Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 8s (2024) 

  

1155 
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Distance Pair Antimagic Labeling on Cycle Related Graphs 

M. Bala1,*, T. Saratha Devi2  

  1Research Scholar (Reg. No. 20222052091003),  

 Manonmaniam Sundaranar University, Abishekapatti - 627 012, Tirunelveli, Tamil Nadu, India.  

    2Department of Mathematics, Research Center,  

 G. Venkataswamy Naidu College, Kovilpatti-628 502, Tamil Nadu, India. 

 E-mails :  1 balamaths27@gmail.com ,   2 rajanvino03@gmail.com 

* Corresponding author: M. Bala 

Article History: 

Received: 10/09/2024 

Revised: 27/10/2024  

Published: 14/11/2024 

Abstract:  

A distance pair antimagic labeling of a graph 𝐺 with 𝑝 vertices is a bijection 

𝑓: 𝑉(𝐺) ⟶ 𝑃 

where 𝑃 = {
±1,±2,⋯ , ±

𝑝

2
, 𝑖𝑓    𝑝  𝑖𝑠  𝑒𝑣𝑒𝑛

0, ±1, ±2,⋯ , ±
𝑝−1

2
, 𝑖𝑓    𝑝  𝑖𝑠  𝑜𝑑𝑑

 such that the induced weight             

function 𝑤: 𝑉(𝐺) ⟶ 𝑊 defined by 𝑤(𝑣) = ∑ 𝑓(𝑢) = 𝑘𝑖𝑢∈𝑁(𝑣) is one-one, where                

 𝑁(𝑣) = {𝑢 ∈ 𝑉: 𝑢𝑣 ∈ 𝐸} is the open neighborhood of 𝑣 and the set of all weights 𝑊 

is either of the form {±𝑘1, ±𝑘2, ±𝑘3, ⋯ ,±𝑘𝑝
2
} or {0, ±𝑘1, ±𝑘2, ±𝑘3, ⋯ ,±𝑘𝑝−1

2

} 

according as 𝑝 is even or odd. In this paper, we explored the result on distance pair 

antimagic labeling of cycle related graphs. Also we investigated the closed distance 

magic labeling of circulant graph and its complement. 

Keywords: Graph labeling, distance antimagic, pair sum labeling, distance pair 

antimagic labeling. 

AMS Subject Classification(2010): 05C12, 05C78. 

1. Introduction 

      A magic square of order 𝑛 is an 𝑛 × 𝑛 array whose entries are an arrangement of the integers 1,2,3, … , 𝑛2 
in which all elements in any row, any column, the main diagonal or the main back diagonal add to the same 

sum 𝑟. Vilfred [9] in his doctoral thesis introduced the concept of Σ-labeling. Following this, Miller et al. [5] 

and B.D. Acharya et al [1] studied the concepts under the name of neighborhood magic graphs. Further, Sugeng 

et al. [7] used the term distance magic labeling for the same concept.  A distance magic labeling of a graph   

𝐺 = (𝑉, 𝐸) of order 𝑛 is a bijection  𝑓: 𝑉(𝐺) → {1,2,… , 𝑛} such that 


=
)(

)(
vNu

kuf  for all 𝑣 ∈ 𝑉. The constant 

𝑘 is called the  magic constant of the labeling 𝑓. A graph which admits a distance magic labeling is called a  

distance magic graph. In 2013, Kamatchi and Arumugam [3] introduced the concept of a distance antimagic 

graph. Motivated by these works and pair sum labeling [6] we introduced the concept of distance pair antimagic 

labeling [4]. In this paper, we present several results of distance pair antimagic labeling on cycle related graphs 

and complement of circulant graph. 

2. Preliminaries  

Definition 2.1  A graph G is said to be distance antimagic if there is a bijection 𝑓: 𝑉(𝐺) → {1,2, . . . , 𝑝} such 

that for every pair of distinct vertices 𝑥 and 𝑦 applies 𝑤(𝑥) ≠ 𝑤(𝑦).   

mailto:rajanvino03@gmail.com


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Definition 2.2  A injective map 𝑓: 𝑉(𝐺) → {±1,±2,⋯ ,±𝑝} is said to be pair sum labeling if the induced 

edge function 𝑓𝑒: 𝐸(𝐺) → 𝑍\{0} defined by 𝑓𝑒(𝑢𝑣) = 𝑓(𝑢) + 𝑓(𝑣) is one-one and 𝑓𝑒(𝐸(𝐺)) is either of the 

form {±𝑘1, ±𝑘2, ±𝑘3, ⋯ ,±𝑘𝑞
2
} or {±𝑘1, ±𝑘2, ±𝑘3, ⋯ ,±𝑘𝑞−1

2

} ∪ {±𝑘𝑞+1
2

} according as q is even or odd.  

