# Total Product and Total Edge Product Cordial Labelings of Dragonfly GraphDgn

Authors: N. Inayah; A. Erfanian; M. Korivand
Journal: Journal of Mathematics
 (2022)

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## Abstract

In this paper, we study the total product and total edge product cordial labeling for dragonfly graphDgn. We also define generalized dragonfly graph and find product cordial and total product cordial labeling for this family of graphs.


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## Body

## 1. Introduction

In this paper, all graphsG=GV,E are simple and finite connected with order p and size q. We will give some definitions and other information, which are useful for this research. Terms that are not defined here, we refer to West [1]. Let function f be a vertex labeling of graph G and f∗, an edge labeling of graph G. Let vfi (respectively ef∗i) denote the number of vertices (edges) labeled with i=0,1.The cordial labeling was introduced in 1987 by Cahit [2], which he defines that a graph G is said to be cordial graph if there exists a vertex labeling f:V⟶0,1 such that induces an edge labeling f∗:E⟶0,1 defined by f∗uv=fu−fv and satisfied vf0−vf1≤1 and ef∗0−ef∗1≤1. In [2], Cahit proved many result for cordial labeling. Prime cordial labeling, A-cordial labeling, product cordial labeling, H-cordial labeling, etc. are some variations of labeling schemes introduced after cordial labeling. For product cordial labeling, it was introduced in 2004 by Sundaram, et al. [3], which f∗uv=fu−fv on cordial labeling is replaced by f∗uv=fufv. In this paper we investigate the total product and total edge product cordial labelings of dragonfly graph Dgn.The product cordial labeling is defined in Definition1.1.Definition 1.1.
A graphG is said to be the product cordial if there exists a vertex labeling f:V⟶0,1 such that induces an edge labeling f∗:E⟶0,1 defined by f∗uv=fu−fv and satisfied vf0−vf1≤1 and ef∗0−ef∗1≤1.In [3], Sundaram et al. proved that unicycle graphs with odd order, trees, helms, triangular snakes, dragons, and unicon with two paths are product cordial. Furthermore, Vaidya and Barasara [4] discussed product cordial labeling of graph fans Fn, Cn with one chord, and Cn with two chord. Gao et al. [5] discussed product cordial labeling of graph Pn+1m.Motivated by definition of product cordial labeling, in [6], Sundaram et al. introduce a total product cordial labeling and investigate the total product cordial of some standard graphs. The total product cordial labeling is defined in Definition 1.2.Definition 1.2.
A graphG is said to be the total product cordial if there exists a vertex labeling f:V⟶0,1 such that induces an edge labeling f∗:E⟶0,1 defined by f∗uv=fu−fv and satisfied vf0+ef∗0−vf1−ef∗1≤1.The total product cordial labeling of cycleC7 is shown in Figure 1.Figure 1
Total product cordial labeling of cycleC7.In [6, 7], Sundaram et al. proved that tree graph Pn, fans graph Fn, graph Cn, except n=4, wheels graph wn, helms graph Hn, and graph Cn with m edges appended at each vertex are total product cordial graph. They also proved that every product cordial graph G is a total product cordial if G has either even size and even order or odd order.In [8], Vaidya and Barasara introduce the concept of edge product cordial labeling, which is defined in Definition 1.3.Definition 1.3 (see [8]).
A graphG is said to be edge product cordial if there exits an edge labeling f:E⟶0,1 such that it induces a vertex labeling f∗:V⟶0,1 defined by f∗v=Πfei for eiei∈E/eiandeiisincidenttov and satisfies ef0−ef1≤1 and vf∗0−vf∗1≤1.In [8–10], Vaidya and Barasara have investigated several results related to edge product cordial labeling.In, Vaidya and Barasara introduce the concept of total edge product cordial labeling, which is defined in Definition1.4.Definition 1.4 (see [10]).
A graphG is said to be the total edge product cordial if there exits an edge labeling f:E⟶0,1, such that it induces a vertex labeling f∗:V⟶0,1 defined by f∗v=Πfei for eiei∈E/eiandeiisincidenttov and satisfies ef0+vf∗0−ef1+vf∗1≤1.The total edge product cordial labeling of graphC43 is shown in Figure 2.Figure 2
Total edge product cordial labeling ofC43.In [4], Vaidya and Barasara have investigated total edge product cordial labeling in the context of various graph operations.Proposition 1.5 (see [10]).
If every edge product cordial graphG has either even size or even order, then graph G is the total edge product cordial.In this paper, we determine the total product and total edge product cordial labelings of dragonfly graph, denoted byDgn, which is defined in Definition 1.6. Also, we generalized dragonfly graph, defined in Definition 3.1, and present two family of graphs in that, which are product and total product cordial graph.Definition 1.6.
For an integern, the dragonfly graph Dgn is the graph with vertex set:
(1)V=ui,vj,wki,j∈1,2,⋯,n+2,k∈1,2,3,and edge set
(2)E=uiui+1,i∈1,2,⋯,n+1∪,uiw0i∈1,2,⋯,n+2∪,vivi+1,i∈1,2,⋯,n+1∪,viw0i∈1,2,⋯,n+2∪,w0wii∈1,2.In Figure3, we give a representation of our definition.Figure 3
The dragonfly graphDgn.


