Journal of Indian Acad. Math.

ISSN: 0970-5120

Vol. 47, No. 1 (2025) pp. 153-167

Chirag Barasara¹

ANTIMAGIC LABELING OF LINE

GRAPH OF SOME GRAPHS

and

Palak Prajapati²

Abstract: Motivated from the study of magic square, Hartsfield and

Ringel defined antimagic labeling as bijection

f : E(G)  {1, 2, 3,  , |E(G)

such that u, v  V(G), uƒ  v , sum

a

}

of f(e) for all e incident to is different from sum of f(e) for all

u

e

incident to . In this paper, we discussed antimagic labeling of the line

v

graph of armed crown, double comb, ladder, wheel and tadpole.

Keywords: Graph Labeling, Antimagic Labeling, Graph Operation, Line

Graph.

Mathematics Subject Classification (2020) No.: 05C78, 05C76.

1. Introduction

All the graphs considered in this paper are simple, finite, connected and

undirected. A graph G  (V(G), E(G)) with edges is said to be antimagic, if

there exist a bijective labeling f from edge set of to 1, 2, 3, , q such that the

q

G

sums of the labels of the edges incident to each vertex is distinct. Hartsfield and

Ringel [12] in ‘Pearls in graph theory’ introduced antimagic labeling and conjecture

that ‘every connected graph different from K₂is antimagic’.

Many authors have tried to attack antimagic conjecture, Alon et al. [1] have

derived conditions on degree of a vertices for graph to be antimagic. Arumugam

154

CHIRAG BARASARA AND PALAK PRAJAPATI

et al. [2] have shown that various pyramid graphs are antimagic graphs. Cheng [6]

has proved that Cartesian products of two or more regular graphs are antimagic.

Joseph and Kureethara[15] have investigated that Cartesian product of wheel

graph and path graph is antimagic. Bača et al. [3] as well as Wang et al .[24]

have discussed antimagic labeling for some join graphs. Latchoumanane and

Varadhan [17] have studied antimagicness for tensor product of wheel and star.

Lozano et al. [18] have proved antimagic labeling of caterpillars. Sethuraman and

Shermily [21] have verified binomial tree and Fibonacci tree are antimagic. Barasara

and Prajapati [4, 5] have obtained antimagic labeling of some degree splitting graphs

as well as for some snake graphs. Although researchers applied various techniques,

still antimagic conjecture remains open.

A detailed survey on antimagic labeling can be found in Jin and Tu [14].

While survey on graph labeling is carried out by Gallian [10].

In this paper, we study antimagic labeling in the context of line graph

operation.

2. Preliminaries

Definition 2.1 ([7]): The line graph L(G) of a graph

whose vertex set is E(G) and two vertices are adjacent in L(G)whenever they

are incident in

G

is the graph

G

.

Definition 2.2 ([23]): The armed crown AC_nis a graph in which path

P

is attached at each vertex of cycle C_nby an edge.

2

Definition 2.3 ([19]): The Cartesian product of graphs G₁and G₂

denoted

by

G₁G₂

is

the

graph

with

vertex

set

V(G₁)  V(G₂)  {(u, v)/u  V(G₁) and v  V(G₂)} and (u, v) is adjacent

to (u', v') if and only if either u  u' and vv'  E(G₂) or v  v' and

uu'  E(G₁)

.

Definition 2.4 ([11]): The ladder graph L_nis defined as L_nP K₂

.

n

ANTIMAGIC LABELING OF LINE GRAPH OF SOME GRAPHS

155

Definition 2.5 ([9]): Let

and , denoted by G  H , is obtained by taking one copy of

V(G) copies of

vertex of

G

and

H

be two graphs. The corona product

of

G

H

G

and

H

, and by joining each vertex of the i^thcopies of

H

to the i^th

G

, for 1, 2, 3, , V(G)

.

Definition 2.6 ([13]): Let

P

be a path graph with n vertices. The

n

double comb graph is defined as P  2K₁

.

n

Definition 2.7 ([22]): The graph obtained by joining cycle C_nto a path

P_mwith an edge is called tadpole graph. It is denoted by T(n, m)

.

Preposition 2.1 ([1]): If

is antimagic.

G

has n  4 vertices and (G)  n  2 then

G

Preposition 2.2 (Exercise in [12]): The cycle C_nis antimagic.

3. Main Results

Theorem 3.1: The armed crown graph AC_nis an antimagic graph.

