Communications on Applied Nonlinear Analysis 

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(𝑺, 𝒅)Magic Labeling of Some Cycle Related Graphs 

 

 P. Sumathi 1, P. Mala2 
1Department of Mathematics, C. Kandaswami College for Men, Anna Nagar, Chennai-102. 

2Department of Mathematics, St Thomas College of Arts and Science, Koyambedu, Chennai-107. 
1sumathipaul@gmail.com, 2mala.vedha@gmail.com 

 

Article History: 

Received: 20-06-2024 

Revised: 21-07-2024 

Accepted: 09-08-2024 

Abstract:  

Objective: To examine the existence of (s,d) Magic Labeling on cycle related graphs. 

Methods: Let G (p,q) be a simple, non-trivial, connected, undirected graph with p vertices 

and q edges. Let f:V(G)→{s,s+d,s+2d,..s+(q+1)d} and g:E(G)→{d,2d,3d…2(q-1)d} be an 

injective function. Then, for any u,v∈V(G) and uv∈E(G),f(u)+g(uv)+f(v) is a constant, and 

the function f is said to be (s, d) magic labeling. If a graph G admits (s,d) magic labeling, 

then it is referred to as a (s,d) magic graph. 

Findings: In this paper the existence of (s,d) magic labeling in some cycle related graphs 

such as a Cycle graph C_(n⊙K_(1,m) )  graph, n -Sunlet graph, Friendship graph Flower 

graph and wheel graph were found.  

Novelty: The labeling of the vertices and edges is done mathematically, and this leads to 

the creation of a new labeling known as (s,d) magic labeling. 

Keywords: Cycle graph, C_(n⊙K_(1,m) )  graph, n-Sunlet graph, Friendship graph, 

Flower graph and wheel graph. 

 

1. Introduction  

[1,3] The graphs that are being studied are simple, undirected, finite, and non-trivial. Graph labeling has 

advanced significantly according to graph theory, which has a wide range of uses. In 1963, Sedl´ȧcek 

developed the first magic-type labeling. He gave a graph's edges real numbers and mandated that the 

total of all the labels on all the edges that are incident to a vertex must remain constant. [5] Hegde SM, 

Shetty S. have introduced (𝑘, 𝑑) - arithmetic labeling of graphs. We have showed (𝑠, 𝑑).magic labeling 

for the path, star, and a few standard graphs. This work established the existence of (𝑠, 𝑑) magic 

labeling in certain cycle-related graphs. 

 
[10] Let 𝐺 (𝑝, 𝑞) be a simple, non-trivial, connected, undirected graph with p vertices and q edges. 

Consider the following: 𝑓: 𝑉(𝐺) → {𝑠, 𝑠 + 𝑑, 𝑠 + 2𝑑, . . 𝑠 + (𝑞 + 1)𝑑} and  𝑔: 𝐸(𝐺) →

{𝑑, 2𝑑, 3𝑑 … 2(𝑞 − 1)𝑑} be an injective function. Then, for any 𝑢, 𝑣 ∈  𝑉(𝐺) and 𝑢𝑣 ∈  𝐸(𝐺), 𝑓(𝑢) +

𝑔(𝑢𝑣) + 𝑓(𝑣) is a constant, and the function f is said to be (𝑠, 𝑑) magic labeling. If a graph G admits 

(𝑠, 𝑑) magic labeling, then it is referred to as a (𝑠, 𝑑) magic graph.  

 

Several studies contribute to graph theory: [2] Farida et al. explore graph labeling, emphasizing magic 

covering and edge super magic labeling, revealing practical applications in secret sharing and ruler 

models. [4,6] Ghodasara G.V. et al. and Baskar Babujee J.et.al. introduce prime cordial labeling, 



Communications on Applied Nonlinear Analysis 

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presenting innovative constructions and advancements in the field. [7] K. Kavitha et al. define group 

magic labeling for connected graphs, focusing on specific structures like cycles with a common vertex 

and chains of even cycles. [8]  S. K. Vaidya et al. introduce prime cordial labeling based on gcd 

properties, demonstrating its applicability to various graph structures such as gear graphs, helms, 

closed helms, and flower graphs. 