## 2. Main Results

Theorem 2.1.
The dragonflyDgn is product cordial graph.Proof.
LetDgn is the dragonfly graph. Define the function f:VDgn⟶0,1, we consider following two cases.
Case 1. Let n be even.
(3)fw0=1,fwi=0,1≤i≤2,fui=1,1≤i≤n+42,fui=0,n+42+1≤i≤n+2,fvi=1,1≤i≤n+42−1,fvi=0,n+42≤i≤n+2.
By of the above labeling, we havevf0=2.n+4/2 and vf1=2.n+4/2−1. On the other hand, the edges of Dgn with labels one are the following:
(4)f∗uiw0=1,1≤i≤n+42,f∗viw0=1,1≤i≤n+22.f∗uiui+1=1,1≤i≤n+22,f∗vivi+1=1,1≤i≤n2,and the edges of Dgn with labels zero are the following:
(5)f∗w0wi=0,1≤i≤2,f∗uiw0=0,n+42+1≤i≤n+2.f∗viw0=0,n+22+1≤i≤n+2.f∗uiui+1=0,n+42≤i≤n+1,f∗vivi+1=0,n+42−1≤i≤n+1.
By of the above labeling, we haveef∗0=2n+4 and ef∗1=2n+4. Hence, vf0−vf1=1 and ef∗0−ef∗1=0. Thus, the graph Dgn is product cordial labeling.
Case 2. Let n be odd.
(6)fw0=1,fwi=0,1≤i≤2,fui=1,1≤i≤n+22,fui=0,n+22+1≤i≤n+2,fvi=1,1≤i≤n+22,fvi=0,n+22+1≤i≤n+2.
By of the above labeling, we havevf0=2.n+2/2 and vf1=2.n+2/2+1. On the other hand, the edges of Dgn with labels one are the following:
(7)f∗uiw0=1,1≤i≤n+22,f∗viw0=1,1≤i≤n+22,f∗uiui+1=1,1≤i≤n+22−1,f∗vivi+1=1,1≤i≤n+22,and the edges of Dgn with labels zero are the following:
(8)f∗w0wi=0,1≤i≤2,f∗uiw0=0,n+22+1≤i≤n+2,f∗viw0=0,n+22+1≤i≤n+2,f∗uiui+1=0,n+22≤i≤n+1,f∗vivi+1=0,n+22+1≤i≤n+1.
By of the above labeling, we haveef∗0=2n+2/2+2 and ef∗1=2n+2/2+2. Hence, vf0−vf1=1 and ef∗0−ef∗1=1. Thus, the graph Dgn is product cordial labeling. Therefore, considering two cases above, we prove that graph Dgn is product cordial graph.Theorem 2.2.
The dragonflyDgn is a total product cordial.Proof.
By: Theorem2.1, ef0+vf∗0−ef1+vf∗1≤1. Thus, the graph Dgn is a total product cordial.
The total product cordial labeling ofDg5 is shown in Figure 4.Theorem 2.3.
The dragonflyDgn is an edge product cordial.Proof.