Definition 2.3  Let 𝐺 be a (𝑝, 𝑞) graph. Let 𝑓: 𝑉(𝐺) ⟶ 𝑃 be a bijection where  

 𝑃 = {
±1, ±2,⋯ , ±

𝑝

2
, 𝑖𝑓    𝑝  𝑖𝑠  𝑒𝑣𝑒𝑛

0, ±1, ±2,⋯ ,±
𝑝−1

2
, 𝑖𝑓    𝑝  𝑖𝑠  𝑜𝑑𝑑

 

Then 𝑓 is called a distance pair antimagic (DPAM) labeling if the induced weight function 𝑤: 𝑉(𝐺) ⟶ 𝑊 

defined by 𝑤(𝑣) = ∑𝑢∈𝑁(𝑣)  𝑓(𝑣) = 𝑘𝑖 is one-one, where 𝑁(𝑣) = {𝑢 ∈ 𝑉: 𝑢𝑣 ∈ 𝐸} is the open neighborhood 

of 𝑣 and the set of all weights 𝑊 is either of the form {±𝑘1, ±𝑘2, ±𝑘3, ⋯ ,±𝑘𝑝
2
} or {0,±𝑘1, ±𝑘2, ±𝑘3, ⋯ ,±𝑘𝑝−1

2

} 

according as 𝑝 is even or odd. A graph which admits distance pair antimagic labeling is called a distance pair 

antimagic graph.  

Definition 2.4 A function 𝑓 is called closed distance pair antimagic labeling, if we take closed neighborhood 

𝑛[𝑣] instead of open neighborhood 𝑁(𝑣) in the previous definition.  

Definition 2.5 The 𝑛-sunlet graph 𝐶𝑛⊙𝐾1 is the graph on 2𝑛 vertices obtained by attaching a pendant edge 

to each vertices of a cycle 𝐶𝑛 and it is denoted by 𝑆𝑛.  

Definition 2.6 The helm graph 𝐻𝑛 is the graph obtained from an wheel graph 𝑊𝑛 by adjoining a pendant edge 

at each vertex of the cycle.  

Definition 2.7 The triangular snake 𝑇𝑛 is obtained from a path 𝑃𝑛 by replacing each edge of the path by a 

triangle 𝐶3.  

Definition 2.8 The Gear graph 𝐺𝑛 is formed by adding a vertex between each pair of adjacent vertices of a 

wheel graph 𝑊𝑛.  

Definition 2.9 The n-book graph is defined as the graph Cartesian product 𝐵𝑛 = 𝑆𝑛+1 × 𝑃2, where 𝑆𝑛+1 is a 

star graph and 𝑃2 is the path graph on two vertices.  

Definition 2.10 The friendship graph is defined as the graph 𝐹𝑛 which consisting of n triangles with a common 

vertex.  

Definition 2.11 The Prism graph 𝐶𝑛 × 𝐾2 is constructed by the Cartesian product of a cycle of length 𝑛 ≥ 3 

and an edge.  

Definition 2.12 Resty [8] defined the following graph. For 𝑛 ≥ 4, the n-crossed prism graph obtained by taking 

two disjoint cycle graphs namely 𝐶𝑛
1 and 𝐶𝑛

2, where 𝑉(𝐶𝑛
1) = {𝑣1, 𝑣2, . . . , 𝑣𝑛} and 𝑉(𝐶𝑛

2) = {𝑢1, 𝑢2, . . . , 𝑢𝑛}, and 

adding edges 𝑢𝑠𝑣𝑠+1 for 𝑠 ∈ {1,3, . . . , 𝑛 − 1} and 𝑢𝑡𝑣𝑡−1 for 𝑡 ∈ {2,4, . . . , 𝑛}. The n-crossed prism graph 

denoted by 𝐶𝑃𝑛 .  

Definition 2.13 The circulant graph 𝐶(𝑛; 𝐷), where 𝐷 ⊆ {1,2,… , ⌊
𝑛

2
⌋} is the graph with vertex set 

{𝑣1, 𝑣2, … , 𝑣𝑛}, where 𝑣𝑖 and 𝑣𝑗  are adjacent if and only if there is a number 𝑑 ∈ 𝐷 such that 𝑖 + 𝑑 ≡ 𝑗(𝑚𝑜𝑑 𝑛) 

or 𝑗 + 𝑑 ≡ 𝑖(𝑚𝑜𝑑 𝑛).  

3. Main Results 

Theorem 3.1 The sunlet graph 𝑆𝑛 is a distance pair antimagic graph if 𝑛 = 4,6,8, …  

Proof. Let 𝑉(𝑆𝑛) = {𝑣𝑖 , 𝑢𝑖: 1 ≤ 𝑖 ≤ 𝑛} be the vertex set and  



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𝐸(𝑆𝑛) = {𝑣𝑖𝑢𝑖: 1 ≤ 𝑖 ≤ 𝑛} ∪ {𝑣𝑖𝑣𝑖+1: 1 ≤ 𝑖 ≤ 𝑛 − 1} ∪ {𝑣1𝑣𝑛} be the edge set of 𝑆𝑛.  

