A study of m -polar neutrosophic graph with applications

1 Introduction

Zadeh [28] introduced the concept of fuzzy set theory as the advanced version of classical set theory to handle many real world problems with uncertainties. Fuzzy set theory is a powerful mathematical tool for dealing with incomplete, inexact, indeterminate and inconsistent information in real world problem. Zhang [29] proposed the concept of bipolar fuzzy sets for handling positive and negative both situation in a given real-world problem. But there are some cases that we cannot solve with the concept of bipolar fuzzy set because they involve multi-agent, multi-attribute, multi-object, multi-index, multi-polar information.

For solving this type of problem, Juanjuan Chen et al. [13] extended the idea of the bipolar fuzzy set to m-polar fuzzy set [7, 15– 19]. Due to the complexity of facts and opacity of the human mind fuzzy set theory is not sufficient to solve many real-life problem. We cannot properly describe the concept of non-membership term in Zadeh’s fuzzy environment. To overcome this shortcoming of the fuzzy set, Atanassov [8] define the idea of intuitionistic fuzzy set(IFS), which is more efficient to handle real life problem compare to fuzzy set. Intuitionistic fuzzy sets handle incomplete information i.e., the grade of membership function and non-membership function but not the indeterminate information and inconsistent information which exists obviously in belief system.

To overcome the demerits of IFS, Smarandache [10, 25] neutrosophic set has been introduced as a remedy. A single-valued neutrosophic set has three components: truth membership degree, indeterminacy membership degree and falsity membership degree. Many papers about neutrosophic set theory and their application have been done by various researchers [1, 22]. In a neutrosophic set, the membership value is associated with truth, false and indeterminacy degrees but there is no restriction on their sum. Deli et al. [2, 14] extended the ideas of bipolar fuzzy sets and neutrosophic sets to bipolar neutrosophic sets and studied its operations and applications in decision making problems.

In 1736, as a new branch of mathematics Graph theory was born with the Eulers paper in which he solved the famous Konigsberg bridge problem. A graph G is defined by a ordered pair of two sets namely vertex set V (G) and edge set E (G) respectively, such that |V (G) | = n and |E (G) | = m. The degree of a vertex v is the number of vertices in G which are connected to v by an edge and denoted by d_G(v). There are a very wide application of graph theory in computer science, image segmentation, clustering, image capturing, netwoking. Single-valued neutrosophic graph is the generalization of graphs, fuzzy graphs and intuitionistic graphs. Akram and Shahzadi [3] defined some operations on single-valued neutrosophic graph and they also showed the representation of graphs using neutrosophic soft sets in [4]. J. Ye [27] described single-valued neutrosophic minimum spanning tree and its clustering method.

Graph is an efficient method for modeling many combinatorial optimization problems especially in areas of expert system, neural network, medical diagnosis, operations research and computer science. The optimization problems in real life scenarios generally involve with multiple agents, multiple attributes, multiple objects and multiple indexes and information about the problems are generated from multiple sources, i.e., multi-polar information. We cannot represent this information properly using bipolar fuzzy graphs or m-polar fuzzy graphs.

The motivation behind the work in this paper is to introduce a new neutrosophic graph model which will be simple enough and effective to model the uncertainties in the real world problems. In this manuscript, we define the m-polar neutrosophic graph and several operations/products between two m-polar neutrosophic graph. These operations/products are illustrated by some examples. Some properties of complement, isomorphism, weak and co weak isomorphism of m-polar neutrosophic graph are presented here. We describe an application of m-polar neutrosophic graph in evaluating the professor performance of a college.

The paper is organized as follows. Section 2 briefly describes some basic ideas and definitions on fuzzy set, fuzzy graph, neutrosophic Set, single-valued neutrosophic graph(SVNG) and complete SVNG. In Section 3, we present the idea of m-polarSVNG. The corresponding some operation on m-polar SVNG are also presented in this Section 4. We present some products on m-polar neutrosophic graphs in Section 5. An application of m-polar SVNG is presented in the Section 6. Finally, we conclude in Section 7.

2 Preliminaries

Definition 2.1. (Fuzzy Set)[28] Let X be a universal set. A fuzzy set M of X is the collection of elememts α in X s.t., T (α) ∈ [0, 1]. Here T is called a membership function of M i.e., T : X → [0, 1].

Definition 2.2. (Fuzzy Graph)[23] A f-graph of the graph G′ = (V _G , E _G ) is a pair G = (ϒ, Γ), where ϒ : V → [0, 1] is a fuzzy set V _G and Γ : V _G  × V _G  → [0, 1] is a fuzzy relation on V _G s.t., Γ (x, y) ≤ ϒ (x) ∧ ϒ (y), ∀ (x, y) ∈ V _G  × V _G .