2. Methodology  

Definition 2.1: Cycle graph is a graph of n vertices with exactly n edges. Cycle graph contains cycle 

of length n 

Definition 2.2: [9]   The graph 𝐶𝑛 ⊙𝐾1,𝑚 is obtained by attaching m leaves to each vertex of the cycle 

𝐶𝑛 

Definition 2.3: The Sunlet graph, represented by 𝑆𝑛, is a graph with two n vertices that is created by 

joining n − pendant edges to the cycle 𝐶𝑛. 

Definition 2.4: A graph called friendship, denoted by 𝐹𝑛 is constructed by n triangles that share a 

vertex. 

Definition 2.5: A flower graph, or 𝐹𝑙𝑛, is a graph that is created by connecting each pendent to the 

helm's central vertex. 

Definition 2.6: The graph 𝐶𝑛+ 𝐾1.denoted by 𝑊𝑛 is called wheel graph, n is a number of vertices in the 

cycle. 

3. Results and Discussion  

Theorem 3.1 

The cycle graph 𝐶𝜂 is (𝑠, 𝑑) magic labeling 

Proof: Let 𝐶𝜂 be the cycle graph. |𝑉(𝐶𝜂)| = |𝐸(𝐶𝜂)|= η. Let 𝑉(𝐶𝜂) = {𝑣𝑙̇; 1 ≤ 𝑙̇ ≤ 𝜂} and 𝐸(𝐶𝜂) =

{𝑣𝑙̇𝑣𝑙̇+1  : 1 ≤ 𝑙̇ ≤ 𝜂 − 1} ∪ {𝑣1𝑣𝜂} 

Define the function f from the vertex set to {{𝑠, 𝑠 + 𝑑, 𝑠 + 2𝑑 … . 𝑠 + (|𝐸| + 1)},  𝑔: 𝐸(𝐺) →

{𝑑, 2𝑑, 3𝑑 … . .2(|𝐸| − 1)𝑑   } 

Labeling of Vertices 

Cases Value of 𝑙 ̇ 𝑓(𝑣𝑙̇) 

η is even 

𝑙̇ = 1 s 

2 ≤ 𝑙̇ ≤
𝜂

2
+ 1 𝑠 + (2𝑙̇ − 3)𝑑 

𝜂

2
+ 2 ≤ 𝑙̇ ≤  𝜂 𝑠 + (18 − 2𝑙)̇𝑑 

η is odd 1 ≤ 𝑙̇ ≤  𝜂 𝑠 + (𝑙̇ − 1)𝑑 



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Table 1: Labeling  of vertices for the graph  𝐶𝜂 

 

 

Labeling of Edges 

Cases Value of 𝑙 ̇ 𝑔(𝑣𝑙̇𝑣𝑙̇+1) 𝑔(𝑣𝑙̇𝑣1) 

η is 

even 

𝑙̇ = 𝜂 - 
2𝑠 + 2(|𝐸| − 1)𝑑

− (𝑓(𝑣𝑙̇) + 𝑓(𝑣1)) 

1 ≤ 𝑙̇ ≤  𝜂 − 1 
2𝑠 + 2(|𝐸| − 1)𝑑 − (𝑓(𝑣𝑙̇)

+ 𝑓(𝑣𝑙̇+1)) 
- 

η is odd 

𝑙̇ = 𝜂 - 
2𝑠 + 2(|𝐸| − 1)𝑑

− (𝑓(𝑣𝑙̇) + 𝑓(𝑣1)) 

1 ≤ 𝑙̇ ≤  𝜂 − 1 
2𝑠 + 2(|𝐸| − 1)𝑑 − (𝑓(𝑣𝑙̇)

+ 𝑓(𝑣𝑙̇+1)) 
- 

Table 2: Labeling of edges for the graph  𝐶𝜂 

                                     

Therefore 𝑓(𝑣𝑙̇) + 𝑓(𝑣𝑙̇+1) +  𝑔(𝑣𝑙̇𝑣𝑙̇+1),  𝑓 (𝑣1) + 𝑓(𝑣𝑙̇) +  𝑔(𝑣1𝑣𝑙̇) are constant equals to 

2(𝑠 + (|𝐸| − 1)𝑑). Hence the Cycle graph   𝐶𝜂 admits (𝑠, 𝑑) magic labeling. 

