Approach of the value of a rent when non-central moments of the capitalization factor are known: an R application with interest rates following normal and beta distributions


Ratio Mathematica                                                                            Volume 42, 2022 

 

              
 

 

 

 

Arboricity and Span in M-Fuzzy 

Chromatic Index 

 

 
 

S. Yahya Mohamed* 

S. Suganthi† 

  
 

 

 

Abstract  

A fuzzy matching is a set of edges in which no two edges incident on a 

same membership value. In this paper, Arboricity and span of a fuzzy 

labeling graph are defined with suitable examples. Also, the relation 

between Arboricity and M-fuzzy Chromatic Index, Span and M-fuzzy 

Chromatic Index are discussed. Some results based on these concepts are 

stated and proved. 

 

Keywords: M-fuzzy coloring; M-fuzzy chromatic index; Arboricity; span 

annuity 

 

2010 AMS subject classification: 03E72, 05C72, 05C78‡  
 

 

 
*Dr. S. Yahya Mohamed, Asst. Professor in Mathematics, Government Arts College, Trichy-22, 

Affiliated to Bharathidasan University, Trichy. email: yahya_md@yahoo.com. 
†Department of Mathematics, Bharathiyar Arts & Science College for Women,  Deviyakurichi,            

Salem – 636112; sivasujithsuganthi@gmail.com. 
‡ Received on January 29th, 2022. Accepted on June 22nd, 2022. Published on June 30th, 2022.  

doi: 10.23755/rm.v39i0.710. ISSN: 1592-7415. eISSN: 2282-8214. ©The Authors. This paper is 

published under the CC-BY licence agreement. 

115



S.Yahya Mohamed and S.Suganthi 

 

 

1. Introduction  

      In 1736, Euler first invited the concept of graph theory. The theory of graphs is 

extremely useful for solving combinatorial problems in different areas such as geometry, 

algebra, number theory, topology, operations  research, optimization and computer 

science. Graph Theory is the main tool to constitute many real-world problems. 

Nowadays, graphs do not represent all systems properly due to the uncertainty or 

nebulousness of the framework of systems. This and many other circumstances 

stimulated to define fuzzy graphs.  

 

      Rosenfeld [8] first established the notion of fuzzy graphs in 1975. Crisp graph and 

fuzzy graph are both systematically the same, but when there is an uncertainty on points 

and or lines then fuzzy graph has a segregated significance. Since the world is brimming 

with uncertainty, so the fuzzy graph occurs in many real-life locations. Fuzzy graph 

theory is elaborated with a large number of branches. The fuzzy graph model is used to 

constitute the traffic network. Also it is used for job allocation, group structure and 

decision-making analysis etc. Seethalakshmi et al. [10] defined perfect fuzzy matching 

with some examples, and they derived that if 𝐺 is a strong regular fuzzy graph with each 
vertex and is of degree at least two, then 𝐸 is not a perfect fuzzy matching in 𝐺.  
 

     Ananthanarayanan et al. [1] discussed the coloring of fuzzy graphs using α-cut with 

appropriate examples. Anjaly Kishore et al.[2] defined the chromatic number of a fuzzy 

graph and they discussed some results based on these chromatic numbers. Arindam 

Deyet al.[3] explained the concept of complement fuzzy graph and their edge coloring. 

R. Jahir Hussain et al.[5] found the fuzzy chromatic number of some special fuzzy 

graphs S.Yahya Mohamed et al.[11-14] introduced the concept of fuzzy matching in 

fuzzy labeling graphs through some illustrations. In this paper, we introduced the new 

concept of Arboricity and Span of a fuzzy labeling graph 𝐺. Also, we discussed some 
properties based on Arboricity and Span of 𝐺. 
 

 

2. Preliminaries 
 
Definition 2.1 

    Let 𝐺 = (𝛼, 𝛽) be a fuzzy graph. The set of all subsets of 𝛽 in which they received the 
same color is called Color Class of  𝐺. It is denoted by 𝐶𝐶(𝐺). 
 

Example 2.2  

   Consider the fuzzy graph 𝐺 given in Figure 2.1 
In Fig 2.1, the edge set is  𝛽 = {𝑒1(0.2), 𝑒2(0.5), 𝑒3(0.35), 𝑒4(0.14)} and all edges 
receive different colors. So all edges belong to the color class of  𝐺. 