We note that, if Γ is not a symmetric relation on V _G , then G is called directed fuzzy graph.

Definition 2.3. (Neutrosophic Set)[3] Let X be a universal set. A neutrosophic set N of X is of the form N = { ( < T N ( ξ ) , I N ( ξ ) , F N ( ξ ) > ξ ) : 0 ≤ T N ( ξ ) + I N ( ξ ) + F N ( ξ ) ≤ 3 , ξ ∈ X } .

Here T _N  : X → [0, 1], I _N  : X → [0, 1], F _N  : X → [0, 1] are called truth membership function, indeterminacy membership function and falsity membership function respectively.

Definition 2.4. (Single-valued Neutrosophic Graph (SVNG))[3] A graph G = (V _G , ϒ, Γ) of the graph G′ = (V _G , E _G ), where ϒ = (T _ϒ , I _ϒ , F _ϒ ), and Γ = (T _Γ , I _Γ , F _Γ ) is said to be SVNG if

0 ≤ T _ϒ  (ξ) + I _ϒ  (ξ) + F _ϒ  (ξ) ≤3, ∀ξ ∈ V _G ;

Definition 2.5. (Complete SVNG) A SVNG G = (V _G , ϒ, Γ) of the graph G′ = (V _G , E _G ), where ϒ = (T _ϒ , I _ϒ , F _ϒ ), Γ = (T _Γ , I _Γ , F _Γ ) is said to be complete if T _Γ  (ξη) = T _ϒ  (ξ) ∧ T _ϒ  (η), I _Γ  (ξη) = I _ϒ  (ξ) ∧ I _ϒ  (η) and F _Γ  (ξη) = F _ϒ  (ξ) ∧ F _ϒ  (η), ∀ ξ η ∈ V ˜ G 2 .

3 The m-polar SVNG

Throughout the paper [0, 1]  ^r (r-th power of [0, 1]) is considered a poset with the point-wise order ≤, where r is an arbitrary ordinal number (we make an appointment that m = {n|n < m} when m > 0), ≤ is defined by u ≤ v ⇔ ϕ _j  (u) ≤ ϕ _j  (y) for each j ∈ m (u, v ∈ [0, 1]  ^m ) and ϕ _j  : [0, 1]  ^m  → [0, 1] is the j-th projection mapping (j ∈ m).

Definition 3.1. (m-Polar Fuzzy Graph)[19] An m-polar f-graph of the graph G′ = (V _G , E _G ) is a pair G = (ϒ, Γ), where ϒ : V _G  → [0, 1]  ^m is a fuzzy set on V _G and Γ : V ˜ G 2 → [ 0 , 1 ] m is a fuzzy relation on V _G s.t., for each j = 1, 2, 3, . . . , m,

ϕ _j  (Γ (xy)) ≤ ϕ _j  (ϒ (x)) ∧ ϕ _j  (ϒ (y)), ∀ xy ∈ V ˜ G 2 .

Definition 3.2. (m-Polar SVNS) Let X be a universal set. An m-polar SVNS N of X is of the form

N = { ( < T N ( ξ ) , I N ( ξ ) , F N ( ξ ) > ξ ) : 0 ≤ ϕ j ( T N ( ξ ) ) + ϕ j ( I N ( ξ ) ) + ϕ j ( F N ( ξ ) ) ≤ 3 , ξ ∈ X and

for each j = 1, 2, 3, . . . , m}.

Here T _N  : X → [0, 1]  ^m , I _N  : X → [0, 1]  ^m , F _N  : X → [0, 1]  ^m are called truth membership function, indeterminacy membership function and falsity membership function respectively.

Definition 3.3. (m-Polar SVNG) A graph G = (V _G , ϒ, Γ) of the graph G′ = (V _G , E _G ), where ϒ = (T _ϒ , I _ϒ , F _ϒ ), and Γ = (T _Γ , I _Γ , F _Γ ) is said to be m-polar SVNG if, for each j = 1, 2, 3, . . . , m,

0 ≤ ϕ _j  (T _ϒ  (ξ)) + ϕ _j  (I _ϒ  (ξ)) + ϕ _j  (F _ϒ  (ξ)) ≤3, ∀ξ ∈ V _G ;

4 Some operation on m-polar SVNG

Definition 4.1. (Cartesian Product (CP) of Two m-Polar SVNG’s) The CP of two m-Polar SVNG’s G = (V _G , ϒ₁, Γ₁) and H = (V _H , ϒ₂, Γ₂) of the graphs G′ = (V _G , E _G ) and H′ = (V _H , E _H ) respectively, is represented by G × H and is defined as (V _G  × V _H , ϒ₁ × ϒ₂, Γ₁ × Γ₂), where ϒ₁ × ϒ₂ = (T _{ϒ 1}  × T _{ϒ 2} , I _{ϒ 1}  × I _{ϒ 2} , F _{ϒ 1}  × F _{ϒ 2} ) and Γ₁ × Γ₂ = (T _{Γ 1}  × T _{Γ 2} , I _{Γ 1}  × I _{Γ 2} , F _{Γ 1}  × F _{Γ 2} ) s.t., for each j = 1, 2, 3, . . . , m,

