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Bi-Magic and Anti-Magic Labeling on M-Fuzzy Graph 

 

Suruthi. S1, Shanmuga Sundari. M2, Sivakami. L3* 

1, 3Department of Mathematics and Statistics, Faculty of Science and Humanities, SRM Institute of Science 

and Technology, Kattankulathur- 603203, Chengalpattu District, Tamil Nadu, India 
2Department of Mathematics, Sir Theagaraya College, Chennai- 600021, Tamil Nadu, India 

1E mail id: ss5112@srmist.edu.in  
2E mail id: shanmugasundarim@stcchennai.edu.in 

3*Corresponding author E mail id: sivakaml@srmist.edu.in 

 

Article History: 

Received: 30-09-2023 

Revised: 04-11-2023 

Accepted: 25-11-2023 

Abstract:  

The conceptions of M-fuzzy anti-magic and M-fuzzy bi-magic labeling are developed 

by the proposed algorithms. M-fuzzy anti-magic labeling for standard graphs such as 

star and path graph explained. M-fuzzy bi-magic labeling for a few graphs such as star 

and bi-star is determined. Also, we explored the characteristics of such labeling 

applied to these graphs.  

Keywords: M-fuzzy graph, M-fuzzy labeling, M-fuzzy edge anti-magic graph, M-

fuzzy vertex anti-magic graph.  

1. Introduction 

Real world problems are often represented with graph theory. Because of uncertainties or haziness in 

system parameters, graphs do not properly represent all systems nowadays. Graphs can represent social 

networks, for example, by representing the accounts (people, institutions, etc.) as vertices and the 

relations between the accounts as edges. The accuracy of representation should be enhanced by 

fuzziness if the relations among accounts are measured by their frequency of contact. Rosenfeld [2] 

introduced fuzzy graphs for the first time. A wealth of research is conducted in fuzzy graph theory 

after that. Crisp graphs and fuzzy graphs have similar structural features. There is a separate 

importance to fuzzy graphs when there is uncertainty about vertices and edges. In real life, fuzzy graphs 

are found in many situations since the world is filled with uncertainty. In fuzzy graph theory, there are 

a variety of branches that can be explored. 

Fuzzy labeling graph (FLG) is a rapidly evolving area of research in the field of graph theory that has 

procured considerable notice in recent years because of its remarkable real-world applications. In 

essence, it involves assigning numerical values, or labels, to the edges or vertices of a fuzzy graph 

(FG), where the labels are allowed to take on imprecise or uncertain values. This enables fuzzy graph 

labeling to model complex systems that are inherently uncertain, such as social networks, biological 

systems, and computer networks. Gallian [1] discussed more graph labeling in his dynamic survey of 

graph labeling. An anti-magic graph was first introduced in 1990 by Hartsfield and Ringel [13]. 

Nagoorgani et al., [4] presented the novel idea of fuzzy labeling, while Basker Babujee [8] presented 

the concept of bi-magic labeling, which involves the identification of two magic values. Prathap et al., 

[15] also postulated bi magic labeling for cycle related graph. Bibi et al., [6] discussed the fuzzy vertex 

graceful labeling for some family of graphs. Devi et al., [7] developed the idea of fuzzy bi-magic 



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(FBM) labeling. Devi et al., [5] postulated the idea of fuzzy anti- magic labeling. Sujatha N, Dharman, 

Thirusangu [9] proposed Triangular anti-magic fuzzy labeling in an algorithmic approach for some 

standard graphs. Thirusangu et al., [10] developed fuzzy bi- magic and anti- magic labeling for star 

graphs. Bharathi et al., [14] developed the fuzzy proper bi magic labeling for tadpole graph.  Jahir 

Hussain et al., [11] proposed a new class of fuzzy graph. Shyamala Malini and Prathika [12] developed 

the idea of M-fuzzy magic labeling for special graphs. In this article, we present the existence of M-

fuzzy edge(vertex) anti- magic labeling (M-FEAM and M-FVAM) of path and star, M- fuzzy bi- magic 

labeling (M-FBML) of star and bi- star. The end of the article will present the conclusion.  