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Arboricity and span in M-fuzzy chromatic index 

 

 

                              
Figure 2.1: Color class of a fuzzy graph 𝐺 

 

Definition 2.3 

    Let 𝐺 =  (𝛼, 𝛽) be a fuzzy labeling graph. The number of elements in the color 
class is called as coloring number of  𝐺. It is denoted by 𝐶𝑁(𝐺). 
 

Definition 2.4 

    A family 𝜆 =  {𝑀1, 𝑀2, 𝑀3 … … … . 𝑀𝑘 } of fuzzy matchings on a set 𝛽 is called 
a 𝑀 − fuzzy coloring of 𝐺 =  (𝛼, 𝛽) if  

(i) ∨  𝜆 =  𝛽. It means single edge does not belong to two distinct color 
classes. 

(ii)   𝜆𝑖 𝛬  𝜆𝑗   =  0. 

(iii) For every edge (𝑥, 𝑦) 𝑜𝑓 𝐺  𝑚𝑖𝑛 {𝜆𝑖  (𝑥), 𝜆𝑗  (𝑦)}  = 0  (1 ≤ 𝑖 ≤ 𝑘).  

        (This means any one of the edges does not receive different color). 

Example 2.5 

Consider the fuzzy graph 𝐺 given in figure 2.2, 

 
Figure 2.2: M-Fuzzy Coloring of 𝐺 

 

In Fig 2.2, the fuzzy matchings are 𝑀1 =  { 𝑒1(0.4), 𝑒3(0.53 }  and                   
𝑀2  = { 𝑒4(0.24), 𝑒2(0.62)}. Here 𝜆 =  {𝑀1, 𝑀2} is 𝑀 −fuzzy coloring of  𝐺. 
 

 

 

 

 v1(0.4) 

   v2(0.7) 

v3(0.9) 

v4(0.3) 

e3(0.3

5) 

   e1(0.2) 

e2(0.5) 

e4(0.14) 

Fig 2.1 

 

v5(0.7) 

v3(0.8) 

v4(0.6) 
     v1(0.5) 

v2(0.7) 

e2(0.62) 

e3(0.53

) 

e4(0.24) 

e1(0.4) 

e5(0.18) 

Fig 2.2 

117



S.Yahya Mohamed and S.Suganthi 

 

 

Definition 2.6  

     Let 𝐺 =  (𝛼, 𝛽) be a fuzzy labeling graph .The minimum number 𝑘 for which there 
exists a 𝑀 −fuzzy coloring is called fuzzy matching chromatic index of 𝐺. It is denoted 
by 𝜒fmci (𝐺). 
 

Example 2.7 

    Consider the fuzzy graph given in Figure 2.3 

 
Figure 2.3: Fuzzy matching chromatic index of 𝐺 

 

     In Fig 2.3, the fuzzy matching of a given graph 𝐺 are as follows 𝑀1 =
{ 𝑒1(0.04), 𝑒3(0.26), 𝑒5(0.48)}, 𝑀2 = { 𝑒2(0.15), 𝑒4(0.37), 𝑒6(0.09)}. Then the fuzzy 
matching chromatic index  𝜒fmci (𝐺) =  2 because 𝜆 =  {𝑀1, 𝑀2} is 𝑀 −fuzzy coloring 
of  𝐺. 
 

 

3. Main Results 

Definition 3.1 

     The minimum number of paths to cover all edges of a fuzzy graph G  is called 

Arboricity of .G  It is denoted by ( ).A G
 

 

Example 3.2 

Consider the following fuzzy graph 𝐺 given in Figure 3.1 
 

 

Figure 3.1:  Arboricity of 𝐺 

 

v4(0.4) 

v5(0.5) 
v6(0.6) e5(0.48) 

e4(0.37) 

e3(0.26) 

v3(0.3) v2(0.2) e2(0.15) 

e1(0.04) 

   v1(0.1) 

e6(0.09) 

Fig 2.3 

118



Arboricity and span in M-fuzzy chromatic index 

 

 

 

In fig 3.1 the path cover all edges of a given graph are 
1 1 2 2 3

v e v e v and 
3 3 4 4 1

.v e v e v  Here 

each colour represents one path and two distinct paths cover all edges of .G  Hence, we 

obtain ( ) 2.A G =  
 

Theorem 3.3 

     For any fuzzy labeling graph G  with n  vertices has Arboricity .
2

n 
 
 

 

Proof 

 Consider a fuzzy labeling graph ( ),G  =  with n  vertices. Now we 
construct a path to cover all edges of G. We obtain many paths to cover all edges of G. 