∀ (α, β) ∈ V _G  × V _H ,

ϕ _j  ((T _{ϒ 1}  × T _{ϒ 2} ) (α, β)) = ϕ _j  (T _{ϒ 1}  (α)) ∧ ϕ _j  (T _{ϒ 2}  (β));

Example 4.1. We consider two graphs G′ = (V _G , E _G ) and H′ = (V _H , E _H ) s.t., V _G  = {α, β}, V _H  = {γ, δ}, E _G  = {αβ} and E _H  = {γδ}. Cosider the 3-polar SVNG’s G = (V _G , ϒ _G , Γ _G ) and H = (V _H , ϒ _H , Γ _H ) of the graphs G′ = (V _G , E _G ) and H′ = (V _H , E _H ) respectively, where ϒ _G  = (T _{ϒ G} , I _{ϒ G} , F _{ϒ G} ), ϒ _H  = (T _{ϒ H} , I _{ϒ H} , F _{ϒ H} ), Γ _G  = (T _{Γ G} , I _{Γ G} , F _{Γ G} ) and Γ _H  = (T _{Γ H} , I _{Γ H} , F _{Γ H} ) s.t.,

T ϒ G = { < 0.3 , 0.4 , 0.6 > α , < 0.3 , 0.5 , 0.7 > β } ,

T Γ G = { < 0.1 , 0.2 , 0.5 > α β } ,

I ϒ G = { < 0.6 , 0.7 , 0.4 > α , < 0.4 , 0.3 , 0.6 > β } ,

I Γ G = { < 0.3 , 0.2 , 0.4 > α β } ,

F ϒ G = { < 0.5 , 0.4 , 0.7 > α , < 0.4 , 0.5 , 0.4 > β } ,

F Γ G = { < 0.5 , 0.5 , 0.6 > α β } ,

T ϒ H = { < 0.7 , 0.8 , 0.9 > γ , < 0.5 , 0.6 , 0.7 > δ } ,

T Γ H = { < 0.4 , 0.5 , 0.7 > γ δ } ,

I ϒ H = { < 0.4 , 0.5 , 0.7 > γ , < 0.3 , 0.4 , 0.2 > δ } ,

I Γ H = { < 0.3 , 0.3 , 0.1 > γ δ } ,

F ϒ H = { < 0.5 , 0.4 , 0.2 > γ , < 0.5 , 0.7 , 0.4 > δ } ,

F Γ H = { < 0.4 , 0.5 , 0.3 > γ δ } .

Compute G × H. Then we have the following:

(T _{Γ G} × T _{Γ H} ) ((α, γ) (α, δ)) = <0.3, 0.4, 0.6 >,

(I _{Γ G} × I _{Γ H} ) ((α, γ) (α, δ)) = <0.4, 0.5, 0.4 >,

(F _{Γ G} × F _{Γ H} ) ((α, γ) (α, δ)) = <0.5, 0.5, 0.7 >.

Similarly, we get (T _{ϒ G} × T _{ϒ G} ) (α, γ) = <0.3, 0.4, 0.6 >. See Fig. 1.

OPEN IN VIEWER

Fig. 1

CP of two 3-polar SVNG’s.

Theorem 4.2. The CP (G × H) = (V _G  × V _H , ϒ _G  × ϒ _H , Γ _G  × Γ _H ), where ϒ _G  × ϒ _H  = (T _{ϒ G}  × T _{ϒ H} , I _{ϒ G}  × I _{ϒ H} , F _{ϒ G}  × F _{ϒ H} ) and Γ _G  × Γ _H  = (T _{Γ G}  × T _{Γ H} , I _{Γ G}  × I _{Γ H} , F _{Γ G}  × F _{Γ H} ) of two m-polar SVNG’s G = (V _G , ϒ _G , Γ _G ) and H = (V _H , ϒ _H , Γ _H ) of the graphs G′ = (V _G , E _G ) and H′ = (V _H , E _H ) is an m-polar SVNG of G′ × H′.