Theorem 3.2    The graph 𝐶𝜂 ⊙𝐾1,𝑚is (𝑠, 𝑑) magic labeling 

Proof: Let 𝑉(𝐶𝑛 ⊙𝐾1,𝑚) = {𝑣𝑖 :1 ≤ 𝑙̇ ≤ 𝜂} ∪ (𝑢
𝑙̇
𝑗

∶ 1 ≤ 𝑙̇ ≤ 𝜂, 1 ≤ 𝑙̇ ≤m} and 𝐸(𝐶𝑛 ⊙𝐾1,𝑚) =

{𝑣𝑙̇ 𝑣𝑙̇+1:1 ≤ 𝑙̇ ≤ 𝜂 − 1} ∪ {𝑣𝜂𝑣1 }∪ (𝑣𝑙̇𝑢𝑙̇
𝑗

∶ 1 ≤ 𝑙̇ ≤ 𝜂, 1 ≤ 𝑙̇ ≤m}  

 |𝑉(𝐶𝑛 ⊙𝐾1,𝑚)|=|𝐸(𝐶𝑛 ⊙𝐾1,𝑚)|= η+ηm.   

Define the function f from the vertex set to {𝑠, 𝑠 + 𝑑, 𝑠 + 2𝑑 … . 𝑠 + (|𝐸| + 1)},  𝑔: 𝐸(𝐺) →

{𝑑, 2𝑑, 3𝑑 … . .2(|𝐸| − 1)𝑑  }        

Labeling of Vertices 

Cases Value of 𝑙 ̇ 𝑓(𝑣𝑙̇) 𝑓(𝑢
𝑙̇
𝑗
) 𝑓(𝑢𝜂

𝑗
) 

η is 

even 

1 ≤ 𝑙̇

≤ 𝜂 − 1 

𝑠 + 

[(𝑙̇ − 1)(𝑚 + 1)]𝑑 
- - 

𝑙̇ = 𝜂 

 

𝑠 + 

[(𝑙̇ − 1)(𝑚 + 1)

+ 1]𝑑 

 - 

1 ≤ 𝑙̇

≤ 𝜂 − 1, 

1 ≤ 𝑗 ≤ 𝑚 

- 

𝑠 + 

[(𝑙̇ − 1)(𝑚 + 1)

+ 𝑗]𝑑 

- 



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 Labeling of edges 

Cases Value of 𝑙 ̇ 𝑔(𝑣𝑙̇𝑣𝑙̇+1) 𝑔(𝑣1𝑣𝑙̇) 𝑔(𝑣𝑙̇𝑢𝑙̇
𝑗
) 

η is 

even 

1 ≤ 𝑙̇

≤ 𝜂 − 1 

2𝑠 + 2(|𝐸| − 1)𝑑

− (𝑓(𝑣𝑙̇)

+ 𝑓(𝑣𝑙̇+1)) 

- - 

𝑙̇ = 𝜂 - 

2𝑠 + 2(|𝐸| − 1)𝑑

− (𝑓(𝑣1)

+ 𝑓(𝑣𝑙̇)) 

- 

1 ≤ 𝑙̇ ≤ 𝜂, 

1 ≤ 𝑗 ≤ 𝑚 
- - 

2𝑠 + 2(|𝐸| − 1)𝑑

− (𝑓(𝑣𝑙̇)

+ 𝑓(𝑢
𝑙̇
𝑗
)) 

η is 

odd 

1 ≤ 𝑙̇ ≤ 𝜂, 

1 ≤ 𝑗 ≤ 𝑚 
- - 

2𝑠 + 2(|𝐸| − 1)𝑑 

−(𝑓(𝑣𝑖) + 𝑓(𝑢𝑖
𝑗
)) 

1 ≤ 𝑙̇

≤  𝜂 − 1 

2𝑠 + 2(|𝐸| − 1)𝑑

− 

(𝑓(𝑣𝑙̇) + 𝑓(𝑣𝑙̇+1)) 

- - 

𝑙̇ =  𝜂 - 

2𝑠 + 2(|𝐸| − 1)