But we need minimum number of path to cover all edges. Then we construct the path 

with maximum number of edges in .G  This path occurs with maximum number of 
edges ( 1)n − . Hence, we can find all paths with ( 1)n −  edges and which is equal to 

2

n 
 
 

. Therefore, G  has Arboricity 
2

n 
 
 

. 

 

Example 3.4 

       Consider the fuzzy graph given in Figure 3.2, 

 

Figure 3.2: Fuzzy Labeling graph 𝐺 

 

In fig 3.2 the path cover all edges of a given graph are 
3 5 1 4 4 6 2

v e v e v e v  and 

1 1 2 2 3 3 4
.v e v e v e v  Here, each color represents one path and there are two paths exist to 

cover all edges of G.  Hence, we get ( )
4

2 .
2

A G
 

= =
 
 

 

 

 

v3(0.9) 

v4(0.8) v1(0.3) 

e1(0.2) 

v2(0.6) 
e2(0.4) 

e3(0.6) 

e4(0.1) 

Fig 3.2 

e6(0.5) 

    e5(0.24) 

119



S.Yahya Mohamed and S.Suganthi 

 

 

Theorem 3.5 

     For any fuzzy labeling graph ( ) ( ), .fmciG A G G    

 

Proof 

Consider a labeling fuzzy graph ( ),G  =  and 1 2 3, , ... kM M M M   be the distinct 

fuzzy matching in .G  Here 
1 2 3
, , ...

k
M M M M contains the set of independent elements 

and each matching receive different colors. Now we find the path to cover all edges of 

.G  Here we need   more than one path to cover all edges of .G  Already we have for 

any fuzzy graph with n  vertices has arboricity  .
2

n 
 
 

 Here we require two matching or 

more than two matching to cover all edges of .G  Then the number of path to cover all 

edges is less than the number of matching in any color class of .G  Hence, we obtain 

( ) ( ).fmciA G G
 

Example 3.6 

    Consider the fuzzy graph given in Figure 3.3, 

 

Figure 3.3: Fuzzy Graph 𝐺 

 

In fig 3.3  the fuzzy matching of a given graph G  is as follows  

( ) ( )  ( ) ( ) 1 4 6 2 1 30.53 , 0.29 , 0.15 , 0.29M e e M e e= =  and   

 v1(0.5) 
v5(0.9) 

v4(0.6) v3(0.3) 

v2(0.2) 

e4(0.41) 

e3(0.29) 

e4(0.53) 

e2(0.12) 

  e6(0.29) 

e1(0.15) 

Fig 3.3 

120



Arboricity and span in M-fuzzy chromatic index 

 

 

( ) ( ) 3 2 40.12 , 0.41M e e= . Also the color class ( )  1 2 3, , .CC G M M M=  Here 

( ) 3fmci G =  and the paths to cover all edges of G  are 5 4 1 1 2 2 3v e v e v e v and 

1 6 3 3 4 4 5
.v e v e v e v  Then ( ) 2.A G =

 
Hence, we obtain ( ) ( ).fmciA G G

 
 

 

 

Definition 3.7 

     The minimum number of interchanging path to cover all vertices of a fuzzy labeling 

graph 𝐺 is called spanning arboricity of G. 
 

Example 3.8 

Consider the following fuzzy graph 𝐺 given in Figure 3.4 

 

Figure 3.4: Spanning arboricity of 𝐺 

In fig 3.4 one of the fuzzy matching of a given graph G  is 

( ) ( ) 1 2 50.3 , 0.05 .M e e=  Then the interchanging path 2 2 3 1 1 5 4 4 5v e v e v e v e v  is to 
cover all points of .G  Here spanning arboricity of G is one. 
 

Theorem 3.9  

     A fuzzy labeling graph ( ),G  =  with n  vertices has spanning arboricity as 
unity. 