All the basic definitions are followed, as in [3], [4], [5], [7], [8], [9], [10], and [12].  

2. Preliminaries 

Definition 2.1. [5] A FLG is called as FEAM graph if 1 1 1( ) ( , ) ( )u u v v  + + has different value for 

all ,u v V .  

Definition 2.2: [9] “A FLG is called an FVAM graph if”’ G is a 1-1 correspondence 

 : ( ) 1, 2,3, ,| ( ) |f E G E G→ in which for any two different vertices v  and w , the total of the labels 

on “edges incident to v  distinct from the total of labels on edges incident to w .”  

Definition 2.3. [10] “An FG is said to admit edge bi-magic labeling if the total of membership values 

of edges and vertices incident at the vertices are either 1k or 2k where 1k and 2k are constants and 

denoted by 0 ( )Bm G . A FLG which admits an edge bi- magic labeling is called a FEBM labeling 

graph.” 

Definition 2.4. [12] “A graph 1 1( , )G   is said to be a M-fuzzy labeling graph if  1 : [0,1]V → and 

1 : [0,1]V V  → is bijective such that the membership value of edges and vertices are distinct and 

satisfied the following conditions 1 1( , ) ( )u v u   or 1 1 1( ) ( , ) ( )u u v v    , then 

1 1 1( , ) ( ) ( ) ,u v u v u v V      ”.  

3. M-Fuzzy Anti-magic Graphs 

Definition 3.1: A M- FLG is said to be a M-FEAM graph if 1 1 1( ) ( , ) ( )u u v v  + + has different value 

for all ,u v V , which is denoted by 0 ( )Am G .  

Definition 3.2: A M- FLG is said to be a M-FVAM graph if  𝐺 is 1-1 correspondence 

 : ( ) 1, 2,3, ,| ( ) |f E G E G→ in which for any two different vertices v  and w ,  the total of labels on 

edges incident to v  distinct from the total of labels on edges incident to w .  

 

 

 

 

 



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M-Fuzzy Labeled Path graph nP : 

_________________________________________________________________________________ 

Algorithm 1 

_________________________________________________________________________________ 

M-FL of edges and vertices of nP  with, where n  is the path length and (0,1]z  

1. Input: Enter values for n and z .  

2. Maximum number of vertices are 1n+  .  

3. Vertex labeling be, 

for 1i = to 1n+  

{ 

1( ) ( )iv n i z = +  where 1 1i n  +  

} 

4. Edge labeling be,  

for 1i = to 1n+      

      if 1 ≤ 𝑖 ≤ 𝑛 

{ 

   1 1 1 1 1 1 1 1( , ) max ( ), ( ) min ( ), ( ) ( 1)i i i i i iv v v v v v i z    + + += − + −  

} 

5. Output: Print the values of vertices and edges.  

_________________________________________________________________________________ 

Theorem 3.1. Let 1 1( , )nP   be the M-fuzzy labeled path graph for all 2n then nP admits M-FEAM 

labeling.  

Proof. Let (0,1]z →  such that 

2

2
1

1

3
1 1

1 1 1

1
, 1 49

10

1
, 49 49 (45 10 ), 1

10

1
, 49 (45 10 ) 49 (45 10 ), 1,2,3,

10

i
t

k
t

i k

i i
t t

k
t t

i k i k

n

z n wherek

n wherek

+
=
 

+
= =
    +



  




=   +  =




+    +  =




 

 

The vertex membership values and edge membership values are defined using algorithm 1.  

1 1: [0,1] ( ) ( ) ,1 1i iV v n i z v V i n →  = +     +  

1 : [0,1]V V  →  such that 



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   1 1 1 1 1 1 1 1( , ) max ( ), ( ) min ( ), ( ) ( 1) ,1i i i i i iv v v v v v i z i n    + + += − + −    

We discuss M-FEAM labeling of path graph, 

0 1 1 1 1 1( ) ( ) ( , ) ( ),1n i i i iAm P v v v v i n  + += + +    

                  1 1 1 1 1 1( ) max ( ), ( ) min ( ), ( ) ( 1) ( 1)i i i in i z v v v v i z n i z   + += + + − + − + + +  

               (2 3 1)n i z= + +  

From the above, it is evident that the M-fuzzy labeled path graph nP  admits M-FEAM labeling.  

