Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 3s (2025) 

 

547 https://internationalpubls.com 

A Novel case on Fuzzy Resolving Sets on Interval-valued Fuzzy 

Graph and its Application on Signal Processing Unit 

 

R. Shanmugapriya1 & Vasuki M1 

1,1Department of Mathematics, Vel Tech Rangarajan Dr.Sagunthala R&D Institute of 

Science and Technology, Chennai. 

Email id: spriyasathish11@gmail.com1 & vasukimani1997@gmail.com1 

 

Article History: 

Received: 23-09-2024 

Revised: 09-11-2024 

Accepted: 27-11-2024 

Abstract:  

A fuzzy graph generalisation known as an interval-valued fuzzy graph (IVFG) reflects 

uncertainty more flexibly by representing the membership values of vertices and edges as 

intervals rather than fixed numbers. We have defined Fuzzy Resolving Set to interval-

valued fuzzy graphs such as interval-valued fuzzy resolving set and interval-valued fuzzy 

super resolving set. We also derived the properties of isomorphism in this topic and 

explained an application based on them. 

Keywords: Fuzzy Graph, Fuzzy resolving set, Interval-valued fuzzy graph, interval valued 

fuzzy resolving set 

 

1. Introduction 

The degree of certainty or uncertainty can be expressed using degrees of membership in fuzzy sets. 

Instead of dealing with precise distances, we deal with fuzzy distances or fuzzy sets of distances in 

an FRS. In summary, an FRS provides a more flexible framework for resolving network vertices 

when precise information is not easily accessible. It is an extension of traditional resolving sets that 

accounts for measurement uncertainty in the vertex-to-vertex distance. FRS can be used to model the 

transmission of illnesses when precise distance measurements between sites or between individuals 

are difficult to obtain. An Interval-Valued Fuzzy Graph (IVFG) is a sophisticated mathematical 

framework that models uncertainty in real-world systems by fusing fuzzy logic and graph theory. 

IVFGs represent uncertainty in the degree of membership of vertices and edges using intervals, as 

opposed to standard graphs or fuzzy graphs, where memberships are discrete or single values. 

While Euler announced graph theory, Zadeh[13] evaluated in 1965. Kauffman developed the fuzzy 

set, which served as the foundation for the fuzzy graph, in 1973.  Moderson and associates [6] 

created a fuzzy graph that has numerous uses. Resolvability and an upper dimension of graphs were 

first suggested by Chartrand[2] in 2000. Shanmugapriya and Mary Jiny [9] subsequently extended 

this concept to resolvability in fuzzy graphs. Additionally, they both introduced modified FRN and 

FRS characteristics. Furthermore, Atanassov[1] extended the FG to create an intuitive fuzzy graph 

(IFG). An interval-valued fuzzy graph has been found by Muhammad akram and A. Dudek in 

2011[7]. It has been further elaborated by Pramanik and others in 2020[10]. Also, the properties of 

interval-valued fuzzy graph has been developed by Xiaoli Qiang and others in the year 2022[12]. 

They also collaborated with numerous others to create a number of research works on fuzzy graphs, 

including studies on the size, order, and strong and weak domination of fuzzy graphs. The parameter 

mailto:spriyasathish@gmail.com1
mailto:vasukimani1997@gmail.com


Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

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of clustering the sets is used in the development of numerous applications. In 2023, fuzzy resolving 

domination set has been introduced by Shanmugapriya and Co [8][11].  

Let 𝐺 be a connected simple finite FG with the number of vertices greater than 3. Standard fuzzy 

graph and interval-valued fuzzy graph definitions were covered in section 2, whereas IVFRS, 

IVFSRS and its theorems were presented in section 3 and 4. We have created an application based on 

the idea in section 5. We are keen to explore into further fuzzy resolving set-related subjects in the 

future studies. 

Names Notations 

Fuzzy Resolving Set FRS 

Interval-valued Fuzzy Graph IVFG 

Interval-valued Fuzzy Resolving Set IVFRS 

Interval-valued Fuzzy Super Resolving Set IVFSRS 

Interval-valued Fuzzy Resolving Matrix IVFRM 

Interval-valued Fuzzy Super Resolving Matrix IVFSRM 

Table 1 

2. Preliminaries 

In this section, we have defined a basic definition to deal with the IVFRS. 