Define 𝑓: 𝑉(𝑆𝑛) ⟶ {±1,±2,⋯ , ±𝑛} by  

 𝑓(𝑣𝑖) = {

(−1)𝑖(𝑖), 𝑖𝑓    1 ≤ 𝑖 ≤
𝑛

2

(−1)𝑖(𝑛 + 1 − 𝑖), 𝑖𝑓    
𝑛

2
< 𝑖 ≤ 𝑛 

 𝑓(𝑢𝑖) =

{
 

 (−1)
𝑖+1   (

𝑛

2
+ 𝑖) , 𝑖𝑓    1 ≤ 𝑖 ≤

𝑛

2

(−1)𝑖+1 (
3𝑛

2
+ 1 − 𝑖) , 𝑖𝑓    

𝑛

2
< 𝑖 ≤ 𝑛 

Then the induced vertex weight labeling are as follows.  

 𝑤(𝑣𝑖) =

{
 
 
 
 

 
 
 
 
𝑛

2
+ 4, 𝑖𝑓    𝑖 = 1

(−1)𝑖+1 (
𝑛

2
+ 3𝑖) , 𝑖𝑓    1 < 𝑖 <

𝑛

2

(−1)𝑖+12𝑛 − 1, 𝑖𝑓    𝑖 =
𝑛

2
,
𝑛

2
+ 1

(−1)𝑖+12𝑛 + 3(
𝑛

2
+ 1 − 𝑖) , 𝑖𝑓    𝑖 =

𝑛

2
+ 2,

𝑛

2
+ 3, . . . , 𝑛 − 1

− (
𝑛

2
+ 4) , 𝑖𝑓    𝑖 = 𝑛

 

𝑤(𝑢𝑖) =

{
 
 

 
 (−1)𝑖(𝑖), 𝑖𝑓    1 ≤ 𝑖 ≤

𝑛

2

(−1)𝑖(𝑛 + 1 − 𝑖), 𝑖𝑓    
𝑛

2
< 𝑖 ≤ 𝑛

 

Hence 𝑆𝑛 is a distance pair antimagic graph if 𝑛 = 4,6,8, …  

Corollary 3.2 The Helm graph 𝐻𝑛 is a distance pair antimagic graph if 𝑛 = 4,6,8,… 

 By using Theorem 2.10 [4], If 𝐺 is a distance pair antimagic graph with even number of vertices, then 

the join graph 𝐺 + 𝐾1 is a distance pair antimagic graph. 

Theorem 3.3 The triangular snake graph 𝑇𝑛 is a distance pair antimagic graph for all 𝑛 ≥ 2. 

Proof. Let 𝑉(𝑇𝑛) = {𝑣𝑖: 1 ≤ 𝑖 ≤ 𝑛} ∪ {𝑢𝑖: 1 ≤ 𝑖 ≤ 𝑛 − 1} be the vertex set and 𝐸(𝑇𝑛) = {𝑣𝑖𝑣𝑖+1: 1 ≤ 𝑖 ≤ 𝑛 −

1} ∪ {𝑢𝑖𝑣𝑖 , 𝑢𝑖𝑣𝑖+1: 1 ≤ 𝑖 ≤ 𝑛 − 1} be the edge set of 𝑇𝑛. 

Consider following two cases. 

Case(i): 𝑛 =is odd  

 

 

 

 

Figure 3.1 DPAM Labeling of 𝑇3. ( The vertex labels are in usual font and weights are in bold font) 



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Clearly from figure 3.1 𝑇3 is a distance pair antimagic graph. 

For 𝑛 = 5,7,9, . .. define 𝑓: 𝑉(𝑇𝑛) ⟶ {0, ±1, ±2,⋯ ,±𝑛 − 1} by  

 𝑓(𝑣𝑖) = (𝑛 + 1) − 2𝑖, if 𝑖 = 1,2,3, . . . , 𝑛 

𝑓(𝑢𝑖) = 𝑛 − 2𝑖, if 𝑖 = 1,2,3, . . . , 𝑛 − 1  

 Then the induced vertex weight labeling are as follows.  

 𝑤(𝑣𝑖) = {

2𝑛 − 5,                             𝑖𝑓    𝑖 = 1
4(𝑛 − 3) − 8(𝑖 − 2),    𝑖𝑓    𝑖 = 2,3,4, . . . , 𝑛 − 1

−(2𝑛 − 5),                       𝑖𝑓    𝑖 = 𝑛
 

 𝑤(𝑢𝑖) = 2(𝑛 − 2𝑖) if 𝑖 = 1,2,3, . . . , 𝑛 − 1  

Case(ii): 𝑛 is even  

  

Figure 3.2 DPAM Labeling of 𝑇2 and 𝑇4.  

Clearly from figure 3.2 𝑇2 and 𝑇4 are distance pair antimagic graphs. 

