# Tree-Antimagicness of Web Graphs and Their Disjoint Union

Authors: Zhijun Zhang; Muhammad Awais Umar; Xiaojun Ren; Basharat Rehman Ali; Mujtaba Hussain; Xiangmei Li
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

In graph theory, the graph labeling is the assignment of labels (represented by integers) to edges and/or vertices of a graph. For a graphG=V,E, with vertex set V and edge set E, a function from V to a set of labels is called a vertex labeling of a graph, and the graph with such a function defined is called a vertex-labeled graph. Similarly, an edge labeling is a function of E to a set of labels, and in this case, the graph is called an edge-labeled graph. In this research article, we focused on studying super ad,d-T4,2-antimagic labeling of web graphs W2,n and isomorphic copies of their disjoint union.


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## Body

## 1. Introduction and Preliminaries

AQ-covering for a finite and simple graph P is a family of subgraphs Q1,Q2,…,Qs with Qt≅Q and Q⊂P, with every edge EQ in one of the subgraphs Qt, t=1,2,…,s. For an l,m-graph, a bijection α:VP∪EP⟶1,2,…,l+m is a total labeling of P, and wtαQ=∑v∈VQαv+∑e∈EQαe is the corresponding weight of the subgraph Q. Such a graph P is an ad,d-Q-antimagic if wtαQ=ad,ad+d,…,ad+s−1d, where ad>0,d≥0 are two integers, s is the number of subgraphs Qt isomorphic to Q, and the labeling α is super if vertices are assigned with the integers 1,2,…,l.In [1], Gutiérrez and Lladó defined Q-supermagic labelings and the results: Pn and Cn which are Ph-supermagic for some h were proved. Lladó and Moragas [2] investigated Ch-magicness of wheels Wn, windmills Wn,r, books Bn, and of prisms Dn for some h. In [3], Baca et al. proved results for tree-antimagicness of disconnected graphs. Noshad Ali et al. in [4] discussed the C3-antimagicness of corona graphs. Umar [5] stated the construction for antimagicness of ladder graphs. Ph-magic labelings of trees Tn, subdivision of shrubs, and banana tree were proved by Maryati, Baskoro, and Salman [6]. The cycle antimagicness of book graphs for differences 1,2,…,13 is given in [7]. Fan graph is cycle-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].Graph labeling has been used in many applications such as coding theory, X-ray crystallography, radar, astronomy, circuit design, communication network addressing, and data base management [10–12]. In the present article, we have studied super T4,2-antimagic labeling of web graphs W2,n for differences d∈0,1,…,12,14.


