-magic covering labeling with magic constants for all covering is .

PELABELAN SELIMUT AJAIB SUPER PADA GRAF LINTASAN

2018 ◽  
Vol 15 (2) ◽  
pp. 118-129
Author(s):  
N Farida ◽  
I W Sudarsana ◽  
Resnawati Resnawati
Keyword(s):  
Simple Graph ◽  
Path Graph ◽  
Covering Graph ◽  
Edge Covering

Let 𝐺 = (𝑉, 𝐸) be a simple graph. An edge covering of 𝐺 is a family of subgraphs 𝐻1 , … , 𝐻𝑘 such that each edge of 𝐸(𝐺) belongs to at least one of the subgraphs 𝐻𝑖 , 1 ≤ 𝑖 ≤ 𝑘. If every 𝐻𝑖 is isomorphic to a given graph 𝐻, then the graph 𝐺 admits an 𝐻 − covering. Let 𝐺 be a containing a covering 𝐻, and 𝑓 the bijectif function 𝑓: (𝑉 ∪ 𝐸) → {1,2,3, … , |𝑉| + |𝐸|} is said an 𝐻 −magic labeling of 𝐺 if for every subgraph 𝐻 ′ = (𝑉 ′ ,𝐸 ′ ) of 𝐺 isomorphic to 𝐻, is obtained that ∑ 𝑓(𝑉) + ∑ 𝑓(𝐸) 𝑒∈𝐸(𝐻′ 𝑣∈𝑉(𝐻 ) ′ ) is constant. 𝐺 is said to be 𝐻 −super magic if 𝑓(𝑉) = {1, 2, 3, … , |𝑉|}. In this case, the graph 𝐺 which can be labeled with 𝐻-magic is called the covering graph 𝐻 −magic. The sum of all vertex labels and all edge labels on the covering 𝐻 − super magic then obtained constant magic is denoted by ∑ 𝑓(𝐻). The duplication graph 2 of graph 𝐷2 (𝐺) is a graph obtained from two copies of graph 𝐺, called 𝐺 and 𝐺 ′ , with connecting each respectively vertex 𝑣 in 𝐺 with the vertexs immediate neighboring of 𝑣 ′ in 𝐺 ′ . The purpose of this study is to obtain a covering super magic labeling for of 𝐷2 (𝑃𝑚) on (𝐷2 (𝑃𝑛 )) for 𝑛 ≥ 4 and 3 ≤ 𝑚 ≤ 𝑛 − 1. In this paper, we have showed that duplication path graph (𝐷2 (𝑃𝑛 )) has 𝐷2 (𝑃𝑚) covering super magic labeling for 𝑛 ≥ 4 and 3 ≤ 𝑚 ≤ 𝑛 − 1 with constant magic for all covering is ∑ 𝑓(𝐷2 (𝑃𝑚) (𝑠) ) = ∑ 𝑓(𝐷2 (𝑃𝑚) (𝑠+1) )

scholarly journals Super (a, d)-H-antimagic labeling of subdivided graphs

2018 ◽  
Vol 16 (1) ◽  
pp. 688-697
Author(s):  
Amir Taimur ◽  
Muhammad Numan ◽  
Gohar Ali ◽  
Adeela Mumtaz ◽  
Andrea Semaničová-Feňovčíková
Keyword(s):  
Simple Graph ◽  
Arithmetic Sequence ◽  
Antimagic Labeling ◽  
Bijective Function

AbstractA simple graphG= (V,E) admits anH-covering, if every edge inE(G) belongs to a subgraph ofGisomorphic toH. A graphGadmitting anH-covering is called an (a,d)-H-antimagic if there exists a bijective functionf:V(G) ∪E(G) → {1, 2, …, |V(G)| + |E(G)|} such that for all subgraphsH′ isomorphic toHthe sums ∑v∈V(H′)f(v) + ∑e∈E(H′)f(e) form an arithmetic sequence {a,a+d, …,a+ (t− 1)d}, wherea> 0 andd≥ 0 are integers andtis the number of all subgraphs ofGisomorphic toH. Moreover, if the vertices are labeled with numbers 1, 2, …, |V(G)| the graph is called super. In this paper we deal with super cycle-antimagicness of subdivided graphs. We also prove that the subdivided wheel admits an (a,d)-cycle-antimagic labeling for somed.

scholarly journals Power graphs and exchange property for resolving sets

2019 ◽  
Vol 17 (1) ◽  
pp. 1303-1309 ◽  
Author(s):  
Ghulam Abbas ◽  
Usman Ali ◽  
Mobeen Munir ◽  
Syed Ahtsham Ul Haq Bokhary ◽  
Shin Min Kang
Keyword(s):  
Present Article ◽  
Linear Algebra ◽  
Finite Groups ◽  
Robot Navigation ◽  
Simple Graph ◽  
Exchange Property ◽  
Metric Dimension ◽  
Sufficient Condition ◽  
Resolving Set ◽  
Dihedral Groups