Definition 2.1.  

A FG G(V, σ, µ) where µ is a symmetric fuzzy relation on σ: V → [0,1] and µ: V × V → [0,1] such 

that µ(x, z) ≤ σ(x)ʌσ(z), ∀ x, z ∈ V. 

Definition 2.2. 

Consider an ordered fuzzy subset H = {(u1, σ(u1)), (u2, σ(u2)), … (uk, σ(uk))} , |H| ≥ 2, the 

representation of (z, σ(z))ϵσ − H = {(uk+1, σ(uk+1)), (uk+2, σ(uk+2)),… (un, σ(un))}  and 

{w(z, u1), w(z, u2),… w(z, uk)}, where the significance of the link between z and y is denoted by 

w(z, y) . The set is called a FRS if every pair of items in the fuzzy subset H  has a different 

representation with respect to H of G. The FRN is represented as Fr(G), the minimal cardinality of 

FRS. 

Definition 2.3. 

A fuzzy graph G's FRS is considered a fuzzy super resolving set (FSRS) if any two of its elements 

have unique representations with respect to H. We also write the FSRN, denoted by Sr(G), and the 

super resolving matrix, denoted by  Sn×k, which is the lowest cardinality of a FG G. 

Definition 2.4. 

A set C of vertices V is a dominating set if every vertex of V\C is adjacent to any vertex of 𝐶. The 

smallest cardinality of this set is called the fuzzy domination number(FDN) and it is denoted by 

Fγ(G). 

 



Communications on Applied Nonlinear Analysis 

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Definition 2.5 

An interval-valued fuzzy relation Q  is a mapping, Q: XxY →  D [0,1]  such that Q (x, y)  =

 [Q− (x, y), Q+ (x, y)]  ∈ D [0,1] ̬for all pairs (x, y) ϵ XxY. 

Definition 2.6 

We define G =  [E, F] , where E =  [μ
E
−, μ

E
+]  and F = [μ

F
−, μ

F
+]  is called an interval-valued fuzzy 

graph (IVFG), if E is an interval-valued fuzzy set on V and F is an interval-valued fuzzy relation 

such that 

μ
F
− (ef)  ≤  min (μ

E
− (e), μ

E
− (f)), μ

F
+ (ef)  ≤  min (μ

E
+ (e), μ

E
+ (f)), ∀ ef ∈  E. 

Definition 2.7 

The order and size of IVFG is defined by  

Ο(G)  =  ∑
1 + μ

E
+ (r) − μ

E
− (r)

2
r∈V

 

𝒮(G)  =  ∑
1 + μ

F
+ (rq) − μ

F
− (rq)

2
r∈V

 

Definition 2.8 

An IVFG is called complete if μ
F
−(ef)  =  min (μ

E
− (e), μ

E
− (f)) , μ

F
+ (ef)  =  max (μ

E
+ (e), μ

E
+ (f)),

∀ ef ∈  E. 

Definition 2.9 

An isomorphism φ: G1  → G2  is a bijective mapping of two IVFGs, φ: X1  → X2,  ∋  μ
E
− (r) =

μ
E
−′(φ (r)), μ

E
+ (r) = μ

E
+′ (φ(r)), μ

F
− (r, u) = μ

F
−′ (φ(r), φ(u)),  μ

F
+ (r, u) = μ

F
+′ (φ(r), φ(u)) ∀r, u ∈ E.    

Definition 2.10 

A homomorphism φ: G1  → G2  is a mapping φ: X1  → X2  of two IVFGs such that μ
E
− (r) ≤

μ
E
−′(φ (r)), μ

E
+ (r) ≤ μ

E
+′ (φ(r)), μ

F
− (r, u) ≤ μ

F
−′ (φ(r), φ(u)),   μ

F
+ (r, u) ≤ μ

F
+′ (φ(r), φ(u))    ∀r, u ∈

E.    