## 2. Main Results

The join graphCn+K1 is called a wheel graph Wn, and a helm graph Hn is obtained by attaching a pendant edge to each cycle vertex of wheel, while the closed helm CHn is constructed from a helm by joining each pendant vertex to form a cycle. A web graph W2,n is the graph obtained from closed helm by attaching a single pendant edge to each vertex of the outer cycle. The vertex and edge set of the web graph are(1)VW2,n=ν,ai,j:1≤i≤n,1≤j≤3,EW2,n=νai,1,ai,jai+1,j,ai,jai,j+1:1≤i≤n,1≤j≤2,where the indices i are taken as modulo n.The vertexν is called the apex or central vertex. The vertices ai,1 are inner cycle vertices, ai,2 are outer cycle vertices, and ai,3 are pendant vertices. The edges νai,1 are spokes, ai,1ai+1,1 are inner rim edges, ai,2ai+1,2 are outer rim edges, ai,1ai,2 are middle spokes, and ai,2ai,3 are outer spokes.Figure1 shows a web graph W2,6.Figure 1
A web graphW2,6.The web graph is covered by a treeT4,2 on 6 vertices. The tree T4,2 is constructed from path P4 by joining a pendant edge with two inner vertices. The vertices and edges of kth tree T4,2k in a web graph W2,n are(2)VT4,2k=ν,ak,1,ak,2,ak,3,ak+1,1,ak+1,2,ET4,2k=νak,1,ak,1ak,2,ak,2ak,3,ak,1ak+1,1,ak,2ak+1,2.Figure2 shows a tree T4,2.Figure 2
A treeT4,2.Each web graphW2,n would be covered by n trees of type T4,2. The weight of kth tree T4,2k under a total labeling α would be(3)wtαT4,2k=∑v∈VT4,2kαv+∑e∈ET4,2kαe=αν++ανak,1+∑p=13αak,p+∑p=12αak+1,p+αak,pak,p+1++∑p=12αak,pak+1,p.Theorem 1.
LetW2,n define a web graph on 3n+1 vertices and 5n edges and T4,2 be a tree on 6 vertices for n≥3. Then, W2,n admits a super ad,d-T4,2-antimagic labeling for differences d∈0,2,…,14.Proof.
The total labelingαd is defined as(4)αdν=1,αdai,j=n+2−i,j=1,n+1+i,j=2,2n+1+i,j=3,αdνai,1=6n+2−i,d=0,2,…,8,3n+i+1,d=10,12,14,αdai,jai,j+1=5n+2−i,j=1,d=0,3n+1+i,j=2,d=0,2,4n+1+i,j=1,d=2,3n+1+2i,j=1,d=4,6,8,3n+2i,j=2,d=4,6,8,3n+i,j=1,d=10,12,14,3n+i−1,j=2,d=10,12,14,αdai,jai+1,j=7n+2−i,j=1,d=0,2,4,7n+1+i,j=2,d=0,2,4,6n+1+i,j=1,d=6,12,7n+1+i,j=2,d=6,12,6n+2i,j=1,d=8,14,6n+2i+1,j=2,d=8,14,6n+1+i,j=1,d=10,8n+2−i,j=2,d=10.
One can easily verify that(5)∑j=13αdai,j+∑j=12αdai+1,j=6n+7+i.
Using (3) and (5) and the corresponding labeling αd, we have(6)wtα0T4,2k=6n+8+i+14n+5−i+14n+3=34n+16,(7)wtα2T4,2k=12n+10+7n+2+2i+14n+3=33n+15+2i,(8)wtα4T4,2k=12n+10+6n+1+4i+14n+3=32n+14+4i,(9)wtα6T4,2k=12n+10+6n+1+4i+13n+2+2i=31n+13+6i,(10)wtα8T4,2k=12n+10+6n+1+4i+12n+1+4i=30n+12+8i,(11)wtα10T4,2k=6n+8+i+9n+9i+14n+3=29n+11+10i,(12)wtα12T4,2k=6n+8+i+9n+9i+13n+2+2i=28n+10+12i,(13)wtα14T4,2k=6n+8+i+9n+9i+2n+1+4i=27n+9+14i.
Equations (6)–(13) show that wtαT4,2k constitute an arithmetic progression for differences d∈0,2,…,14. This completes the proof.Figures3 and 4 show the super 210,4-T4,2-antimagic labeling of W2,6 and super 190,12-T4,2-antimagic labeling of W2,6, respectively.Figure 3
Super210,4-T4,2-antimagic labeling of W2,6.Figure 4
Super190,12-T4,2-antimagic labeling of W2,6.Theorem 2.
LetW2,n define a web graph on 3n+1 vertices and 5n edges and T4,2 be a tree on 6 vertices for n≥3. Then, W2,n admits a super ad,d-T4,2-antimagic labeling for differences d∈1,3,…,11 and for n≡1 (mod 2).Proof.
The total labelingα is defined as(14)αdν=1,αdai,j=1+i,j=1,for alld,2n+1−i,j=2,for alld,2n+1+i,j=3,d≠9,3n+2−i,j=3,d=9,αdνai,1=22n+1−i,d=1,3,3n+1+i,d=5,3n+2i,d=7,3n+i−1,d=9,11,αdai,jai,j+1=4n+1+i,j=1,d=1,3,5,3n+1+2i,j=1,d=7,3n+i,j=1,d=9,11,6n+2−i,j=2,d=1,5n+1+i,j=2,d=3,5,7,3n+i+1j=2,d=9,11,αdai,1ai+1,1=6n+1+i2,ifi≡0,mod2,6n+1+n−12+i+12,ifi≡1,mod2,αdai,2ai+1,2=7n+1+n+12+i2,ifi≡0,mod2,7n+1+i+12,ifi≡1,mod2.