Proof  

     Let us consider a fuzzy labeling graph ( ),G  = with n  vertices. Then we 
identify the fuzzy matching of G  and an interchanging path with maximum lines to 
cover all vertices of G . Now we can find an interchanging path to cover all vertices of 

.G   An interchanging path is a path in which the edges are alternatively in M and 

( ).M −
 
But we can able to construct a fuzzy matching with maximum number of 

edges in the boundary of G . Then an interchanging path in the boundary path of any 
graph is only enough to cover all vertices of G  

 

v2(0.8) 
v5(0.5) 

v4(0.6) v3(0.7) e3(0.2) 

e4(0.1) 

v1(0.9) 

e5(0.05) e1(0.4) 

e2(0.3) 

Fig 3.4 

121



S.Yahya Mohamed and S.Suganthi 

 

 

Example 3.10 

     Consider the following fuzzy graph given in Figure 3.5 

 

Figure 3.5: Spanning arboricity of 𝐺 

    In fig 3.5 the fuzzy matching of a given graph G  is ( ) ( ) 1 2 40.04 , 0.08M e e=  
and 

1 1 2 2 3 3 4 4 5
v e v e v e v e v  is an interchanging path to cover all vertices of .G   Hence, the 

spanning arboricity of G  is one. 

Definition 3.11 

    The number of edges in an interchanging path to cover all vertices of a fuzzy graph 

G  is called span of G . It is denoted by ( ).GS
 

Example 3.12 

Consider the following fuzzy graph given in Figure 3.6, 

 

Figure 3.6 : Span of 𝐺 

 

v4(0.45) 

v5(0.56) 

e4(0.08) 

e3(0.06) 

v3(0.34) 

e2(0.04) 

v1(0.12) 

e5(0.03) e1(0.02) 

e6(0.07) 

v2(0.23) 

Fig 3.5 

 

v4(0.4) 

v3(0.3) 

e3(0.24) 

e4(0.07) 
v1(0.1) 

e1(0.05) 

v2(0.2) 
e2(0.11) 

e5(0.09) 

Fig 3.6 

122



Arboricity and span in M-fuzzy chromatic index 

 

 

In fig 3.6 the fuzzy matching of a given graph G  is  ( ) ( ) 1 1 30.05 , 0.24M e e=  and 
the number of edges in an interchanging path 

1 1 2 2 3 3 4
v e v e v e v is 3. Hence, we have 

( ) 3.G =S
 

 

 

Theorem 3.13  

      For any fuzzy labeling graph ( ),G  =  with n  vertices, we have 

( ) ( ).fmciG GS
 

Proof  

      Let us consider a fuzzy labeling graph ( ),G  =  with n  vertices. Now we can 
form fuzzy matching of G  and we find an interchanging path with maximum edges to 
cover all vertices of .G  Then the interchanging path in the boundary of any graph is 
only enough to cover all vertices of .G  to cover all vertices of .G  But this path occurs 

with maximum number of edges ( )1n −  because each fuzzy matching receives distinct 

colors. Therefore, span of G  is ( )1 .n −  Here the fuzzy matching chromatic index is 

less than ( )1 .n −  and  hence, we obtain ( ) ( ).fmciS G G  
 

Example 3.14 

Consider the following fuzzy graph given in Figure 3.7, 

 

Figure 3.7: Span of 𝐺 

In fig 3.7 the fuzzy matching is ( ) ( ) ( ) 1 2 5 80.05 , 0.11 , 0.17M e e e=  and 

1 1 2 2 3 4 4 5 5 6 6
v e v e v e v e v e v  is an interchanging path to cover all vertices of G . Then we 

have ( ) ( )5 2.fmciS G G=  =
 

 

v4(0.45) 

v2(0.23) 

v5(0.56) 

v3(0.34)  v1(0.19) 

v6(0.67) 
e6(0.13) 

e5(0.11) 

e4(0.09) 

e2(0.05) e1(0.03) 

e3(0.07) 

e7(0.15) 

 e8(0.17) 

Fig 3.7 

123



S.Yahya Mohamed and S.Suganthi 

 

 

4. Conclusion 

       Fuzzy set theory proposes to be a useful tool for handling vagueness and degrees of 

certainty, and for giving a consistent representation of linguistically formulated 

knowledge which allows the use of precise properties and algorithms. In this paper, we 

introduced the new concept of Arboricity and span of fuzzy labeling graph. Also we 

compared these values with M-fuzzy Chromatic Index with related examples. In future, 

we plan to extend our research work to operations like union, intersection and Cartesian 

product of two fuzzy labeling graphs. 

 

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