Definition 2.11 

A co-weak isomorphism φ: G1  → G2  is a bijective mapping, φ: X1  → X2, such that 

(i)  φ is homomorphism 

(ii) μ
F
− (r, u) = μ

F
−′ (φ(r), φ(u)), μ

F
+ (r, u) = μ

F
+′ (φ(r), φ(u)) ∀r, u ∈ E. 

3. Fuzzy Resolving Sets of Interval-valued Fuzzy Graphs 

Consider a fuzzy subset ℋ , then the representation of ℋ  and (u, σ−(u), σ+(u))  ∈ σ − ℋ  are all 

distinct, where ℋ = {(x1, σ
−(x1), σ+(x1)), (x2, σ

−(x2), σ+(x2)), . . . (xk, σ
−(x2), σ

+(x2))}  and σ −

ℋ = {(xk+1, σ
−(xk+1), σ+(xk+1)), (xk+2, σ

−(xk+2), σ+(xk+2)), . . . (xn, σ
−(xn), σ

+(xn))} , then ℋ  is 



Communications on Applied Nonlinear Analysis 

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known as an interval-valued fuzzy resolving set (IVFRS). The minimum cardinality of this 

concerned set is called as Interval-valued Resolving Number (IVRN), denoted by ℐr(G). 

3.1 Definition 

The representation of σ − ℋ with respect to ℋ is written as [Aij
−, Aij

+], where 

Aij
− = [w−(uj, x1), w

−(uj, x2), . . . w
−(uj, xk)],   

Aij
+ = [w+(uj, x1), w

+(uj, x2), . . . w
+(uj, xk)] for j =  k + 1, k + 2, … n are written in a form 𝐴𝑛−𝑘×𝑘 . 

This matrix is called Interval-valued Fuzzy Resolving Matrix(IVFRM). 

3.2 Illustration                  

 

Fig 1 : IVFG 

𝝁𝑬 𝑸𝟏 𝑸𝟐 𝑸𝟑 𝑸𝟒 

𝝁𝑬
− 

 

𝝁𝑬
+ 

0.4 

 

0.5 

0.6 

 

0.6 

0.8 

 

0.7 

0.8 

 

0.7 

 

𝝁𝑭 𝑸𝟏𝑸𝟐 𝑸𝟐𝑸𝟑 𝑸𝟏𝑸𝟑 𝑸𝟑𝑸𝟒 

𝝁𝑭
− 

 

𝝁𝑭
+ 

0.4 

 

0.4 

0.5 

 

0.3 

0.3 

 

0.5 

0.7 

 

0.6 

 

Let ℋ = {Q
1
, Q

3
}, then σ − ℋ = {Q

2
, Q

4
} 

where 𝑄1 = {𝜎1
−, 𝜎1

+}, 𝑄2 = {𝜎2
−, 𝜎2 

+},  𝑄3 = {𝜎3
−, 𝜎3

+} 𝑎𝑛𝑑 𝑄4 = {𝜎4
−, 𝜎4

+}  

then, 𝜎2
−/ℋ =  {μ∞(𝜎2

−, 𝜎1
−), μ∞(𝜎2

−, 𝜎3
−)}  =  (0.4, 0.5) 

𝜎2
+/ℋ =  {μ∞(𝜎2

+, 𝜎1
+), μ∞(𝜎2

+, 𝜎3
+)}  =  (0.4, 0.4) 

𝜎4
−/ℋ =  {μ∞(𝜎4

−, 𝜎1
−), μ∞(𝜎4

−, 𝜎3
−)}  =  (0.4, 0.7) 

𝜎4
+/ℋ =  {μ∞(𝜎4

+, 𝜎1
+), μ∞(𝜎4

+, 𝜎3
+)}  =  (0.5, 0.6) 

Therefore, Aij
− = [

0.4 0.5
0.4 0.7

], Aij
+ = [

0.4 0.4
0.5 0.6

] 



Communications on Applied Nonlinear Analysis 

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Hence, the values are all distinct 

Therefore, ℐr(G) = 2 

3.3 Theorem 

If G and G′ are isomorphic, then ℐr(G) = ℐr(G′) 