− 

(𝑓(𝑣1) + 𝑓(𝑣𝑙̇)) 

- 

Table 4: Labeling of edges of the graph 𝐶𝜂 ⊙𝐾1,𝑚 

1 ≤ 𝑗 ≤ 𝑚 - - 

𝑠 + 

[(𝜂 − 1)(𝑚 + 1)

+ 𝑗]𝑑 

η is 

odd 

1 ≤ 𝑙̇ ≤ 𝜂 

𝑠

+ [(𝑙̇ − 1)(𝑚

+ 1)]𝑑 

-  

1 ≤ 𝑙̇ ≤ 𝜂 

1 ≤ 𝑗 ≤ 𝑚 
- 

𝑠 + [(𝑙̇ − 1)(𝑚

+ 1) + 𝑗]𝑑 
- 

Table 3: Labeling of vertices of the graph 𝐶𝜂 ⊙𝐾1,𝑚 

 



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Therefore 𝑓(𝑣𝑙̇) + 𝑓(𝑢
𝑙̇
𝑗
) +𝑔(𝑣𝑙̇𝑢𝑙̇

𝑗
),  𝑓(𝑣𝑙̇) + 𝑓(𝑣𝑙̇+1) + 𝑔(𝑣𝑙̇𝑣𝑙̇+1)  , 𝑓(𝑣1) + 𝑓(𝑣𝑙̇) + 𝑔(𝑣1𝑣𝑙̇ )are 

constant equals to 2(𝑠 + (|𝐸| − 1)𝑑) .Hence the  𝐶𝜂 ⊙𝐾1,𝑚 graph  admits (𝑠, 𝑑) magic labeling. 

Theorem 3.3 The η-Sunlet graph 𝑆𝜂 is a (𝑠, 𝑑) magic labeling. 

Proof : Let G be a 𝑆𝜂 graph. 𝑉(𝑆𝜂) = {𝑢𝑙̇} ∪ {𝑣𝑙̇} ; 1 ≤ 𝑙̇ ≤ 𝜂 and 𝐸(𝑆𝜂) =  {(𝑢𝑙̇𝑣𝑙̇); 1 ≤ 𝑙̇ ≤ η}∪

(𝑣𝑙̇𝑣𝑙̇+1); 1 ≤ 𝑙̇ ≤ η-1}∪ {𝑣1𝑣𝑙̇); 𝑙̇ =η}.  

 Define the function f from the vertex set to {𝑠, 𝑠 + 𝑑, 𝑠 + 2𝑑 … . 𝑠 + (|𝐸| + 1)},  𝑔: 𝐸(𝐺) →

{𝑑, 2𝑑, 3𝑑 … . .2(|𝐸| − 1)𝑑 } 

Labeling of vertices 

Cases Value of 𝑙 ̇ 𝑓(𝑢𝑙̇) 𝑓(𝑢𝑙̇+1) 𝑓(𝑣𝑙̇) 𝑓(𝑣𝑙̇+1) 

η is 

odd 

𝑙̇ = 0 - s - 𝑠 + 𝑑 

1 ≤ 𝑙̇

≤  𝜂 − 1 
- 𝑠 + 2𝑙�̇� - 

𝑠 + (2𝑙̇

+ 1)𝑑 

η is 

even 

𝑙̇ = 0 - s - 𝑠 + 𝑑 

𝑙̇ = 𝜂 
𝑠 + (2(𝜂 − 1)

+ 1)𝑑 
- 

𝑠 + (2(𝜂

− 1))𝑑 
- 

1 ≤ 𝑙̇

≤  𝜂 − 2 
- − - 

𝑠 + (2𝑙̇

+ 1)𝑑 

Table 5: Labeling of vertices of the graph 𝑆𝜂 

Labeling of edges 

Cases Value of 𝑙 ̇ 𝑔(𝑢𝑙̇𝑣𝑙̇) 𝑔(𝑣𝑙̇𝑣𝑙̇+1) 𝑔(𝑣1𝑣𝑙̇) 

η is 

odd 

 

1 ≤ 𝑙̇ ≤ 𝜂 
2𝑠 + 2(|𝐸| − 1)𝑑

− (𝑓(𝑢𝑖) + 𝑓(𝑣𝑖)) 
- 

 