Proof:

Let

AC_n

be

an

armed

crown

graph

with

'

''

V(AC_n)  {v ,v,v /i  1,2,,n}

and

E(AC_n)  {v_iv_{i 1}/i  1, 2, , n  1}

i

' ''

 v₁v_n  v_iv^'_i/i  1, 2, , n  v v /i  1, 2, , n.

i i

Then V(AC_n)  3n and E(AC_n)  3n

.

We define f : E(AC_n)  {1, 2, , 3n} as follows.

f(v₁v_n)  2n  1

,

f(v_iv_{i 1})  3n  i  1;

For 1  i n  1

,

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CHIRAG BARASARA AND PALAK PRAJAPATI

f(v_iv^'_i)  n  i;

For 1  i  n

,

.

' ''

f(v v )  i;

i i

For 1  i  n

Above define edge labeling function will generate distinct vertex labels for

all the vertices of AC_n. Thus, is an antimagic labeling.

f

Hence, the armed crown graph AC_nis an antimagic graph.

Illustration 3.1: The graph AC₈and its antimagic labeling is shown in

Figure 1.

Figure 1: AC₈and its antimagic labeling.

Theorem 3.2: The graph L(AC_n) is an antimagic graph.

Proof: Let AC_nbe an armed crown graph with V(AC_n)  {v_i, v^'_i, v^'_i^'/i 

1, 2, , n} and E(AC_n)  {e_i v_iv_{i 1}/i  1, 2, , n  1}  {e_n v₁v_n}

'

''

 e_i v_iv_i^'/i  1, 2, , n  e_i v v /i  1, 2, , n.To construct L(AC_n)

i i

' ''

,

ANTIMAGIC LABELING OF LINE GRAPH OF SOME GRAPHS

157

let the vertices corresponding to e_ibe x_i

,

e^'_ibe x^'_iand e^'_i^'be x^'_i^'for each

i

. Then

V(L(AC_n)  3n and E(L(AC_n)  4n

.

We define f : E(L(AC_n))  {1, 2, , 4n} as follows.

' ''

f(x x )  i

i i

;

For 1  i  n

,

f(x_nx₁^')  n  1

,

f(x_ix^'_i1)  n  1  i

;

For 1  i  n  1

For 1  i  n

For 1  i  n  1

,

f(x_ix^'_i)  3n  1  i

;

,

f(x_ix_i1)  3n  1  i

f(x₁x_n)  3n  1

.

Above define edge labeling function will generate distinct vertex labels for

all the vertices of L(AC_n). Thus, is an antimagic labeling.

f

Hence, the graph L(AC_n) is an antimagic graph.



Illustration 3.2: The graph L(AC₆) and its antimagic labeling is shown

in Figure 2.

Figure 2: L(AC₆) and its antimagic labeling.

158

CHIRAG BARASARA AND PALAK PRAJAPATI

Theorem 3.3: The graph L(P  2K₁) is an antimagic graph.

n

Proof: Let P  2K₁be a double comb graph with V(P  2K₁) 

n

{v_i, v^'_i, v^'_i^'/i  1, 2, , n} and E(P  2K₁)  {e_i v_iv_i1/i  1, 2, , n  1}

n

 {e^'_i v_iv^'_i/i  1, 2,  , n}  {e^'_i^' v_iv^'_i^'/i  1, 2,  , n}.

e^'_ibe x^'_iand e^'_i^'be x^'_i^'for

. Then V(L(P  2K₁)  3n  1 and E(L(P  2K₁)  6n  6

To

construct

L(P  2K₁) , let the vertices corresponding to e_ibe x_i

,

n

each

i

.

n

We define f : E(P  2K1)  {1, 2, , 6n  6} as per following two

n

cases.

Case 1: For n  2

.

The graph L(P  2K₁) has 5 vertices and (L(P  2K₁))  4 . Thus,

2

by Preposition 2.1,L(P  2K₁) is an antimagic graph.

2

Case 2: For n  3

.

' ''

f(x x )  1

1 1

,

f(x^'_nx^'_n^')  2

f(x_ix^'_i)  2i  1

f(x_i1x^'_i)  2i

f(x_ix^'_i^')  4n  2i

f(x_ix^'_i^'_1)  4n  1  2i

,

;

For 1  i  n  1

For 2  i  n

,

;

,

;

For 1  i  n  1

For 1  i  n  2

For 2  i  n  1

,

;

f(x_ix_i1)  5n  4  i

;

,

' ''

f(x x )  4n  3  i

;

.

i i

Above define edge labeling function will generate distinct vertex labels for

ANTIMAGIC LABELING OF LINE GRAPH OF SOME GRAPHS

159

all the vertices of L(P  2K₁) . Thus,

f

is an antimagic labeling.

n

Hence, the graph L(P  2K₁) is an antimagic graph.



n

Illustration 3.3: The graph L(P  2K₁) and its antimagic labeling is

7

shown in Figure 3.