Proof: 

Let us assume that V =  {r1, r2, . . . , rn}, ℐr(G)= k, and let ℋ = {(σ1
−, σ1

+), (σ2
−, σ2

+), . . . , (σk
−, σk

+) is the 

corresponding FRS. We denote (r1, σ
−(r1)) = σ1

− , (r1, σ
+(r1)) = σ1

+ , (φ(r1), σ
−(φ(r

1
)) = σ1

−′ , 

(φ(r1), σ
+(φ(r

1
)) = σ1

+′. The representation, 𝜎𝑘+𝑖
− /ℋ =(w− (rk+i, r1), w

− (rk+i, r2), . . . , w
− (rk+i, rk),  

𝜎𝑘+𝑖
+ /ℋ =(w+ (rk+i, r1), w

+ (rk+i, r2), . . . , w
+ (rk+i, rk) all are distinct, ∀ i =  1,2, . . . n − k. If G and G′ 

are isomorphic, then ∃  a bijection φ: V → V′  which satisfies, μ
E
− (r) = μ

E
−′(φ (r)) , μ

E
+ (r) = 

μ
E
+′ (φ(r)) , μ

F
− (r, u) = μ

F
−′ (φ(r), φ(u)) and μ

F
+ (r, u) = μ

F
+′ (φ(r), φ(u)) ∀r, u ∈ E.  Now, we define 

ℋ1 = {(σ1
−′, σ1

+′), (σ2
−′, σ2

+′), . . . , (σk
−′, σk

+′)} for  i =  1,2, . . . n − k, 

𝜎𝑘+𝑖
−′ /ℋ = {w− (φ(r

k+i
), φ(r

1
)), w− (φ(r

k+i
), φ(r

2
)), . . . , w− (φ(r

k+i
), φ(r

k
)} 

               = {((μ−′)∞(φ(r
k+i

), (μ−′)∞(φ(r
1
)), ((μ−′)∞(φ(r

k+i
), (μ−′)∞(φ(r

2
)), . .. 

((μ−′)∞(φ(r
k+i

), (μ−′)∞(φ(r
k
)) 

              = μ−(rk+i, r1), μ
−(rk+i, r2), . . . , μ

−(rk+i, rk)  

              = w− (rk+i, r1), w
− (rk+i, r2), . . . , w

− (rk+i, rk)  

𝜎𝑘+𝑖
+′ /ℋ = {w+ (φ(r

k+i
), φ(r

1
)), w+(φ(r

k+i
), φ(r

2
)), . . . , w−+(φ(r

k+i
), φ(r

k
)} 

               = {((μ+′)∞(φ(r
k+i

), (μ+′)∞(φ(r
1
)), ((μ+′)∞(φ(r

k+i
), (μ+′)∞(φ(r

2
)), . .. 

((μ+′)∞(φ(r
k+i

), (μ+′)∞(φ(r
k
)) 

              = μ+(rk+i, r1), μ
+(rk+i, r2), . . . , μ

+(rk+i, rk)  

              = w+(rk+i, r1), w
+ (rk+i, r2), . . . , w

+ (rk+i, rk)  

 which are all distinct for i =  1 to n =  k. Therefore, ℋ1 is the IVFRS of G′ and │ℋ1│ =  k. Now 

to prove that ℋ1 is the minimum IVFRS OF G’. Assume that there exists a FRS R of G’ such that 

R = {(σ1
−, σ1

+), (σ2
−, σ2

+), . . . , (σk1

− , σk1

+ )} and│ℋ1│ =  k >  │R│ = k1, then 𝜎𝑘+𝑖
− /𝑅 and 𝜎𝑘+𝑖

+ /R are 

all distinct for n =  1, 2, . . . , n − k1. Let R1 = {(σ1
−, σ1

+), (σ2
−, σ2

+), . . . , (σk1

− , σk1

+ )}, 

𝜎𝑘+𝑖
− /R1  = w− (rk+i, r1), w

− (rk+i, r2), . . . , w
− (rk+i, rk1

)  

                 = {(μ−)∞(rk+i, r1), (μ−)∞(rk+i, r2), . .. (μ−)∞(rk+i, rk1
)} 

                 = {((μ−′)∞(φ(r
k+i

), (μ−′)∞(φ(r
1
)), ((μ−′)∞(φ(r

k+i
), (μ−′)∞(φ(r

2
)), . ..  