LetDgn is dragonfly graph. Define the function f:EDgn⟶0,1, we consider following two cases.
Case 1. Let n be even.
(9)fw0wi=1,1≤i≤2,fw0ui=0,1≤i≤n+42,fw0ui=1,n+42+1≤i≤n+2,fw0vi=0,1≤i≤n+22,fw0vi=1,n+22+1≤i≤n+2,fuiui+1=0,1≤i≤n+22,fuiui+1=1,n+22≤i≤n+1,fvivi+1=0,1≤i≤n2,fvivi+1=1,n2≤i≤n+1.
By of the above labeling, we haveef0=4.n+2/2 and ef1=4.n+2/2. On the other hand, the vertices of Dgn with labels zero are the following:
(10)f∗w0=0,f∗ui=0,1≤i≤n+42,f∗vi=0,1≤i≤n+22,and the vertices of Dgn with labels one are the following:
(11)f∗wi=1,1≤i≤2,f∗ui=1,n+42+1≤i≤n+2,f∗vi=1,n+22+1≤i≤n+2.
By of the above labeling, we havevf∗0=n+4 and vf∗1=n+3. Hence, ef0−ef1=0 and vf∗0−vf∗1=1. Thus, the graph Dgn is an edge product cordial labeling.
Case 2. Let n be odd.
(12)fw0wi=1,1≤i≤2,fw0ui=0,1≤i≤n+32,fw0ui=1,n+32+1≤i≤n+2,fw0vi=0,1≤i≤n+32,fw0vi=1,n+32+1≤i≤n+2,fuiui+1=0,1≤i≤n+12,fuiui+1=1,n+12≤i≤n+1,fvivi+1=0,1≤i≤n+12,fvivi+1=1,n+12≤i≤n+1.
By of the above labeling, we haveef0=n+2 and ef1=n+2. On the other hand, the vertices of Dgn with labels zero are the following:
(13)f∗w0=0,f∗ui=0,1≤i≤n+32,f∗vi=0,1≤i≤n+32,and the vertices of Dgn with labels one are the following:
(14)f∗wi=1,1≤i≤2,f∗ui=1,n+32+1≤i≤n+2,f∗vi=1,n+32+1≤i≤n+2.
By of the above labeling, we havevf∗0=n+4 and vf∗1=n+3. Hence, ef0−ef1=0 and vf∗0−vf∗1=1. Thus, the graph Dgn is an edge product cordial labeling. Therefore, considering two cases above, we prove that graph Dgn is edge product cordial.Corollary 2.4.
The dragonflyDgn is a total edge product cordial graph.Proof.
LetDgn is dragonfly graph. Here, graph Dgn has even size and in Theorem 2.3, Dgn is edge product cordial. Then, by proposition 3.2, the result holds.Figure 4
Total product cordial labeling ofDg5.The total edge product cordial labeling of Dg5 is shown in Figure5.Figure 5
Total edge product cordial labeling ofDg5.