Figure 3: L(P  2K₁)and its antimagic labeling.

7

Theorem 3.4: The graph L(L_n) is an antimagic graph.

Proof: Let L_nbe a ladder graph with V(L_n)  {v_i, v^'_i/i  1, 2,, n} and

E(L_n)  {e^'_i v_iv_{i 1}/i  1, 2, , n  1}  {e_i v_iv_i^'/i  1, 2, , n}

' '

 {e^'_i^' v v

/i  1, 2,  , n  1}. To construct L(L_n), let the vertices

i i 1

corresponding to e_ibe x_i

,

e^'_ibe x^'_iand e^'_i^'be x^'_i^'for each

i

.

Then V(L(L_n)  3n  2 and E(L(L_n)  6n  8

.

We define f : E(L(Ln))  {1, 2, , 6n  8} as per following four cases.

Case 1: For n  2

.

The graph L(L₂) is also known as cycle C₄. Thus, by Preposition 2.2,

L(L₂) is an antimagic graph.

160

CHIRAG BARASARA AND PALAK PRAJAPATI

Case 2: For n  4

.

The antimagic labeling of graph L(L₄)is demonstrated in following Figure 4.

Figure 4: L(L₄) and its antimagic labeling.

Case 3: For n  0, 1, 3(mod 4) and n  4

.

f(x_ix^'_i)  i

;

For 1  i  n  1

For 1  i  n  2

,

f(x_i1x^'_i)  n  1  i

;

f(x_ix^'_i^')  2(n  1)  i

;

f(x_i1x^'_i^')  3(n  1)  i

;

' '

f(x x

)  4(n  1)  i

;

,

i i1

f(x^'_i^'x^'_i^'_1)  5(n  1)  1  i

;

.

Case 4: For n  2(mod 4) and n  2

.

f(x_ix^'_i)  i

f(x_{i 1}x^'_i)  2(n  1)  i

f(x_ix^'_i^')  n  1  i

;

For 1  i  n  1

,

;

,

;

ANTIMAGIC LABELING OF LINE GRAPH OF SOME GRAPHS

161

f(x_i1x^'_i^')  3(n  1)  i

;

For 1  i  n  1

For 1  i  n  2

,

' '

f(x x

)  4(n  1)  i

;

,

i i1

f(x^'_i^'x^'_i^'_1)  5(n  1)  1  i

;

.

Above define edge labeling function will generate distinct vertex labels for

all the vertices of L(L_n). Thus, is an antimagic labeling.

f

Hence, the graph L(L_n)is an antimagic graph.



Illustration 3.4: The graph L(L₆) and its antimagic labeling is shown

in Figure 5.

Figure 5: L(L₆)and its antimagic labeling.

Theorem 3.5: The graph L(W_n) is an antimagic graph.

Proof: Let W_nbe a wheel graph with V(W_n)  {v, v_i/i  1, 2, , n} and

E(W )  {e_i v_iv_i1/i  1, 2,, n  1}  {e_n v₁v_n}  {e_i^' vv_i/i  1, 2,, n}.

n

To construct L(W_n), let the vertices corresponding to e_ibe x_iand e^'_ibe x^'_i

for each

i

.

n² 5n

Then V(L(W_n)  2n and E(L(W_n) 

.

2









n  5n

We define f : E(L(W_n))  1, 2, ,

as follows.



2



162

CHIRAG BARASARA AND PALAK PRAJAPATI

f(x_ix_i1)  i

;

For 1  i  n  1

,

f(x_nx₁)  n

f(x_ix^'_i)  n  i

f(x^'_ix_i1)  3n  1  i

f(x^'_nx₁)  2n  1

,

;

For 1  i  n

,

;

For 1  i  n  1

,

1  i  2,



f(x^'_ix^'_{i j})  i(n  1)  2n  j  1

;

For



1  j  n  i,





3  i  n  1,

(i  1)(i  2)



f(x^'_ix^'_{i j})  i(n  1)  2n  j  1 

;

For



1  j  n  i,

2





Above define edge labeling function will generate distinct vertex labels for

all the vertices of L(W_n)

.

Thus,

f

is an antimagic labeling.

Hence, the graph L(W_n) is an antimagic graph.



Illustration 3.5: The graph L(W₅) and its antimagic labeling is shown

in Figure 6.

Figure 6: L(W₅)and its antimagic labeling.