Example 3.1.  

  

Fig. 1. 499P  M-FEAM path graph 

Theorem 3.2. Let 1 1( , )nP    be the M-fuzzy labeled path graph for all 2n  and if n  is even then nP  

admits M-FVAM labeling.  

Proof. Given 1 1( , )nP    be the fuzzy labeled path graph.  

To prove that M-fuzzy labeled path graph 1 1( , )nP    satisfies the condition of M-FVAM 

labeling.  

That is to prove that for any two vertices u  and v  in nP , the total of the grade membership on 

the edges incident at the vertex u  is distinct from the total of the grade membership on the edges 

incident at the vertex v .  

The vertex membership values and edge membership values are defined using algorithm 1.  

 Let (0,1]z →  such that 

2

2
1

1

3
1 1

1 1 1

1
, 1 49

10

1
, 49 49 (45 10 ), 1

10

1
, 49 (45 10 ) 49 (45 10 ), 1,2,3,

10

i
t

k
t

i k

i i
t t

k
t t

i k i k

n

z n wherek

n wherek

+
=
 

+
= =
    +



  




=   +  =




+    +  =




 

 



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1 1: [0,1] ( ) ( ) ,1 1i iV v n i z v V i n →  = +     +                (3.1) 

1 : [0,1]V V  →  such that 

   1 1 1 1 1 1 1 1( , ) max ( ), ( ) min ( ), ( ) ( 1) ,1i i i i i iv v v v v v i z i n    + + += − + −                 (3.2) 

Also, the total of the edge labels incident at 

1

( )

( ) ( , )
i

i i i

u N v

v Wt v u v


= =                                                                                             (3.3) 

Where ( )iN v  be the neighbourhood vertices of iv  for all 1i = to 1n+  and ( )iWt v be the weight of iv . 

From equation (3.1), (3.2) and (3.3) for any two vertices pv  and qv with p q , ( )pWt v and ( )qWt v  

have distinct values.  

Thus, a path graph of even length admits M-FVAM labeling.  

Example 3.2.  

 

Fig. 2. 500P  M-FVAM path graph 

M-Fuzzy Labeled Star graph 1,nS : 

_________________________________________________________________________________ 

Algorithm 2 

_________________________________________________________________________________ 

M-Fuzzy labeling of edges and vertices of 1,nS with 3n , n  is the no. of pendent edges and (0,1]z  

1. Input: Enter values for n and z .  

2. Maximum number of vertices are 1n+ .  

3. Vertex labeling be,  

 for 1i = to 1n+  

{ 

 if 1i = or 2i =  

 1( ) ( )iv n i z = +  

 if 3 i n   

 1( ) ( 1)iv n i z = + +  

} 

if 3n  

 1( ) ( 3)w n z = +  



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4. Edge labeling be,  

 for 1i = to 1n+  

 { 

 if 1i =  

    1 1 1 1 1( , ) max ( ), ( ) min ( ), ( )i i iw v w v w v z    = − −  

 if 2i =  

    1 1 1 1 1( , ) max ( ), ( ) min ( ), ( )i i iw v w v w v z    = − +  

 if 3 i n   

    1 1 1 1 1( , ) max ( ), ( ) min ( ), ( ) 2i i iw v w v w v z    = − +  

} 

5. Output: Print the values of vertices and edges.  

_________________________________________________________________________________ 

Theorem 3.3. Let 1, 1 1( , )nS    be the M-fuzzy labeled star graph for all 𝑛 ≥ 3. Then 1,nS  is a M-FEAM 

labeling.  

Proof. Let (0,1]z →  such that 

2

2
1

1

3
1 1

1 1 1

1
, 1 49

10

1
, 49 49 (45 10 ), 1

10

1
, 49 (45 10 ) 49 (45 10 ), 1,2,3,

10

i
t

k
t

i k

i i
t t

k
t t

i k i k

n

z n wherek

n wherek

+
=
 

+
= =
    +



  




=   +  =




+    +  =




 

 

The vertex membership values and edge membership values are defined using algorithm 2.  