One can easily verify that(15)∑j=13αdai,j+∑j=12αdai+1,j=4n+1+i,d≠9,5n+5−i,d=9.
Using (3), the above relation, and the corresponding labeling, we have(16)wtα1T4,2k=20n+13+13n+2+i+n+12=67n+312+i,(17)wtα3T4,2k=19n+12+2i+13n+2+i+n+12=65n+292+3i,(18)wtα5T4,2k=18n+11+4i+13n+2+i+n+12=63n+272+5i,(19)wtα7T4,2k=17n+10+6i+13n+2+i+n+12=61n+252+7i,(20)wtα9T4,2k=16n+8i+9+13n+2+i+n+12=59n+232+9i,(21)wtα11T4,2k=15n+10i+8+13n+2+i+n+12=57n+212+11i.
Equations (16)–(21) show that wtαT4,2k constitute an arithmetic progression for differences d∈1,3,…,11. This completes the proof.Figures5 and 6 show the super 172,7-T4,2-antimagic labeling of W2,5 and super 168,9-T4,2-antimagic labeling of W2,5, respectively.In view of Theorems1 and 2 and the construction of the web graph, helm graph, and closed helm graph, one can deduce the following corollaries.Figure 5
Super172,7-T4,2-antimagic labeling of W2,5.Figure 6
Super168,9-T4,2-antimagic labeling of W2,5.Corollary 1.
LetHn define a helm graph on 2n+1 vertices and 3n edges and T3,1 define a tree on 4 vertices for n≥3. Then, Hn admits a super ad,d-T3,1-antimagic labeling for differences d∈0,2,…,14.Corollary 2.
LetHn define a helm graph on 2n+1 vertices and 3n edges and T3,1 define a tree on 4 vertices for n≥3. Then, Hn admits a super ad,d-T3,1-antimagic labeling for differences d∈1,3,…,11 and for n≡1 (mod 2).Corollary 3.
LetCHn define a closed helm graph on 2n+1 vertices and 3n edges and T3,2 define a tree on 5 vertices for n≥3. Then, CHn admits a super ad,d-T3,2-antimagic labeling for differences d∈0,2,…,14.Corollary 4.
LetCHn define a closed helm graph on 2n+1 vertices and 3n edges and T3,2 define a tree on 5 vertices for n≥3. Then, CHn admits a super ad,d-T3,2-antimagic labeling for differences d∈1,3,…,11 and for n≡1 (mod 2), where tree T3,2 is obtained from P3 by attaching two pendant edges: one pendant edge with the inner vertex and one with the end vertex.In [3], Bača et al. proved the tree-antimagicness of disconnected graphs. For the complete reference, the results are as follows.Theorem 3.
(see [3]). Let m,t≥1 be positive integers. Let Gi with a Ti1,Ti2,…,Tit-covering be a super a,1-T-antimagic graph of order p and size q, i=1,2,…,m, where T is a tree and every tree Tij, j=1,2,…,t is isomorphic to T. Then, the disjoint union ∪i=1mGi is also a super b,1-T-antimagic graph.Corollary 5.
(see [3]). Let G be a super a,1-T-antimagic graph, where T is a tree. Then, the disjoint union of the arbitrary number of copies of G, i.e., mG, m≥1, also admits a super b,1-T-antimagic total labeling.


## 3. Conclusion

The first polynomial time construction procedure for different graphs is being discussed in artificial intelligence, for example, see [13]. These procedures and constructions have a wide range of practical applications in radio astronomy, X-ray crystallography, cryptography, and experimental design. In general, many results in theoretical computer science can be generalized and constructed using the graph structures and labeling of the graphs. Here, in this manuscript, we have discussed labeling of web graphs, their associated graphs, and their disjoint union. As a matter of fact, disjoint union of a graph is the relationship between its isomorphic copies keeping the structural properties intact and used to create new larger structures having the same properties as the base graph have. In future, one can extend this idea to find the results for larger differences and for more general structure with applications in theoretical computer science [14–16].



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