((μ−′)∞(φ(r
k+i

), (μ−′)∞(φ(r
k1

)) 

Similarly, 



Communications on Applied Nonlinear Analysis 

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𝜎𝑘+𝑖
+ /R1  = w+ (rk+i, r1), w

+ (rk+i, r2), . . . , w
+ (rk+i, rk1

)  

                 = {(μ+)∞(rk+i, r1), (μ+)∞(rk+i, r2), . .. (μ+)∞(rk+i, rk1
)} 

                 = {((μ+′)∞(φ(r
k+i

), (μ+′)∞(φ(r
1
)), ((μ+′)∞(φ(r

k+i
), (μ+′)∞(φ(r

2
)), . ..  

((μ+′)∞(φ(r
k+i

), (μ+′)∞(φ(r
k1

)) 

which are all distinct for i =  1,2, . . . n − k. Therefore, R1 is the IVFRS of G and ℐr(G)=𝑘1 [𝑘1 <  k], 

which is a contradiction to our assumption that ℐr(G) = k. Hence, ℋ1 is the minimum IVRS of G′. 

Therefore,ℐr(G)  ≅ ℐr(G′). 

3.4 Theorem 

If G and G′ are homomorphic functions, then ℐr(G) and ℐr(G′) are homomorphic. 

Proof: 

Let us assume that V =  {r1, r2, . . . , rn}, ℐr(G)= k, and let ℋ = {(σ1
−, σ1

+), (σ2
−, σ2

+), . . . , (σk
−, σk

+) is the 

corresponding FRS. We denote (r1, σ
−(r1)) = σ1

− , (r1, σ
+(r1)) = σ1

+ , (φ(r1), σ
−(φ(r

1
)) = σ1

−′ , 

(φ(r1), σ
+(φ(r

1
)) = σ1

+′. The representation, 𝜎𝑘+𝑖
− /ℋ =(w− (rk+i, r1), w

− (rk+i, r2), . . . , w
− (rk+i, rk),  

𝜎𝑘+𝑖
+ /ℋ =(w+ (rk+i, r1), w

+ (rk+i, r2), . . . , w
+ (rk+i, rk) all are distinct, ∀ i =  1,2, . . . n − k. If G and G′ 

are homomorphic, then ∃  a bijection φ: V → V′  which satisfies, μ
E
− (r) ≤ μ

E
−′(φ (r)) , μ

E
+ (r) ≤ 

μ
E
+′ (φ(r)) , μ

F
− (r, u) ≤ μ

F
−′ (φ(r), φ(u)) and μ

F
+ (r, u) ≤ μ

F
+′ (φ(r), φ(u)) ∀r, u ∈ E.  Now, we define 

ℋ1 = {(σ1
−′, σ1

+′), (σ2
−′, σ2

+′), . . . , (σk
−′, σk

+′)} for  i =  1,2, . . . n − k, 

𝜎𝑘+𝑖
−′ /ℋ = {w− (φ(r

k+i
), φ(r

1
)), w− (φ(r

k+i
), φ(r

2
)), . . . , w− (φ(r

k+i
), φ(r

k
)} 

               = {((μ−′)∞(φ(r
k+i

), (μ−′)∞(φ(r
1
)), ((μ−′)∞(φ(r

k+i
), (μ−′)∞(φ(r

2
)), . .. 

((μ−′)∞(φ(r
k+i

), (μ−′)∞(φ(r
k
)) 

              ≥ μ−(rk+i, r1), μ
−(rk+i, r2), . . . , μ

−(rk+i, rk)  

              = w− (rk+i, r1), w
− (rk+i, r2), . . . , w

− (rk+i, rk)  

𝜎𝑘+𝑖
+′ /ℋ = {w+ (φ(r

k+i
), φ(r

1
)), w+(φ(r

k+i
), φ(r

2
)), . . . , w−+(φ(r

k+i
), φ(r

k
)} 

               = {((μ+′)∞(φ(r
k+i

), (μ+′)∞(φ(r
1
)), ((μ+′)∞(φ(r

k+i
), (μ+′)∞(φ(r

2
)), . .. 