For 𝑛 = 6,8,10, . .. define 𝑓: 𝑉(𝑇𝑛) ⟶ {0,±1,±2,⋯ , ±𝑛 − 1} by  

 𝑓(𝑣1) = 𝑛 − 1 = −𝑓(𝑣𝑛) 

 and remaining vertices has following labeling  

 𝑓(𝑣𝑖) = {

𝑛 − 2(𝑖 − 1), 𝑖𝑓    1 < 𝑖 <
𝑛

2

𝑛 − 2𝑖, 𝑖𝑓    
𝑛

2
≤ 𝑖 < 𝑛 

 𝑓(𝑢𝑖) =

{
 

 
𝑛 − 1 − 2𝑖, 𝑖𝑓    𝑖 = 1,2,3, . . . ,

𝑛

2
− 1

0, 𝑖𝑓    𝑖 =
𝑛

2

𝑛 − 1 − 2(𝑖 − 1) 𝑖𝑓    𝑖 =
𝑛

2
+ 1,

𝑛

2
+ 2, . . . , 𝑛 − 1

 

Then the induced vertex weight labeling are as follows.  

 𝑣𝑖  𝑤(𝑣𝑖) 

 𝑖 = 1  2𝑛 − 5 

 𝑖 = 2 3 + 4(𝑛 − 4) 

 𝑖 = 3,4,5, . . . ,
𝑛

2
− 1  4(𝑛 + 1 − 2𝑖) 



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 𝑖 =
𝑛

2
  3 

𝑖 =
𝑛

2
+ 1 −3 

𝑖 =
𝑛

2
+ 2,

𝑛

2
+ 3, . . . , 𝑛 − 2 4(𝑛 + 1 − 2𝑖) 

 𝑖 = 𝑛 − 1 −(3 + 4(𝑛 − 4)) 

 𝑖 = 𝑛 −(2𝑛 − 5) 

  

 𝑢𝑖  𝑤(𝑢𝑖) 

 𝑖 = 1  2𝑛 − 3 

 𝑖 = 2,3,4, . . . ,
𝑛

2
− 1  2(𝑛 + 1) − 4𝑖 

 𝑖 =
𝑛

2
   0 

 
𝑛

2
+ 1,

𝑛

2
+ 2, . . . , 𝑛 − 2 2(𝑛 − 1) − 4𝑖 

 𝑖 = 𝑛 − 1 −(2𝑛 − 3) 

 Hence 𝑇𝑛 is a distance pair antimagic graph for all 𝑛 ≥ 2.  

Theorem 3.4 The n-gear graph 𝐺𝑛 is a distance pair antimagic graph, if 𝑛 = 4,6,8,…  

Proof. Let 𝑉(𝐺𝑛) = {𝑢0, 𝑢𝑖 , 𝑣𝑖: 𝑖 = 1,2,3, . . . , 𝑛} and 𝐸(𝐺𝑛) = {𝑢𝑖𝑣𝑖 , 𝑣𝑖𝑢𝑖+1: 𝑖 = 1,2,3, . . . , 𝑛 − 1} ∪ {𝑣𝑛𝑢1} ∪

{𝑢0𝑢𝑖: 𝑖 = 1,2,3, . . . , 𝑛} be vertex set and edge set of 𝐺𝑛. 

Define 𝑓: 𝑉(𝐺𝑛) ⟶ {0, ±1, ±2,⋯ ,±𝑛} by the following two cases 

Case : (i) 𝑛 ≡ 2 𝑚𝑜𝑑 4  

 𝑢𝑖  𝑓(𝑢𝑖) 

 𝑖 = 0  0 

 𝑖 = 1,2,3,4, . . . ,
𝑛

2
− 1  𝑖 

 𝑖 =
𝑛

2
   𝑛 

 𝑖 =
𝑛

2
+ 1,

𝑛

2
+ 2, . . . , 𝑛 − 1  

𝑛

2
− 𝑖 

 𝑖 = 𝑛   −𝑛 

  

 𝑣𝑖  𝑓(𝑣𝑖) 

 𝑖 = 1,2,3,4, . . . ,
𝑛

2
  

𝑛

2
+ (𝑖 − 1) 



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 𝑖 =
𝑛

2
+ 1,

𝑛

2
+ 2, . . . , 𝑛  −(𝑖 − 1) 

 Then the induced vertex weight labeling are as follows.  

 𝑢𝑖  𝑤(𝑢𝑖) 

 𝑖 = 1  −(
𝑛

2
− 1) 

 𝑖 = 2,3,4, . . . ,
𝑛

2
  (𝑛 − 1) + 2(𝑖 − 1) 

 𝑖 =
𝑛

2
+ 1   

𝑛

2
− 1 

 𝑖 =
𝑛

2
+ 2,

𝑛

2
+ 3, . . . , 𝑛  3 − 2𝑖 

   

 𝑣𝑖  𝑤(𝑣𝑖) 

 𝑖 = 1,2,3, . . . ,
𝑛

2
− 2  2𝑖 + 1 

 𝑖 =
𝑛

2
− 1   

3𝑛

2
− 1 

 𝑖 =
𝑛

2
   𝑛 − 1 

 𝑖 =
𝑛

2
+ 1,

𝑛

2
+ 2, . . . , 𝑛 − 2  𝑛 − 1 − 2𝑖 

 𝑖 = 𝑛 − 1   −(
3𝑛

2
− 1) 