1 ≤ 𝑙̇ ≤  𝜂 −

1  
- 

2𝑠 + 2(|𝐸| − 1)𝑑

− (𝑓(𝑣𝑙̇) + 𝑓(𝑣𝑙̇+1)) 

 

𝑙̇ =  𝜂 - 
 

- 

2𝑠 + 2(|𝐸| − 1)𝑑 

−(𝑓(𝑣1) + 𝑓(𝑣𝑙̇)) 

η is 

even 
1 ≤ 𝑙̇ ≤ 𝜂 

2𝑠 + 2(|𝐸| − 1)𝑑

− (𝑓(𝑢𝑙̇) + 𝑓(𝑣𝑙̇)) 
- 

- 

1 ≤ 𝑙̇

≤  𝜂 − 1 
- 

2𝑠 + 2(|𝐸| − 1)𝑑

− (𝑓(𝑣𝑙̇) + 𝑓(𝑣𝑙̇+1)) 

- 



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𝑙̇ =  𝜂  - 

2𝑠 + 2(|𝐸| − 1)𝑑

− (𝑓(𝑣1)

+ 𝑓(𝑣𝑙̇)) 

Table 6: Labeling of edges of the graph 𝑆𝜂 

 

Therefore𝑓(𝑢𝑙̇) + 𝑓(𝑣𝑙̇) + 𝑔(𝑢𝑙̇𝑣𝑙̇),𝑓(𝑢𝑙̇) + 𝑓(𝑢𝑙̇+1) + 𝑔(𝑢𝑙̇𝑢𝑙̇+1) are constant equals to 

2(𝑠 + (|𝐸| − 1)𝑑).Hence the η-sunlet graph 𝑆𝜂 admits (𝑠, 𝑑) magic labeling. 

Theorem 3.4 The friendship graph 𝐹𝜂 is a (𝑠, 𝑑) magic labeling for  η≥ 3. 

Proof: Let 𝐹𝜂 be a friendship graph. V (𝐹𝜂)={{𝑢𝑙̇}; 1 ≤ 𝑙̇ ≤ 𝜂} ∪ {𝑣𝑙̇}; 1 ≤ 𝑙̇ ≤ 𝜂} ∪ {𝑣′} 

E (𝐹𝜂)={(𝑢𝑙̇𝑣𝑙̇); 1 ≤ 𝑙̇ ≤ 𝜂} ∪ {(𝑣′𝑢𝑙̇); 1 ≤ 𝑙̇ ≤ 𝜂} ∪ (𝑣′𝑣𝑙̇); 1 ≤ 𝑙̇ ≤ 𝜂 

Here |V(𝐹𝜂)|=2𝜂 + 1,|E(𝐹𝜂)|=3𝜂 .Define the function f from the vertex set to  

{𝑠, 𝑠 + 𝑑, 𝑠 + 2𝑑 … . 𝑠 + (|𝐸| + 1)},  𝑔: 𝐸(𝐺) → {𝑑, 2𝑑, 3𝑑 … . .2(|𝐸| − 1)𝑑  } 

Labeling of vertices 

𝑓(𝑣′)=𝑢𝜂−1 + 𝜂𝑑 

Value 𝑙 ̇ 𝑓(𝑢𝑙̇) 𝑓(𝑢𝑙̇+1) 𝑓(𝑣𝑙̇) 

𝑙̇ = 1 𝑠 + (𝜂 + 1)𝑑 - - 

𝑙̇ = 𝜂 𝑠 + 𝜂𝑑 -  𝑠 

1 ≤ 𝑙̇ ≤ 𝜂 − 1 - -       𝑠 + 𝑙�̇� 

1 ≤ 𝑙̇ ≤ 𝜂 − 2 - 𝑢𝑙̇ + 𝑑 - 

Table 7: Labeling of vertices of the graph 𝐹𝜂 

     

Labeling of edges 

Value 𝑙 ̇ 𝑔(𝑢𝑙̇𝑣𝑙̇) 𝑔(𝑣′𝑢𝑙̇) 𝑔(𝑣′𝑣𝑙̇) 