Abstract Classical applications of resolving sets and metric dimension can be observed in robot navigation, networking and pharmacy. In the present article, a formula for computing the metric dimension of a simple graph wihtout singleton twins is given. A sufficient condition for the graph to have the exchange property for resolving sets is found. Consequently, every minimal resolving set in the graph forms a basis for a matriod in the context of independence defined by Boutin [Determining sets, resolving set and the exchange property, Graphs Combin., 2009, 25, 789-806]. Also, a new way to define a matroid on finite ground is deduced. It is proved that the matroid is strongly base orderable and hence satisfies the conjecture of White [An unique exchange property for bases, Linear Algebra Appl., 1980, 31, 81-91]. As an application, it is shown that the power graphs of some finite groups can define a matroid. Moreover, we also compute the metric dimension of the power graphs of dihedral groups.

scholarly journals The Irregularity and Modular Irregularity Strength of Fan Graphs

Symmetry ◽  
2021 ◽  
Vol 13 (4) ◽  
pp. 605
Author(s):  
Martin Bača ◽  
Zuzana Kimáková ◽  
Marcela Lascsáková ◽  
Andrea Semaničová-Feňovčíková
Keyword(s):  
Simple Graph ◽  
Isolated Vertex ◽  
Positive Integers ◽  
Irregularity Strength

For a simple graph G with no isolated edges and at most, one isolated vertex, a labeling φ:E(G)→{1,2,…,k} of positive integers to the edges of G is called irregular if the weights of the vertices, defined as wtφ(v)=∑u∈N(v)φ(uv), are all different. The irregularity strength of a graph G is known as the maximal integer k, minimized over all irregular labelings, and is set to ∞ if no such labeling exists. In this paper, we determine the exact value of the irregularity strength and the modular irregularity strength of fan graphs.

scholarly journals k-Zumkeller labeling of super subdivision of some graphs

2021 ◽  
Vol 29 (1) ◽  
Author(s):  
M. Basher
Keyword(s):  
Simple Graph ◽  
Natural Numbers ◽  
Injective Function ◽  
Planar Grid ◽  
Circular Ladder

AbstractA simple graph $$G=(V,E)$$ G = ( V , E ) is said to be k-Zumkeller graph if there is an injective function f from the vertices of G to the natural numbers N such that when each edge $$xy\in E$$ x y ∈ E is assigned the label f(x)f(y), the resulting edge labels are k distinct Zumkeller numbers. In this paper, we show that the super subdivision of path, cycle, comb, ladder, crown, circular ladder, planar grid and prism are k-Zumkeller graphs.

scholarly journals Total Roman {3}-Domination: The Complexity and Linear-Time Algorithm for Trees

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 293
Author(s):  
Xinyue Liu ◽  
Huiqin Jiang ◽  
Pu Wu ◽  
Zehui Shao
Keyword(s):  
Linear Time ◽  
Minimum Weight ◽  
Domination Number ◽  
Time Algorithm ◽  
Simple Graph ◽  
Linear Time Algorithm ◽  
Domination Problem ◽  
Np Complete ◽  
Chordal Bipartite Graphs ◽  
Dominating Function

For a simple graph G=(V,E) with no isolated vertices, a total Roman {3}-dominating function(TR3DF) on G is a function f:V(G)→{0,1,2,3} having the property that (i) ∑w∈N(v)f(w)≥3 if f(v)=0; (ii) ∑w∈N(v)f(w)≥2 if f(v)=1; and (iii) every vertex v with f(v)≠0 has a neighbor u with f(u)≠0 for every vertex v∈V(G). The weight of a TR3DF f is the sum f(V)=∑v∈V(G)f(v) and the minimum weight of a total Roman {3}-dominating function on G is called the total Roman {3}-domination number denoted by γt{R3}(G). In this paper, we show that the total Roman {3}-domination problem is NP-complete for planar graphs and chordal bipartite graphs. Finally, we present a linear-time algorithm to compute the value of γt{R3} for trees.

scholarly journals On the price of stability of some simple graph-based hedonic games

2020 ◽  
Author(s):  
Christos Kaklamanis ◽  
Panagiotis Kanellopoulos ◽  
Konstantinos Papaioannou ◽  
Dimitris Patouchas
Keyword(s):  
Simple Graph ◽  
Price Of Stability ◽  
Hedonic Games

ON THE REGULAR GRAPH RELATED TO THE G-CONJUGACY CLASSES

2021 ◽  
pp. 1-5
Author(s):  
SH. RAHIMI ◽  
Z. AKHLAGHI
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Conjugacy Class ◽  
Regular Graph ◽  
Simple Graph ◽  
Conjugacy Classes ◽  
A Finite Group

Abstract Given a finite group G with a normal subgroup N, the simple graph $\Gamma _{\textit {G}}( \textit {N} )$ is a graph whose vertices are of the form $|x^G|$ , where $x\in {N\setminus {Z(G)}}$ and $x^G$ is the G-conjugacy class of N containing the element x. Two vertices $|x^G|$ and $|y^G|$ are adjacent if they are not coprime. We prove that, if $\Gamma _G(N)$ is a connected incomplete regular graph, then $N= P \times {A}$ where P is a p-group, for some prime p, $A\leq {Z(G)}$ and $\textbf {Z}(N)\not = N\cap \textbf {Z}(G)$ .

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