 𝑖 = 𝑛   −(𝑛 − 1) 

Case : (ii) 𝑛 ≡ 0 𝑚𝑜𝑑 4 

 𝑢𝑖  𝑓(𝑢𝑖) 

 𝑖 = 0  0 

 𝑖 = 1,2,3,4, . . . ,
𝑛

4
  2𝑖 − 1 

 𝑖 =
𝑛

4
+ 1,

𝑛

4
+ 2, . . . ,

𝑛

2
  2(𝑖 −

𝑛

4
) 

 𝑖 =
𝑛

2
+ 1,

𝑛

2
+ 2, . . . ,

3𝑛

4
  𝑛 + 1 − 2𝑖 

 𝑖 =
3𝑛

4
+ 1,

3𝑛

4
+ 2, . . . , 𝑛  

3𝑛

2
− 2𝑖 

   

 𝑣𝑖  𝑓(𝑣𝑖) 

 𝑖 = 1,2,3, . . . ,
𝑛

4
  𝑛 − 2(𝑖 − 1) 

 𝑖 =
𝑛

4
+ 1,

𝑛

4
+ 2, . . . ,

𝑛

2
  2(𝑖 − 1) + 1 



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 𝑖 =
𝑛

2
+ 1,

𝑛

2
+ 2, . . . ,

3𝑛

4
  −2(𝑛 + 1 − 𝑖) 

 𝑖 =
3𝑛

4
+ 1,

3𝑛

4
+ 2, . . . , 𝑛  𝑛 + 1 − 2𝑖 

Then the induced vertex weight labeling are as follows.  

  𝑢𝑖  𝑤(𝑢𝑖) 

 𝑖 = 0  0 

 𝑖 = 1  1 

 𝑖 = 2,3, . . . ,
𝑛

4
  2(𝑛 + 1) − 4(𝑖 − 1) 

 𝑖 =
𝑛

4
+ 1  𝑛 + 3 

 𝑖 =
𝑛

4
+ 2,

𝑛

4
+ 3, . . . ,

𝑛

2
  4(𝑖 − 1) 

 𝑖 =
𝑛

2
+ 1  −1 

 𝑖 =
𝑛

2
+ 2,

𝑛

2
+ 3, . . . ,

3𝑛

4
  −(4(𝑛 − 𝑖) + 6) 

 𝑖 =
3𝑛

4
+ 1  −(𝑛 + 3) 

 𝑖 =
3𝑛

4
+ 2,

3𝑛

4
+ 3, . . . , 𝑛  −(4 (𝑖 −

𝑛

2
− 1)) 

    

𝑣𝑖  𝑤(𝑣𝑖) 

 𝑖 = 1,2,3, . . . ,
𝑛

4
− 1  4𝑖 

 𝑖 =
𝑛

4
  

𝑛

2
+ 1 

 𝑖 =
𝑛

4
+ 1,

𝑛

4
+ 2, . . . ,

𝑛

2
− 1  4(𝑖 −

𝑛

4
) + 2 

 𝑖 =
𝑛

2
  

𝑛

2
− 1 

 𝑖 =
𝑛

2
+ 1,

𝑛

2
+ 2, . . . ,

3𝑛

4
− 1 4 (

𝑛

2
− 𝑖) 

 𝑖 =
3𝑛

4
  −(

𝑛

2
+ 1) 

 𝑖 =
3𝑛

4
+ 1,

3𝑛

4
+ 2, . . . , 𝑛 − 1  −(4 (𝑖 −

3𝑛

4
) + 2) 

 𝑖 = 𝑛  −(
𝑛

2
− 1) 

Hence the n-gear graph 𝐺𝑛 is a distance pair antimagic graph, if 𝑛 = 4,6,8,… 

 

 



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Theorem 3.5 The book graph 𝐵𝑛 = 𝑆𝑛+1 × 𝑃2 is a distance pair antimagic graph if 𝑛 ≥ 2.  

Proof. Let the vertex set of book graph 𝑉(𝐵𝑛) be {𝑢𝑖, 𝑣𝑖 : 𝑖 = 0,1,2,3, . . . , 𝑛} and  

the edge set be {𝑢0𝑢𝑖 , 𝑣0𝑣𝑖: 𝑖 = 1,2,3, . . . , 𝑛} ∪ {𝑢𝑖𝑣𝑖: 𝑖 = 0,1,2,3, . . . , 𝑛}. 

Define 𝑓: 𝑉(𝑆𝑛+1 × 𝑃2) ⟶ {±1, ±2,⋯ ,±𝑛 + 1} by  

 𝑓(𝑢0) = 1;  𝑓(𝑣0) = −1; 

 𝑓(𝑢𝑖) = 𝑖 + 1,   𝑖𝑓    𝑖 = 1,2,3, . . . , 𝑛; 

 𝑓(𝑣𝑖) = −(𝑖 + 1), 𝑖𝑓   𝑖 = 1,2,3, . . . , 𝑛 

 Then the induced vertex weight labeling are as follows.  