1 ≤ 𝑙̇ ≤ 𝜂 
2𝑠 + 2(|𝐸| − 1)𝑑 

−(𝑓(𝑢𝑙̇) + 𝑓(𝑣𝑙̇)) 

2𝑠 + 2(|𝐸| − 1)𝑑

− (𝑓(𝑣′) + 𝑓(𝑢𝑙̇)) 

2𝑠 + 2(|𝐸| − 1)𝑑 

−(𝑓(𝑣′) + 𝑓(𝑣𝑙̇)) 

                                     Table 8: Labeling of edges of the graph 𝐹𝜂 

Therefore 𝑓(𝑢𝑙̇) + 𝑓(𝑣𝑙̇) + 𝑔(𝑢𝑙̇𝑣𝑙̇),𝑓(𝑣′) + 𝑓(𝑢𝑙̇) + 𝑔(𝑣′𝑢𝑙̇) and 𝑓(𝑣′) + 𝑓(𝑣𝑙̇) + 𝑔(𝑣′𝑣𝑙̇) are 

constant equals to 2(𝑠 + (|𝐸| − 1)𝑑).Hence the friendship graph 𝐹𝜂 admits (𝑠, 𝑑) magic labeling. 

Theorem 3.5 The Flower graph 𝐹𝑙𝜂 is a (𝑠, 𝑑) magic labeling. 

Proof: Let 𝐺 be a graph 𝐹𝑙𝜂 



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|𝑉(𝐺)|=2𝜂 + 1  , |𝐸(𝐺)|=4𝜂  

𝑉(𝐺)= {𝑢𝑙̇; 1 ≤ 𝑙̇ ≤ 𝜂} ∪ {𝑣𝑙̇ ; 1 ≤ 𝑙̇ ≤ 𝜂} ∪ {𝑣} and 𝐸(𝐺)= {𝑢𝑙̇𝑢𝑙̇+1:1≤ 𝑖 ≤η-1}∪ {𝑢1𝑢𝜂} ∪

{𝑢𝑙̇𝑣𝑙̇; 1 ≤ 𝑙̇ ≤ 𝜂} ∪ {𝑣𝑢𝑙̇; ≤ 𝑙̇ ≤ 𝜂} ∪ {𝑣𝑣𝑙̇; ≤ 𝑙̇ ≤ 𝜂} 

Define the function f from the vertex set to {{𝑠, 𝑠 + 𝑑, 𝑠 + 2𝑑 … . 𝑠 + (|𝐸| + 1)},  𝑔: 𝐸(𝐺) →

{𝑑, 2𝑑, 3𝑑 … . .2(|𝐸| − 1)𝑑   } 

Labeling of vertices 

Cases Value of 𝑙 ̇ 𝑓(𝑢𝑙̇) 𝑓(𝑣𝑙̇) 𝑓(𝑣𝑙̇+1) 𝑓(𝑢𝑙̇+1) 

η is 

odd 

f(v)=s+(4η-2)d 

𝑙̇ =  0 - - 𝑠 𝑠 + 𝑑 

1 ≤ 𝑙̇ ≤ 𝜂 − 1 - - 𝑠 + 2𝑙�̇� 𝑠 + (2𝑙̇ + 1)𝑑 

η is 

even 

𝑓(𝑣) = 𝑠 + (𝜂 − 2)𝑑 

𝑙̇ = 0 - - 𝑠 𝑠 + 𝑑 

1 ≤ 𝑙̇ ≤ 𝜂 − 2  - 𝑠 + 2𝑙�̇� 𝑠 + (2𝑙̇ + 1)𝑑 

𝑙̇  =  𝜂 𝑢𝜂−1 + 𝑑 𝑣𝜂−1 + 3𝑑 - - 

Table 9: Labeling of vertices of the graph 𝐹𝑙𝜂 

Labeling of  edges 

Cases Value of 𝑙 ̇ 𝑔(𝑢𝑙̇𝑢𝑙̇+1) 𝑔(𝑢𝑙̇𝑣𝑙̇) 𝑔(𝑢𝑙̇𝑢1) 𝑔(𝑣𝑢𝑙̇) 𝑔(𝑣𝑣𝑙̇) 