1 : [0,1]V →  such that 

1( ) ( ) , 1, 2iv n i z i = + =  

1( ) ( 1) ,3iv n i z i n = + +    

1( ) ( 3) , 3w n z n = +    

1 : [0,1]V V  →  such that 

   1 1 1 1 1( , ) max ( ), ( ) min ( ), ( ) , 1i i iw v w v w v z i    = − − =  

   1 1 1 1 1( , ) max ( ), ( ) min ( ), ( ) , 2i i iw v w v w v z i    = − + =  



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   1 1 1 1 1( , ) max ( ), ( ) min ( ), ( ) 2 ,3i i iw v w v w v z i n    = − +    

We discuss M-FEAM labeling in three cases, 

Case 1:  

0 1, 1 1 1( ) ( ) ( , ) ( ), 1n i iAm S w w v v i  = + + =  

                   1 1 1 1( 3) max ( ), ( ) min ( ), ( ) ( )i in z w v w v z n i z   = + + − − + +  

                (2 4 )n i z= + +  

Case 2: 

0 1, 1 1 1( ) ( ) ( , ) ( ), 2n i iAm S w w v v i  = + + =  

                   1 1 1 1( 3) max ( ), ( ) min ( ), ( ) ( )i in z w v w v z n i z   = + + − + + +  

                (2 5 )n i z= + +  

Case 3:  

0 1, 1 1 1( ) ( ) ( , ) ( ),3n i iAm S w u v v i n  = + +    

                   1 1 1 1( 3) max ( ), ( ) min ( ), ( ) 2 ( 1 )i in z w v w v z n i z   = + + − + + + +  

                (2 4 2 )n i z= + +  

Hence, in Case 1, Case 2, and Case 3, the M-fuzzy labeled star graph admits M-FEAM labeling.  

Theorem 3.4. Let 1, 1 1( , )nS   be the M-fuzzy labeled star graph for all 3n . Then 1,nS  is a M-FVAM 

labeling. 

Proof. Given 1, 1 1( , )nS    be the fuzzy labeled star graph.  

To prove that fuzzy labeled  1, 1 1( , )nS   satisfies the condition of M-FVAM labeling.  

That is to prove that for any two vertices u  and v  in 1,nS , the total of the grade membership 

on the edges incident at the vertex u  is distinct from the total of the grade membership on the edges 

incident at the vertex v .  

Let (0,1]z →  such that 



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2

2
1

1

3
1 1

1 1 1

1
, 1 49

10

1
, 49 49 (45 10 ), 1

10

1
, 49 (45 10 ) 49 (45 10 ), 1,2,3,

10

i
t

k
t

i k

i i
t t

k
t t

i k i k

n

z n where k

n where k

+
=
 

+
= =
    +



  




=   +  =




+    +  =




 

 

The vertex membership values and edge membership values are defined using algorithm 2.  

1 : [0,1]V →  such that         

1( ) ( ) , 1, 2iv n i z i = + =          (3.4) 

1( ) ( 1) ,3iv n i z i n = + +            (3.5) 

1( ) ( 3) , 3w n z n = +            (3.6) 

1 : [0,1]V V  →  such that 

   1 1 1 1 1( , ) max ( ), ( ) min ( ), ( ) , 1i i iw v w v w v z i    = − − =      (3.7) 

   1 1 1 1 1( , ) max ( ), ( ) min ( ), ( ) , 2i i iw v w v w v z i    = − + =     (3.8) 

   1 1 1 1 1( , ) max ( ), ( ) min ( ), ( ) 2 ,3i i iw v w v w v z i n    = − +       (3.9) 

Also, the total of the edge labels incident at 

1

( )

( ) ( , )
i

i i i

w N v

v Wt v w v


= =                                                                              (3.10) 

Where ( )iN v  be the neighbourhood vertices of iv  for all 1i = to 1n+  and ( )iWt v be the weight of iv .  