 𝑤(𝑢0) =
(𝑛+1)(𝑛+2)

2
− 1 

 𝑤(𝑢𝑖) = −𝑖,    𝑖𝑓    𝑖 = 1,2,3, . . . , 𝑛 

 𝑤(𝑣0) = 1 −
(𝑛+1)(𝑛+2)

2
 

 𝑤(𝑣𝑖) = 𝑖,    𝑖𝑓    𝑖 = 1,2,3, . . . , 𝑛 

 Hence 𝐵𝑛 is a distance pair antimagic graph if 𝑛 ≥ 2.  

Theorem 3.6 The friendship graph 𝐹𝑛 is a distance pair antimagic graph.   

Proof. Let the vertex set of friendship graph 𝑉(𝐹𝑛) be {𝑣0, 𝑣1, 𝑣2, . . . , 𝑣2𝑛} and  

the edge set be {𝑣𝑖𝑣𝑖+1: 𝑖 = 1,3, . . . ,2𝑛 − 1} ∪ {𝑣0𝑣𝑖: 𝑖 = 1,2,3, . . . ,2𝑛}. 

Define 𝑓: 𝑉(𝐹𝑛) ⟶ {0,±1,±2,⋯ , ±𝑛} by  

 𝑓(𝑣𝑖) =

{
 
 

 
 
0, 𝑖𝑓    𝑖 = 0
𝑖+1

2
, 𝑖𝑓    𝑖    𝑖𝑠    𝑜𝑑𝑑

−(
𝑖

2
) , 𝑖𝑓    𝑖    𝑖𝑠    𝑒𝑣𝑒𝑛

 

Then the induced vertex weight labeling are as follows.  

 𝑤(𝑣𝑖) =

{
 
 

 
 
0, 𝑖𝑓    𝑖 = 0

− (
𝑖+1

2
) , 𝑖𝑓    𝑖    𝑖𝑠    𝑜𝑑𝑑

𝑖

2
, 𝑖𝑓    𝑖    𝑖𝑠    𝑒𝑣𝑒𝑛

 

Hence 𝐹𝑛 is a distance pair antimagic graph.  

Theorem 3.7 The prism graph 𝐶𝑛 × 𝐾2 is a distance pair antimagic graph.   

Proof. Let 𝑉(𝐶𝑛 × 𝐾2) = {𝑣𝑖 , 𝑢𝑖: 1 ≤ 𝑖 ≤ 𝑛} be the vertex set and  

𝐸(𝐶𝑛 × 𝐾2) = {𝑣𝑖𝑣𝑖+1, 𝑢𝑖𝑢𝑖+1, 𝑣𝑖𝑢𝑖: 1 ≤ 𝑖 ≤ 𝑛 − 1} ∪ {𝑢𝑛𝑢1, 𝑣𝑛𝑣1, 𝑣𝑛𝑢𝑛} be the edge set of 𝐶𝑛 × 𝐾2. 

 



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Define 𝑓: 𝑉(𝐶𝑛 × 𝐾2) ⟶ {±1,±2,⋯ , ±𝑛} by 𝑓(𝑣𝑖) = 𝑖,  if 𝑖 = 1,2,3, . . . , 𝑛  

𝑓(𝑢𝑖) = {
−𝑛,

−(𝑖 − 1),   
𝑖𝑓  𝑖 = 1

                 𝑖𝑓  𝑖 = 2,3,4, . . . , 𝑛
      

Then the induced vertex weight labeling are as follows.  

 𝑤(𝑣𝑖) = 𝑖 + 1,    𝑖𝑓    𝑖 = 1,2,3, . . . , 𝑛 − 1;     𝑤(𝑣𝑛) = 1 

 𝑤(𝑢𝑖) = {

−(𝑛 − 1), 𝑖𝑓    𝑖 = 1
−𝑛, 𝑖𝑓    𝑖 = 2

−(𝑖 − 2), 𝑖𝑓    𝑖 = 3,4,5, . . . , 𝑛
 

Hence 𝐶𝑛 × 𝐾2 is a distance pair antimagic graph.  

Theorem 3.8 The crossed prism 𝐶𝑃𝑛 is a distance pair antimagic graph if 𝑛 ≡ 0 (𝑚𝑜𝑑 4). 

Proof. Let 𝑉(𝐶𝑃𝑛) = {𝑣𝑖 , 𝑢𝑖: 1 ≤ 𝑖 ≤ 𝑛} be the vertex set of 𝐶𝑃𝑛 and edge sets as follows: 

𝐸(𝐶𝑃𝑛) = {𝑣𝑖𝑣𝑖+1, 𝑣1𝑣𝑛: 1 ≤ 𝑖 ≤ 𝑛 − 1} ∪ {𝑢𝑖𝑢𝑖+1, 𝑢1𝑢𝑛: 1 ≤ 𝑖 ≤ 𝑛 − 1} and adding edges 𝑢𝑠𝑣𝑠+1 for 𝑠 ∈

{1,3, . . . , 𝑛 − 1} and 𝑢𝑡𝑣𝑡−1 for 𝑡 ∈ {2,4, . . . , 𝑛}, where 𝑣𝑖 , 𝑢𝑖 are inner and outer vertices of  𝐶𝑃𝑛. 