η is 

odd 

1 ≤ 𝑙̇ ≤ 𝜂 − 1 

2𝑠 + 

2(|𝐸| −

1)𝑑 −

(𝑓(𝑢𝑙̇) +

𝑓(𝑢𝑙̇+1) 

- - - - 

1 ≤ 𝑙̇ ≤ 𝜂 - 

2𝑠 + 

2(|𝐸| − 1)𝑑

− (𝑓(𝑢𝑙̇)

+ 𝑓(𝑣𝑙̇)) 

- 

2𝑠 + 

2(|𝐸| − 1)𝑑

− (𝑓(𝑣)

+ 𝑓(𝑢𝑙̇)) 

2𝑠 + 

2(|𝐸| − 1)𝑑

− (𝑓(𝑣)

+ 𝑓(𝑣𝑙̇)) 

𝑙̇  =  𝜂 - - 

2𝑠 + 

2(|𝐸| − 1)𝑑

− (𝑓(𝑢𝑙̇)

+ 𝑓(𝑢1)) 

- - 



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η is 

even 

1 ≤ 𝑙̇ ≤ 𝜂 − 1 

2𝑠 + 

2(|𝐸| − 1)𝑑

− (𝑓(𝑢𝑙̇)

+ 𝑓(𝑢𝑙̇+1) 

- - - - 

1 ≤ 𝑙̇ ≤ 𝜂 - 

2𝑠 + 

2(|𝐸| − 1)𝑑

− (𝑓(𝑢𝑙̇)

+ 𝑓(𝑣𝑙̇)) 

- 

2𝑠 + 

2(|𝐸| − 1)𝑑

− (𝑓(𝑣)

+ 𝑓(𝑢𝑙̇)) 

2𝑠 + 

2(|𝐸| − 1)𝑑

− (𝑓(𝑣)

+ 𝑓(𝑣𝑙̇)) 

𝑙̇ = η - - 

2𝑠 + 

2(|𝐸| − 1)𝑑

− (𝑓(𝑢𝑙̇)

+ 𝑓(𝑢1)) 

- - 

Table 10 Labeling of Edges of the graph 𝐹𝑙𝜂 

Therefore𝑓(𝑢𝑙̇) + 𝑓(𝑣𝑙̇) + 𝑔(𝑢𝑙̇𝑣𝑙̇),𝑓(𝑢𝑙̇) + 𝑓(𝑢𝑙̇+1) + 𝑔(𝑢𝑙̇𝑢𝑙̇+1), (𝑓(𝑢𝑙̇) + 𝑓(𝑢1) + 𝑔(𝑢𝑙̇𝑢1), 

𝑓(𝑣) + 𝑓(𝑢𝑙̇) + 𝑔(𝑣𝑢𝑙̇)  and  𝑓(𝑣) + 𝑓(𝑣𝑙̇) + 𝑔(𝑣𝑣𝑙̇)    are constant equals to  2(𝑠 +

(|𝐸| − 1)𝑑).Hence the Flower  graph 𝐹𝑙𝜂 admits (𝑠, 𝑑) magic labeling. 

Figure1  : Flower graph 𝑭𝒍𝟕 

 

 

 

 

 

 

 

 

 

 

 

Theorem 3.6 The Wheel graph  𝑊𝜂 is a (𝑠, 𝑑) magic labeling η≥ 4 



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Proof : Let G = 𝑊𝜂 . |𝑉(𝑊𝜂)|=𝜂 + 1  , |𝐸(𝑊𝜂)|=2𝜂. 

Define the function f from the vertex set to {{𝑠, 𝑠 + 𝑑, 𝑠 + 2𝑑 … . 𝑠 + (|𝐸| + 1)},  𝑔: 𝐸(𝐺) →

{𝑑, 2𝑑, 3𝑑 … . .2(|𝐸| − 1)𝑑 } 

Labeling of vertices 

Cases Value of 𝑙 ̇ f (𝑣𝑙̇) f(𝑣𝜂+1−𝑙̇) 