((μ+′)∞(φ(r
k+i

), (μ+′)∞(φ(r
k
)) 

              ≥ μ+(rk+i, r1), μ
+(rk+i, r2), . . . , μ

+(rk+i, rk)  

              = w+(rk+i, r1), w
+ (rk+i, r2), . . . , w

+ (rk+i, rk)  

 which are all distinct for i =  1 to n =  k. Therefore, ℋ1 is the IVFRS of G′ and │ℋ1│ =  k. Now 

to prove that ℋ1 is the minimum IVFRS OF G’. Assume that there exists a FRS R of G’ such that 

R = {(σ1
−, σ1

+), (σ2
−, σ2

+), . . . , (σk1

− , σk1

+ )} and│ℋ1│ =  k >  │R│ = k1, then 𝜎𝑘+𝑖
− /𝑅 and 𝜎𝑘+𝑖

+ /R are 

all distinct for n =  1, 2, . . . , n − k1. Let R1 = {(σ1
−, σ1

+), (σ2
−, σ2

+), . . . , (σk1

− , σk1

+ )}, 

𝜎𝑘+𝑖
− /R1  = w− (rk+i, r1), w

− (rk+i, r2), . . . , w
− (rk+i, rk1

)  



Communications on Applied Nonlinear Analysis 

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553 https://internationalpubls.com 

                 = {(μ−)∞(rk+i, r1), (μ−)∞(rk+i, r2), . .. (μ−)∞(rk+i, rk1
)} 

                 ≤ {((μ−′)∞(φ(r
k+i

), (μ−′)∞(φ(r
1
)), ((μ−′)∞(φ(r

k+i
), (μ−′)∞(φ(r

2
)), . ..  

((μ−′)∞(φ(r
k+i

), (μ−′)∞(φ(r
k1

)) 

Similarly, 

𝜎𝑘+𝑖
+ /R1  = w+ (rk+i, r1), w

+ (rk+i, r2), . . . , w
+ (rk+i, rk1

)  

                 = {(μ+)∞(rk+i, r1), (μ+)∞(rk+i, r2), . .. (μ+)∞(rk+i, rk1
)} 

                 ≤ {((μ+′)∞(φ(r
k+i

), (μ+′)∞(φ(r
1
)), ((μ+′)∞(φ(r

k+i
), (μ+′)∞(φ(r

2
)), . ..  

((μ+′)∞(φ(r
k+i

), (μ+′)∞(φ(r
k1

)) 

which are all distinct for i =  1,2, . . . n − k. Therefore, R1 is the IVFRS of G and ℐr(G)=𝑘1 [𝑘1 <  k], 

which is a contradiction to our assumption that ℐr(G) = k. Hence, ℋ1 is the minimum IVRS of G′. 

Therefore,ℐr(G) and ℐr(G′) are homomorphic functions. 

3.5 Corollary 

If G and G′ are co-weak isomorphic, then ℐr(G) and ℐr(G′) also have a co-weak isomorphism.  

3.6 Theorem: 

For a connected IVFG with atleast three different membership value with both 𝜎− and 𝜎+, there 

exists an IVFRSs, such that 2 ≤  ℐr(G)  ≤  n − 1. [n ≥  3] 

Proof:  

Let us take n =  3, then the defined three vertices are a, b, c. This implies that, 𝜎−(a) = α1, 𝜎
+(a) =

β
1
, 𝜎−(b) = α2, 𝜎

+(b) = β
2
, 𝜎−(c) = α3, 𝜎

+(c) = β
3

, as they exist three different membership 

values, then the representation of any two vertices in H will be different. Hence ℐr(G) = 2 , for n =

 3. Similarly, if n =  k, then ∃ atleast three different membership values of both 𝜎− and 𝜎+. Hence 

the representation will be unique with atleast (n − 1) vertices. That is, ℐr(G)  ≤  n − 1. Hence 2 ≤

 ℐr(G)  ≤  n − 1. 