Define 𝑓: 𝑉(𝐶𝑃𝑛) ⟶ {±1,±2,⋯ , ±𝑛} by 

  𝑣𝑖  𝑓(𝑣𝑖)   𝑢𝑖 𝑓(𝑢𝑖) 

  𝑖 = 1 2 𝑖 = 1 1 

  𝑖 = 2 -2 𝑖 = 2 -1 

 𝑖 = 3,4,5, . . . ,
𝑛

2
 (−1)𝑖+1(2𝑖 − 3) + 1 𝑖 = 3,4,5, . . . ,

𝑛

2
 (−1)𝑖+1    (2𝑖 − 3) 

 𝑖 =
𝑛

2
+ 1 𝑛 𝑖 =

𝑛

2
+ 1 𝑛 − 1 

  𝑖 =
𝑛

2
+ 2 −𝑛 𝑖 =

𝑛

2
+ 2 −(𝑛 − 1) 

𝑖 =
𝑛

2
+ 3,

𝑛

2
+ 4, . . . , 𝑛 (−1)𝑖+1[2(𝑛 − 𝑖) + 4] 𝑖 =

𝑛

2
+ 3,

𝑛

2
+ 4, . . . , 𝑛 (−1)𝑖+1    2(𝑛 − 𝑖) + 3 

Then the induced vertex weight labeling are as follows. 

  𝑣𝑖  𝑤(𝑣𝑖)   𝑢𝑖 𝑤(𝑢𝑖) 

  𝑖 = 1 7 𝑖 = 1 6 

  𝑖 = 2 -7 𝑖 = 2 -6 

 𝑖 = 3,5, . . . ,
𝑛

2
− 1 −[6(𝑖 − 1) + 1] 𝑖 = 3,5, . . . ,

𝑛

2
− 1 −[6(𝑖 − 1)] 

  𝑖 = 4,6, . . . ,
𝑛

2
 6𝑖 − 9 𝑖 = 4,6, . . . ,

𝑛

2
 6𝑖 − 10 

 𝑖 =
𝑛

2
+ 1 3 − 6(𝑖 − 1) 𝑖 =

𝑛

2
+ 1 4 − 6(𝑖 − 1) 



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  𝑖 =
𝑛

2
+ 2 6(𝑖 − 2) − 3   𝑖 =

𝑛

2
+ 2 6(𝑖 − 2) − 4 

  𝑖 =
𝑛

2
+ 3,

𝑛

2
+ 5, . . . , 𝑛 − 1 −[6(𝑛 − 𝑖 + 1) + 3]    𝑖 =

𝑛

2
+ 3,

𝑛

2
+ 5, . . . , 𝑛 − 1 −[6(𝑛 − 𝑖 + 1) + 2] 

𝑖 =
𝑛

2
+ 4,

𝑛

2
+ 6, . . . , 𝑛 6(𝑛 − 𝑖 + 2) + 1 𝑖 =

𝑛

2
+ 4,

𝑛

2
+ 6, . . . , 𝑛 6(𝑛 − 𝑖 + 2) 

Hence the crossed prism 𝐶𝑃𝑛 is a distance pair antimagic graph if  𝑛 ≡ 0 (𝑚𝑜𝑑 4). 

Theorem 3.9 The circulant graph 𝐶(2𝑛; {𝑑1, 𝑑2, 𝑛}) is a distance pair antimagic if 𝑛 = 𝑑1 + 𝑑2, 𝑑1 < 𝑑2 and 

𝑛 > 3. 

Proof. Let 𝐺 = 𝐶(2𝑛; {𝑑1, 𝑑2, 𝑛}) and 𝑉(𝐺) = {𝑢1, 𝑢2, 𝑢3, . . . , 𝑢2𝑛} 

Define 𝑓: 𝑉(𝐺) ⟶ {±1,±2,⋯ ,±𝑛} by following two cases. 

Case (i): n is odd   

 𝑓(𝑢𝑖) =

{
 
 

 
 
−𝑖, 𝑖𝑓    𝑖 = 1,3,5, . . . , 𝑛
𝑖, 𝑖𝑓    𝑖 = 2,4,6, . . . , 𝑛 − 1
𝑛 − 𝑖, 𝑖𝑓    𝑖 = 𝑛 + 2, 𝑛 + 4, 𝑛 + 6, . . . ,2𝑛 − 1

𝑖 − 𝑛, 𝑖𝑓    𝑖 = 𝑛 + 1, 𝑛 + 3, 𝑛 + 4, . . . ,2𝑛

 

Then the induced vertex weight labeling are as follows.  