η is 

odd 

1 ≤ 𝑙̇ ≤ 𝜂 𝑠 + (𝑙̇ − 1)𝑑 - 

𝑙̇  =  𝜂 + 1 𝑠 + 2(𝜂 − 1)𝑑 - 

η is 

even 

𝑙̇ = 1 𝑠 - 

𝑙̇ = 𝜂 + 1 𝑠 + 2(𝜂 − 1)𝑑 - 

2 ≤ 𝑙̇

≤
𝜂 + 2

2
 

- 𝑠 + (2𝑙̇ − 3)𝑑 

1 ≤ 𝑙̇

≤
𝜂 − 2

2
 

𝑠 + 2𝑙�̇� - 

Table 11: Labeling  of vertices of the graph 𝑊𝜂 

                             Labeling of Edges 

Cases Value of 𝑙 ̇ 𝑔(𝑣𝑙̇𝑣𝑙̇+1) 𝑔(𝑣𝑙̇𝑣1) 𝑔(𝑣𝜂+1𝑣𝑙̇) 

η is 

odd 

𝑙̇ =  𝜂 - 
2𝑠 + 2(|𝐸| − 1)𝑑

− (𝑓(𝑣𝑙̇) + 𝑓(𝑣1)) 
- 

1 ≤ 𝑙̇ ≤ 𝜂 - - 
2𝑠 + 2(|𝐸| − 1)𝑑

− (𝑓(𝑣𝜂+1) + 𝑓(𝑣𝑙̇)) 

1 ≤ 𝑙̇

≤ 𝜂 − 1 

2𝑠 + 2(|𝐸| − 1)𝑑

− (𝑓(𝑣𝑙̇) + 𝑓(𝑣𝑙̇+1)) 
- - 

η is 

even 

1 ≤ 𝑙̇

≤ 𝜂 − 1 

2𝑠 + 2(|𝐸| − 1)𝑑

− (𝑓(𝑣𝑙̇) + 𝑓(𝑣𝑙̇+1)) 
- - 

1 ≤ 𝑙̇ ≤ 𝜂 - - 
2𝑠 + 2(|𝐸| − 1)𝑑

− (𝑓(𝑣𝜂+1) + 𝑓(𝑣𝑙̇)) 

𝑙̇ =  𝜂 - 
2𝑠 + 2(|𝐸| − 1)𝑑

− (𝑓(𝑣𝑙̇) + 𝑓(𝑣1)) 
- 

Table 12: Labeling of edges of the graph 𝑊𝜂 

Therefore𝑓(𝑣𝑙̇) + 𝑓(𝑣𝑙̇+1) + 𝑔(𝑣𝑙̇𝑣𝑙̇+1), 𝑓(𝑣𝑙̇) + 𝑓(𝑣1) + 𝑔(𝑣𝑙̇𝑣1)and𝑓(𝑣𝜂+1) + 𝑓(𝑣𝑙̇) + 𝑔(𝑣𝜂+1𝑣𝑙̇)  

are constant equals to 2(𝑠 + (|𝐸| − 1)𝑑) . Hence the wheel graph   𝑊𝜂 admits (𝑠, 𝑑) magic labeling. 

 



Communications on Applied Nonlinear Analysis 

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Figure 2 : wheel graph   𝑾𝟔  

 

4. Conclusion  

The discovery of the (s,d) Magic Labeling for diverse cycle graphs underscores the versatility and 

applicability of this labeling scheme across different graph structures. This finding not only expands 

our understanding of labeling techniques but also opens up possibilities for further exploration in graph 

theory and its applications. Moreover, the ability to apply this labeling to various cycle graphs suggests 

potential advancements in areas such as network design, communication systems, and algorithm 

development. Further research could focus on refining the (s,d) Magic Labeling method or exploring 

its implications in other graph theoretic contexts. Future research will examine the (𝑠, 𝑑) Magic   

labeling of additional graphs and some graph families. 

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[3]. Gallian JA. A Dynamic Survey of Graph Labeling. The Electronic Journal of Combinatorics. 2022. Available 

from: http://www.combinatorics.org 

[4]. Ghodasara, G. V., & Jena, J. P.. Prime cordial labeling of some special graph families. International Journal of 

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[6]. J. Baskar Babujee and S. Babitha, Prime Cordial Labeling on Graphs, International Journal Of Mathematical Sciences, 

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https://doi.org/10.1063/5.0140418