From equation (3.4), (3.5) (3.6), (3.7), (3.8), (3.9) and (3.10) for any two vertices pv  and qv with  

p q , ( )pWt v and ( )qWt v  have distinct values.  

Thus, a star graph 1, 3nS n  admits M-FVAM labeling.  

 

 

 

 



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Example 3.3.  

 

Fig. 3. 1,300S  M-FEAM and M-FVAM star graph 

4. M-Fuzzy Bi-magic Graph 

Definition 4.1: A M-FLG is said to be a M-FEBM graph if 1 1 1( ) ( , ) ( )u u v v  + +  have either 1k  or 

2k  where 1k  and 2k  are constants. 

M-fuzzy labeled Star 1,nS : 

_________________________________________________________________________________ 

Algorithm 3 

 

M-Fuzzy labeling of edges and vertices of 1,nS  with 4n  , n  is the no. of pendent edges and (0,1]z  

 

1. Input: Enter values for n and z .  

2. Maximum number of vertices are 1n+ .  

3. Vertex labeling be,  

 for 1i = to 1n+  

 { 

if 1i =   or 2i =  or 3i =  

 1( ) (2 2 )iv n i z = + −  

 if 4 i n   

 1( ) (2 1 )iv n i z = + −  

 } 

if 4n   

 1( ) (2 2)d n z = −  

4. Edge labeling be,  

 for 1i = to 1n+  



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 { 

 if 1i =  

    1 1 1 1 1( , ) max ( ), ( ) min ( ), ( ) 2i i id v d v d v z    = − −  

 if 2i =  

    1 1 1 1 1( , ) max ( ), ( ) min ( ), ( )i i id v d v d v    = −  

 if 3i =  

    1 1 1 1 1( , ) max ( ), ( ) min ( ), ( ) 2i i id v d v d v z    = − +  

 if 4 i n   

    1 1 1 1 1( , ) max ( ), ( ) min ( ), ( ) 3i i id v d v d v z    = − +  

 }  

5. Output: Print the values of vertices and edges.  

_________________________________________________________________________________ 

Theorem 4.1. Let 1, 1 1( , )nS    be the M-fuzzy labeled star graph for all 4n . Then 1,nS  is a M-FBML.  

Proof. Let (0,1]z →  such that 

2

2
1

1

3
1 1

1 1 1

1
, 1 49

10

1
, 49 49 (45 10 ), 1

10

1
, 49 (45 10 ) 49 (45 10 ), 1,2,3,

10

i
t

k
t

i k

i i
t t

k
t t

i k i k

n

z n where k

n where k

+
=
 

+
= =
    +



  




=   +  =




+    +  =




 

 

The vertex membership values and edge membership values are defined using algorithm 3.  

1 : [0,1]V →  such that 

1( ) (2 2 ) , 1, 2,3iv n i z i = + − =  

1( ) (2 1 ) , 4iv n i z i n = + −    

1( ) (2 2) , 4d n z n = −    

1 : [0,1]V V  →  such that 

   1 1 1 1 1( , ) max ( ), ( ) min ( ), ( ) 2 , 1i i id v d v d v z i    = − − =  

   1 1 1 1 1( , ) max ( ), ( ) min ( ), ( ) , 2i i id v d v d v i    = − =  



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   1 1 1 1 1( , ) max ( ), ( ) min ( ), ( ) 2 , 3i i id v d v d v z i    = − + =  

   1 1 1 1 1( , ) max ( ), ( ) min ( ), ( ) 3 ,3i i id v d v d v z i n    = − +   . 

For all edges idv  in 1,nS , 1 1 1( ) ( , ) ( ) 4i id d v v nz  + + = or (4 1)n z− . Fix 1 4k nz= ; and 2 (4 1)k n z= −

. So, M-fuzzy 1,nS  satisfies all conditions of a bi-magic graph, and M-fuzzy star graph 1,nS  is bi-magic.  

Example 4.1.  

