Research Article
Tree-Antimagicness of Web Graphs and Their Disjoint Union
Zhijun Zhang,
1
Muhammad Awais Umar ,
2
Xiaojun Ren ,
3
Basharat Rehman Ali,
4
Mujtaba Hussain,
5
and Xiangmei Li
6
1
School of Computer and Software, Weifang University of Science and Technology, Shouguang 262700, China
2
Govt. Degree College (B), Sharqpur Sharif, Pakistan
3
School of Computer and Software, Weifang University of Science and Technology, Shouguang 262700, China
4
Abdus Salam School of Mathematical Sciences, GC University, Lahore, Pakistan
5
Department of Mathematics, COMSATS University Islamabad, Lahore, Pakistan
6
School of Cybersecurity, Chengdu University of Information Technology, Chengdu 610225, China
Correspondence should be addressed to Xiaojun Ren; renxiaojun_05@yeah.net
Received 26 January 2020; Accepted 7 March 2020; Published 9 April 2020
Academic Editor: Elio Masciari
Copyright © 2020 Zhijun Zhang et al. is is an open access article distributed under the Creative Commons Attribution
License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is
properly cited.
Ingraphtheory,thegraphlabelingistheassignmentoflabels(representedbyintegers)toedgesand/orverticesofagraph.Fora
graph G �(V, E),withvertexset V andedgeset E,afunctionfrom V toasetoflabelsiscalledavertexlabelingofagraph,andthe
graphwithsuchafunctiondefinediscalledavertex-labeledgraph.Similarly,anedgelabelingisafunctionof E toasetoflabels,
and in this case, the graph is called an edge-labeled graph. In this research article, we focused on studying super
(a
d
,d)-T
(4,2)
-antimagic labeling of web graphs W(2,n) and isomorphic copies of their disjoint union.
1.Introduction and Preliminaries
A Q-covering for a finite and simple graph P is a family of
subgraphs Q
1
,Q
2
, ... ,Q
s
with Q
t
� Q and Q ⊂ P,withevery
edge E(Q) inoneofthesubgraphs Q
t
, t � 1, 2, ... ,s.Foran
(l, m)-graph,abijection α: V(P)∪E(P) ⟶ 1, 2, ... ,l + m { }
is a total labeling of P, and wt
α
(Q)�
v∈V(Q)
α(v)
+
e∈E(Q)
α(e) is the corresponding weight of the subgraph
Q. Such a graph P is an (a
d
,d)-Q-antimagic if
wt
α
(Q)� a
d
,a
d
+ d, ... ,a
d
+(s − 1)d , where a
d
> 0,d ≥ 0
aretwointegers, s isthenumberofsubgraphs Q
t
isomorphic
to Q,andthelabeling α issuperifverticesareassignedwith
the integers 1, 2, ... ,l { }.
In [1], Guti´ errez and Llad´ o defined Q-supermagic
labelings and the results: P
n
and C
n
which are
P
h
-supermagic for some h were proved. Llad´ o and
Moragas [2] investigated C
h
-magicness of wheels W
n
,
windmills W(n, r),books B
n
,andofprisms D
n
forsome h.
In[3],Bacaetal.provedresultsfortree-antimagicnessof
disconnected graphs. Noshad Ali et al. in [4] discussed
the C
3
-antimagicness of corona graphs. Umar [5] stated
the construction for antimagicness of ladder graphs.
P
h
-magiclabelingsoftrees T
n
,subdivisionofshrubs,and
banana tree were proved by Maryati, Baskoro, and Sal-
man [6]. e cycle antimagicness of book graphs for
differences 1, 2, ... , 13 { } isgivenin[7].Fangraphiscycle-
antimagic for differences depending upon the length of
the cycle which is proved in [8]. Q-antimagicness of the
Cartesian product of graphs was discussed by Baca et al.
in [9].
Graphlabelinghasbeenusedinmanyapplicationssuch
as coding theory, X-ray crystallography, radar, astronomy,
circuitdesign,communicationnetworkaddressing,anddata
base management [10–12]. In the present article, we have
studied super T
(4,2)
-antimagic labeling of web graphs
W(2,n) for differences d ∈ 0, 1, ... , 12, 14 { }.
2.MainResults
e join graph C
n
+ K
1
is called a wheel graph W
n
, and a
helm graph H
n
is obtained by attaching a pendant edge to
Hindawi
Mathematical Problems in Engineering
Volume 2020, Article ID 4565829, 6 pages
https://doi.org/10.1155/2020/4565829