 𝑤(𝑢𝑖) =

{
 
 

 
 
𝑖, 𝑖𝑓    𝑖 = 1,3,5, . . . , 𝑛
−𝑖, 𝑖𝑓    𝑖 = 2,4,6, . . . , 𝑛 − 1
𝑖 − 𝑛, 𝑖𝑓    𝑖 = 𝑛 + 2, 𝑛 + 4, 𝑛 + 6, . . . ,2𝑛 − 1

𝑛 − 𝑖, 𝑖𝑓    𝑖 = 𝑛 + 1, 𝑛 + 3, 𝑛 + 4, . . . ,2𝑛

 

Case (ii): n is even  

 𝑓(𝑢𝑖) =

{
 
 

 
 
−𝑖, 𝑖𝑓    𝑖 = 1,3,5, . . . , 𝑛 − 1
𝑖, 𝑖𝑓    𝑖 = 2,4,6, . . . , 𝑛
𝑖 − 𝑛, 𝑖𝑓    𝑖 = 𝑛 + 1, 𝑛 + 3, 𝑛 + 5, . . . ,2𝑛 − 1

𝑛 − 𝑖, 𝑖𝑓    𝑖 = 𝑛 + 2, 𝑛 + 4, 𝑛 + 6, . . . ,2𝑛

 

Then the induced vertex weight labeling are as follows.  

 𝑤(𝑢𝑖) =

{
 
 

 
 
𝑖, 𝑖𝑓    𝑖 = 1,3,5, . . . , 𝑛 − 1
−𝑖, 𝑖𝑓    𝑖 = 2,4,6, . . . , 𝑛
𝑛 − 𝑖, 𝑖𝑓    𝑖 = 𝑛 + 1, 𝑛 + 3, 𝑛 + 5, . . . ,2𝑛 − 1

𝑖 − 𝑛, 𝑖𝑓    𝑖 = 𝑛 + 2, 𝑛 + 4, 𝑛 + 6, . . . ,2𝑛

 

Hence 𝐶(2𝑛; {𝑑1, 𝑑2, 𝑛}) is a distance pair antimagic if 𝑛 = 𝑑1 + 𝑑2,  𝑑1 < 𝑑2 and 𝑛 > 3.  

Corollary 3.10 The circulant graph 𝐶̅(2𝑛; {𝑑1, 𝑑2, 𝑛}) is a closed distance pair antimagic, if 𝑛 = 𝑑1 +

𝑑2,  𝑑1 < 𝑑2 and 𝑛 > 3.  

Corollary 3.11 The circulant graph 𝐶̅(2𝑛; {𝑑1, 𝑑2}) is a distance pair antimagic, if 𝑛 = 𝑑1 + 𝑑2,  𝑑1 < 𝑑2 

and 𝑛 > 3.  



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Corollary 3.12 The circulant graph 𝐶(2𝑛;𝐷) is a distance pair antimagic, if 𝑛 > 3 where 𝐷 ⊆ 𝐷𝑛 =

{(𝑑𝑖 , 𝑑𝑗): 𝑑𝑖 + 𝑑𝑗 = 𝑛} and 𝑑𝑖 < 𝑑𝑗  .  

4.  Conclusion  

The distance pair antimagic labeling on cycle related graphs is discussed in this paper. In 

particular the distance pair antimagicness of prism and crossed prism is investigated for particular n. 

Also it is proved that the circulant graph 𝐶(2𝑛; {𝑑1, 𝑑2, 𝑛}) is a distance pair antimagic if 𝑛 = 𝑑1 + 𝑑2,

d1 < d2 and 𝑛 > 3 while its complement is closed distance pair antimagic with the same n.   

References 

[1] B.D. Acharya, S.B. Rao, T. Singh and V. Parameswaran,  Neighborhood magic graphs, In 

National Conference on Graph Theory, Combinatorics and Algorithm, (2004). 

[2] F. Harary, Graph Therory, Narosa Publishing House, New Delhi, 1998. 

[3] N. Kamatchi , S. Arumugam,  Distance antimagic graphs, JCMCC, 84 (2013), 61-67. 

[4] M. Bala , T. Saratha Devi,  On Distance Pair Antimagic Labeling of Graphs, Indian Journal of 

Natural Sciences, Vol.15, Issue 86, Oct 2024. 

[5] M. Miller, C. Rodger and R. Simanjuntak, Distance magic labelings of graphs,  Australas. J. 

Combin.,  28 (2003), 305-315 

[6] R. Ponraj, J.V.X. Parthipan,  Pair Sum Labeling of Graphs, The Journal of Indian Acad. 

Math,Vol 32, No. 2(2010), 587-595. 

[7] K.A. Sugeng, D. Froncek, M. Miller, J. Ryan and J. Walker, On distance magic labeling of 

graphs,  J. Combin. Math. Combin. Comput.,  71 (2009), 39-48. 

[8] Resty D and Salman A N M 2015  The rainbow connection number of an n-crossed prism 

graph and its corona product with a trivial graph, Procedia Computer Science, 2015 74 143-

150. 

[9] V. Vilfred,  𝛴-labelled graph and circulant graphs, Ph.D. Thesis, University of Kerala, 

Trivandrum, India, 1994.