4. Fuzzy Super Resolving Sets of Interval-valued Fuzzy Graphs 

An IVFRS is called Interval-valued Fuzzy Super Resolving Set (IVFSRS) if any two elements of 

[𝜎− , 𝜎+ ] have different representation with regards to ℋ . Here we take w− (r, r) = σ−(r)  and 

w+(r, r) = σ+(r), and the least cardinality of this set is known as Interval-valued Super Resolving 

Number (IVSRN) which is denoted by ℐsr(G), If we arrange the obtained set in a row form, then it 

is named as Interval-valued Fuzzy Super Resolving Matrix denoted as [𝑆𝑛×𝑘
− , 𝑆𝑛×𝑘

+  ]. 

 

 

 

 



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4.1 Illustration 

                                      

                                                        Fig 2: IVFG 

𝑴𝑬 𝑺𝟏 𝑺𝟐 𝑺𝟑 𝑺𝟒 𝑺𝟓 

𝑴𝑬
− 

 

𝑴𝑬
+ 

0.7 

 

0.4 

0.8 

 

0.5 

0.6 

 

0.6 

0.5 

 

0.7 

0.4 

 

0.8 

 

𝑴𝑭 𝑺𝟏𝑺𝟐 𝑺𝟏𝑺𝟑 𝑺𝟒𝑺𝟐 𝑺𝟑𝑺𝟒 𝑺𝟒𝑺𝟓 

𝑴𝑭
− 

 

𝑴𝑭
+ 

0.6 

 

0.4 

0.5 

 

0.3 

0.5 

 

0.4 

0.3 

 

0.6 

0.4 

 

0.7 

 

The strength of connectedness matrix is  

S1 S2 S3 S4 S5 

     S1 0.7 0.6 0.5 0.5 0.4    

     S2 0.6 0.8 0.5 0.5 0.4 

𝐶𝑖𝑗
− =           S3 0.5   0.5 0.6 0.5 0.4  

              S4 0.5 0.5 0.5 0.5 0.4 

     S5 0.4 0.4 0.4 0.4 0.4 

   S1 S2 S3 S4 S5 

      S1 0.4 0.4 0.4 0.4 0.4    

      S2 0.4 0.5 0.4 0.4 0.4 

𝐶𝑖𝑗
− =             S3 0.4 0.4 0.6 0.6 0.6 

               S4 0.4 0.4 0.6 0.7 0.7 

      S5 0.4 0.4 0.6 0.7 0.8 

  



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The IVFSRS is {𝑆2, 𝑆4, 𝑆5} and IVFSRM is written as, 

[
 
 
 
 
0.6 0.5 0.4
0.8 0.5 0.4
0.5 0.6 0.4
0.5 0.5 0.4
0.4 0.4 0.4]

 
 
 
 

 and 

[
 
 
 
 
0.4 0.4 0.4
0.5 0.4 0.4
0.4 0.6 0.6
0.4 0.6 0.7
0.4 0.6 0.8]

 
 
 
 

 

 

Hence  ℐsr(G) = 3. 

4.2 Corollary 

If G and G′ are isomorphic, then ℐsr(G) ≅ ℐsr(G′). 

4.3 Theorem 

An IVERS is also an IVFRS but the converse may or may not be true.  

Proof: 

Let G be an IVFG with ‘n’ vertices. Let ℋ = {(σ1
−, σ1

+), (σ2
−, σ2

+), . . . , (σm
− , σm

+)} be an IVFRS of G, 

then the representation of 𝜎𝑖
−/ℋ and 𝜎𝑖

+/ℋ, for i =  m + 1, m + 2, . . . n are all distinct. Also, 𝜎𝑖
−/ℋ 

and 𝜎𝑖
+/ℋ need not to be distinct for i =  1,2, . . . n. (i.,e) ℋ does not need to be an IVFSRS. 