ANTIMAGIC LABELING OF LINE GRAPH OF SOME GRAPHS

163

Theorem 3.6: The graph L(T(n, m)) is an antimagic graph.

Proof: Let T(n, m) be a tadpole with V(T(n, m))  {v₁, v₂, , v_mn

andE(T(n, m))  {e_i v_iv_{i 1}/i 1, 2,  , n  m}  e_{n m} v_{n m m 1}

To construct L(T(n, m)), let the vertices corresponding to e_ibe x_ifor each

}

v



.

i

.

Then V(L(T(n, m))  n  m and E(L(T(n, m))  n  m  1

.

We define f : E(L(T(n, m))  {1, 2, , n  m  1} as per following

seven cases.

Case 1: For n  3 and m  n  1

.

f(x_ix_i1)  i For 1  i  n  m  1

;

,

f(x_nmx_m)  n  m

,

f(x_{m1 n m})  n  m  1

x

.

Case 2:For n  4 and (m  n  1 or m  n  3)

.

f(x_ix_i1)  i For 1  i  n  m  1

;

,

f(x_{n m}x_m)  n  m

,

f(x_{m1 n m})  n  m  1

x

.

Case 3: For (n  3 or n  4) and m  n  2

.

f(x_ix_i1)  i For 1  i  n  m  1

;

f(x_{n m m}

f(x_{m1 n m})  n  m

Case 4: For odd n  5 and m  1

x )  n  m  1

,

x

.

n  m

f(x_ix_i1)  n  2i  2

;

For 2  i 

2

n  m

f(x_ix_i1)  2i  (n  m)

;

For

 1  i  n  m  1

2

164

CHIRAG BARASARA AND PALAK PRAJAPATI

f(x₁x₂)  n  m

,

f(x₁x_{n m})  n  m  1

,

f(x₂x_{n m})  n  m  1

.

Case 5: For even n  6 and m  1

.

n  m  1

f(x_ix_i1)  n  2i  2

;

For 2  i 

2

n  m  1

f(x_ix_i1)  2i  (n  m)

f(x₁x₂)  n  m

;

For

2

 i  n  m  1,

,

f(x₁x_{n m})  n  m  1

,

f(x₂x_{n m})  n  m  1

.

Case 6: For n  5 and (1  m  n  4 or m  n  2) and n  m is

even.

n  m

f(x_ix_i1)  2i

;

For 1  i 

,

2

n  m

f(x_ix_i1)  2i  (n  m)  1

;

For

+ 1  i  n  m  1

2

f(x_{n m}x_m)  n  m  1

,

f(x

x

_{m1 n m})  n  m  1

.

Case 7: For n  6 and (1  m  n  4 or m  n  2) and n  m is

odd.

n  m  1

f(x_ix_i1)  2i

;

For 1  i 

2

n  m  1

f(x_ix_i1)  2i  (n  m)  2

;

For

+ 1  i  n  m  1

2

f(x_{n m}x_m)  n  m  2

,

f(x_{m1 n m})  n  m

x

.

ANTIMAGIC LABELING OF LINE GRAPH OF SOME GRAPHS

165

Above define edge labeling function will generate distinct vertex labels for

all the vertices of L(T(n, m)). Thus, is an antimagic labeling.

f

Hence, the graph L(T(n, m)) is an antimagic graph.



Illustration 3.6: The graph L(T(5, 5)) and its antimagic labeling is

shown in Figure 7.

Figure 7: L(T(5, 5)) and its antimagic labeling.

4. Applications of Antimagic Labeling

Labeled graph has many applications in computer science, applied sciences,

social sciences and cryptography. Development of encryption and decryption

algorithm using antimagic labeling was studied by Krishnaa [16], Femina and Xavier

[8] and Selvakumar and Gupta [20].

5. Conclusions

It is quite difficult to verify that the given connected graph admits antimagic

labeling. Many authors [3, 4, 6, 15, 17, 24] have studied antimagic labeling for

various graph operations.

While in this paper, antimagic labeling for the line graph of armed crown,

double comb, ladder, wheel and tadpole is verified.

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166

CHIRAG BARASARA AND PALAK PRAJAPATI

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1. Department of Mathematics,

(Received, December 19, 2024)

Hemchandracharya North Gujarat University,

Patan - 384265, Gujarat, INDIA

(Revised, January 17, 2025)

E-mail: chirag.barasara@gmail.com

2. Department of Mathematics,

Hemchandracharya North Gujarat University,

Patan - 384265, Gujarat, INDIA

E-mail: palakprajapati733@gmail.com