 

Fig. 4. 1,300S M-FBM star graph 

M-Fuzzy Labeled Bi-star graph ,n nB : 

_________________________________________________________________________________ 

Algorithm 4 

_________________________________________________________________________________ 

M-fuzzy labeling of edges and vertices of bi-star graph ,n nB  with 1n  where n is the number of 

pendent edges and (0,1]z  

1. Input: Enter values for n and z .  

2. Maximum number of vertices are 2 2n+ .  

3. Vertex labeling be,  

 for 1i = to 2n  

 

 { 

 if 1 i n   

 1( ) ( )iv n i z = +  

 if 1 2n i n+    

 1( ) (2 1 )iv n i z = + +  



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 } 

if 1n  

 1( ) (2 1)u n z = +  

if 1n  

 1( ) (4 3)w n z = +  

4. Edge labeling be,  

 for 1i = to 2n  

 { 

 if 1 i n   

    1 1 1 1 1( , ) max ( ), ( ) min ( ), ( )i i iu v u v u v    = −  

 if 1 2n i n+    

    1 1 1 1 1( , ) max ( ), ( ) min ( ), ( ) 2i i iw v w v w v nz    = − +  

 } 

if 1n  

    1 1 1 1 1( , ) max ( ), ( ) min ( ), ( ) 2u w u w u w nz    = − +  

5. Output: Print the values of vertices and edges.  

_________________________________________________________________________________ 

Theorem 4.2. Let , 1 1( , )n nB    be the M-fuzzy labeled star graph for all 𝑛 ≥ 1. Then ,n nB is a M-

FBML.  

Proof. Let (0,1]z →  such that 

2

3
0

0

4
0 0

0 0 1

1
, 1 24

10

1
, 24 24 (225 10 ), 0

10

1
, 24 (225 10 ) 24 (225 10 ), 0,1,2,3,

10

i
t

k
t

i k

i i
t t

k
t t

i k i k

n

z n where k

n where k

+
=
 

+
= =
    +



  




=   +  =




+    +  =




 

 

 

The vertex membership values, and edge membership values are defined using algorithm 4.  

1 : [0,1]V →  such that 

1( ) ( ) ,1iv n i z i n = +     

1( ) (2 1 ) , 1 2iv n i z n i n = + + +    



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1( ) (2 1) , 1u n z n = +   

1( ) (4 3) , 1w n z n = +   

1 : [0,1]V V  →  such that 

   1 1 1 1 1( , ) max ( ), ( ) min ( ), ( ) ,1i i iu v u v u v i n    = −    

   1 1 1 1 1( , ) max ( ), ( ) min ( ), ( ) 2 , 1 2i i iw v w v w v nz n i n    = − + +  

   1 1 1 1 1( , ) max ( ), ( ) min ( ), ( ) 2 , 1u w u w u w nz n    = − +   

For all edges iuv  in ,n nB , 1 1 1( ) ( , ) ( ) (4 2)iu u v v n z  + + = +  or (10 6)n z+ . Fix 1 (4 2)k n z= + and

2 (10 6)k n z= + . So, M-fuzzy ,n nB  satisfies all conditions of a bi-magic graph, and the M-fuzzy bistar 

graph ,n nB is bi-magic.  

Example 4.2.  

 

Fig.5. 249,249B M-FBM bi-star graph 

5. Findings 

1. Every M-FBM graph is a FLG, but the converse is not true.  

2. Every M-FEAM and M-FVAM graph is a FLG, but the converse is not true.  

3. If 𝐺 is a M-FBM graph then ( ) ( )d u d v  for any pair of vertices , ( )u v V G .  

4. If 𝐺 is a M-FEAM and M-FVAM path graph of even length then ( ) ( )d u d v  for any pair of 

vertices , ( )u v V G .  



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5. Every M-FEAM path graph of odd length must have one pair of vertices whose degrees are 

same.  

 

6. Conclusion 

The proposed article introduces the M-FBM, M-FEAM, and M-FVAM labeling concepts, discussing 

M-FBM labeling for bi-star and star graphs, as well as M-FEAM and M-FVAM labeling for star and 

path graphs. Additionally, we examined the properties of M-FBML, M-FEAM, and M-FVAM graphs, 

and will expand our analysis to include some other specialised graph classes. 

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