## 3. The Generalized Dragonfly Graphs

In this section, we present a generalization of dragonfly graph and show that some of those graphs are total product cordial graphs.Definition 3.1.
For everym≥2 and k≥1, the generalized dragonfly graph, denoted by Dgnm,k, is the graph with vertex set
(15)V=vi1,vi2,⋯,vim,wji∈1,2,⋯,n+2,j∈1,2,⋯,k,and edge set
(16)E=viℓvi+1ℓ,i∈1,2,⋯,n+1,forℓ∈1,2,⋯,m∪,viℓw0i∈1,2,⋯,n+2,forℓ∈1,2,⋯,m∪,w0wjj∈1,2,⋯k.It is clear thatDgn2,1=Dgn (see Figure 6, for the case m=k=3).Theorem 3.2.
Fork≥1, graph Dg2k3,2 is a product cordial graph.Proof.
We define vertex labelingf:VDg2k3,2⟶0,1, of vertices Dg2k3,2 as follow.
(17)fw0=1,fwi=0,1≤i≤2,fviℓ=1,1≤i≤k+1,1≤ℓ≤2,fviℓ=0,k+2≤i≤2k+2,1≤ℓ≤2,fvi3=1,1≤i≤k+1,fvi3=0,k+2≤i≤2k+2.
By the above labeling, we havevf0=3k+1+1 and vf1=3k+1+2. On the other hand, the edges of Dg2k3,2 with labels one are the following:
(18)f∗viℓw0=1,1≤i≤k+1,1≤ℓ≤2,f∗vi3w0=1,1≤i≤k+2,f∗viℓvi+1ℓ=1,1≤i≤k,1≤ℓ≤2,f∗vi3vi+13=1,1≤i≤k+1,and the edges of Dg2k3,2 with labels zero are the following:
(19)f∗w0wi=0,1≤i≤2,f∗viℓw0=0,k+2≤i≤2k+2,1≤ℓ≤2,f∗vi3w0=0,k+3≤i≤2k+2,f∗viℓvi+1ℓ=0,k+1≤i≤2k+1,1≤ℓ≤2,f∗vi3vi+13=0,k+2≤i≤2k+1.
By the above labeling, we haveef∗0=6k+1 and ef∗1=6k+1−1. Hence, vf0−vf1=1 and ef∗0−ef∗1=1. Thus, labeling f is a product cordial labeling for Dg2k3,2, and the proof is completed.Corollary 3.3.
Fork≥1, graph Dg2k3,2 is a total product cordial.Proof.
By: Theorem3.2, ef0+vf∗0−ef1+vf∗1≤1. Therefore, the graph Dg2k3,2 is a total product cordial.Theorem 3.4.
Fork≥1, graph Dg2k+13,3 is a product cordial graph.Proof.
We define vertex labelingf:VDg2k+13,3⟶0,1, of vertices Dg2k+13,3 as follow.
(20)fw0=1,fwi=0,1≤i≤3,fviℓ=1,1≤i≤k+2,1≤ℓ≤3,fviℓ=0,k+3≤i≤2k+3,1≤ℓ≤3.
By the above labeling, we havevf0=3k+2+1 and vf1=3k+1+3. On the other hand, the edges of Dg2k+13,3 with labels one are the following:
(21)f∗viℓw0=1,1≤i≤k+2,1≤ℓ≤3,f∗viℓvi+1ℓ=1,1≤i≤k+1,1≤ℓ≤3,and the edges of Dg2k+13,3 with labels zero are the following:
(22)f∗w0wi=0,1≤i≤3,f∗viℓw0=0,k+3≤i≤2k+3,1≤ℓ≤3,f∗viℓvi+1ℓ=0,k+2≤i≤2k+2,1≤ℓ≤3,
By the above labeling, we haveef∗0=6k+1+3 and ef∗1=3k+2+3k+1. Hence, vf0−vf1=1 and ef∗0−ef∗1=0. Thus, labeling f is a product cordial labeling for Dg2k+13,3, and the proof is completed.Corollary 3.5.
Fork≥1, graph Dg2k+13,3 is a total product cordial.Figure 6
The generalize dragonfly graphDgn3,3.It is interesting to find all valuesm,k, and n such that generalized dragonfly Dgnm,k is cordial product graph. We end the paper with the following question.Question.
Find all valuesm,k, and n, such that Dgnm,k is (edge) cordial product graph.



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