Conversely, let ℋ = {(σ1
−, σ1

+), (σ2
−, σ2

+), . . . , (σm
− , σm

+)} is an IVFSRS of G, then 𝜎𝑖
−/ℋ and 𝜎𝑖

+/ℋ, 

for i =  1,2, . . . m + 1, . . . n, are all distinct which means that the representation of 𝜎𝑖
−/ℋ and 𝜎𝑖

+/ℋ, 

for i =  m + 1, m + 2, . . . n, are all distinct, Hence, ℋ is also an IVFRS. 

4.4 Theorem 

For an interval-valued Fuzzy Star Graph (IVFSG) with distinct membership values, ∃  any two 

vertices ∋ r, u ∈ E, ℐsr(G) = ℐr(G) = 2 

Proof: 

By the definition of IVFSG, 𝜇𝐸
−(𝑟, 𝑢𝑖) > 0, 𝜇𝐸

+(𝑟, 𝑢𝑖) > 0, 𝜇𝐸
−(𝑢𝑖, 𝑢𝑖+1) = 0, 𝜇𝐸

−(𝑢𝑖, 𝑢𝑖+1) = 0 for i =

 1,2, . . . n. It is evident that we can find any two vertices with distinct membership values in the 

IVFSRM as well as IVFRM. Hence, ℐsr(G) = ℐr(G) = 2. 

5. Application 

In several domains, interval-valued fuzzy graphs (IVFGs) are employed to manage imprecision and 

uncertainty.Compared to crisp or normal fuzzy graphs, interval-valued fuzzy graphs (IVFGs) are 

better at capturing imprecision because they use intervals for edge and vertex values to depict 

interactions with uncertainty. They are used to efficiently handle varying or imprecise data in 

uncertain systems such as networks, decision-making, biology, and transportation. IVFGs use 

interval values to represent relationships in social networks that have unclear strengths, such as 

differing degrees of influence, friendship, or trust. This aids in identifying important influencers or 

clusters and analysing dynamic, imprecise interactions. IVFGs are used in signal processing to 

describe uncertain relationships between elements, such as varying noise levels or signal intensities. 

They help with data transmission uncertainty management, pathway optimisation, and robust system 

design. 



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      Transportation networks, such as airline or urban transit systems, are a real-world example of an 

interval-valued fuzzy graph (IVFG) with a significant number of vertices. We simulate the 

unpredictability of passenger demand and flight connectivity between cities. Every vertex denotes a 

city, such as Tokyo, London, or New York. This might include hundreds or thousands of cities for a 

global airline network. Every edge depicts a flight path between two cities, with an interval 

signifying the degree of popularity or dependability of the route. The uncertainty of a city's traffic 

capacity or significance in the network is indicated by the vertex membership value. Uncertainty in 

passenger demand or flight dependability is represented by the edge membership value. The benefits 

includes Assisting airlines in optimising their schedules in spite of demand fluctuations, helps 

identify important cities or routes for investment, beneficial for managing disruptions (such as delays 

and bad weather). These models effectively handle big, complicated systems with a lot of uncertainty. 

For example, 𝑀𝐸( 𝐴1: London) = (0.5, 0.4), 𝑀𝐸( 𝐴2: New york) = (0.7, 0.6), 𝑀𝐸( 𝐴3: Malaysia) = 

(0.8, 0.6) and so on. Also, 𝑀𝐹( 𝐴1𝐴2) = (0.4, 0.4), 𝑀𝐹( 𝐴1𝐴3) = (0.4, 0.3) and so on. Using IVFRS, 

we can obtain the few set of vertices such that the resulted set will have a shortest route between 

them considering all the traffic routes and the passenger dependancy. 

5. Conclusion 

One effective paradigm for simulating uncertainty in complex systems is offered by IVFGs. IVFGs 

are well suited for real-world applications like as social systems, biological interactions, and 

transportation networks because they incorporate interval-based membership values for vertices and 

edges, which effectively express imprecise relationships. In this article, we have defined interval-

valued fuzzy resolving sets, interval-valued fuzzy super resolving sets and its properties. Also, we 

have defined an application based on interval-valued fuzzy resolving set. In future, we may look into 

more problems related to interval-valued fuzzy resolving set. 

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