Proof. Let γ ∈ V _G and αβ ∈ E _H . Then for each j = 1, 2, 3, . . . , m,

ϕ j ( ( T Γ G × T Γ H ) ( ( γ , α ) ( γ , β ) ) ) = ϕ j ( T Υ G ( γ ) ) ∧ ϕ j ( T Γ H ( α β ) ) ≤ ϕ j ( T Υ G ( γ ) ) ∧ { ϕ j ( T Υ H ( α ) ) ∧ ϕ j ( T Υ H ( β ) ) } = { ϕ j ( T Υ G ( γ ) ) ∧ ϕ j ( T Υ H ( α ) ) } ∧ { ϕ j ( T Υ G ( γ ) ) ∧ ϕ j ( T Υ H ( β ) ) } = ϕ j ( ( T Υ G × T Υ H ) ( γ , α ) ) ∧ ϕ j ( ( T Υ G × T Υ H ) ( γ , β ) ) .

∴ ϕ _j  ((T _{Γ G}  × T _{Γ H} ) ((γ, α) (γ, β))) ≤ ϕ _j  ((T _{ϒ G}  × T _{ϒ H} ) (γ, α)) ∧ ϕ _j  ((T _{ϒ G}  × T _{ϒ H} ) (γ, β)).

Let γ ∈ V _H and αβ ∈ E _G . Then for each j = 1, 2, 3, . . . , m,

ϕ j ( ( T Γ G × T Γ H ) ( ( α , γ ) ( β , γ ) ) ) = ϕ j ( T Γ G ( α β ) ) ∧ ϕ j ( T Υ H ( γ ) ) ≤ { ϕ j ( T Υ G ( α ) ) ∧ ϕ j ( T Υ G ( β ) ) } ∧ ϕ j ( T Υ H ( γ ) ) = { ϕ j ( T Υ G ( α ) ) ∧ ϕ j ( T Υ H ( γ ) ) } ∧ { ϕ j ( T Υ G ( β ) ) ∧ ϕ j ( T Υ H ( γ ) ) } = ϕ j ( ( T Υ G × T Υ H ) ( α , γ ) ∧ ϕ j ( ( T Υ G × T Υ H ) ( β , γ ) ) .

∴ ϕ _j  ((T _{Γ G}  × T _{Γ H} ) ((α, γ) (β, γ))) ≤ ϕ _j  ((T _{ϒ G}  × T _{ϒ H} ) (α, γ)) ∧ ϕ _j  ((T _{ϒ G}  × T _{ϒ H} ) (β, γ)).

Let ( α , β ) ( γ , δ ) ∈ ( V G × ˜ V H ) 2 - E . Then for each j = 1, 2, . . . , m,

ϕ _j  ((T _{Γ G}  × T _{Γ H} ) ((α, β) (γ, δ))) =0 ≤ ϕ _j  ((T _{ϒ G}  × T _{ϒ H} ) (α, β)) ∧ ϕ _j  ((T _{ϒ G}  × T _{ϒ H} ) (γ, δ)).

Let γ ∈ V _G and αβ ∈ E _H . Then for each j = 1, 2, 3, . . . , m,

ϕ j ( ( I Γ G × I Γ H ) ( ( γ , α ) ( γ , β ) ) ) = ϕ j ( I Υ G ( γ ) ) ∧ ϕ j ( I Γ H ( α β ) ) ≤ ϕ j ( I Υ G ( γ ) ) ∧ { ϕ j ( I Υ H ( α ) ) ∧ ϕ j ( I Υ H ( β ) ) } = { ϕ j ( I Υ G ( γ ) ) ∧ ϕ j ( I Υ H ( α ) ) } ∧ { ϕ j ( I Υ G ( γ ) ) ∧ ϕ j ( I Υ H ( β ) ) } = ϕ j ( ( I Υ G × I Υ H ) ( γ , α ) ) ∧ ϕ j ( ( I Υ G × I Υ H ) ( γ , β ) ) .

∴ ϕ _j  ((I _{Γ G}  × I _{Γ H} ) ((γ, α) (γ, β))) ≤ ϕ _j  ((I _{ϒ G}  × I _{ϒ H} ) (γ, α)) ∧ ϕ _j  ((I _{ϒ G}  × I _{ϒ H} ) (γ, β)).

Let γ ∈ V _H and αβ ∈ E _G . Then for each j = 1, 2, 3, . . . , m,

ϕ j ( ( I Γ G × I Γ H ) ( ( α , γ ) ( β , γ ) ) ) = ϕ j ( I Γ G ( α β ) ) ∧ ϕ j ( I Υ H ( γ ) ) ≤ { ϕ j ( I Υ G ( α ) ) ∧ ϕ j ( I Υ G ( β ) ) } ∧ ϕ j ( I Υ H ( γ ) ) = { ϕ j ( I Υ G ( α ) ) ∧ ϕ j ( I Υ H ( γ ) ) } ∧ { ϕ j ( I Υ G ( β ) ) ∧ ϕ j ( I Υ H ( γ ) ) } = ϕ j ( ( I Υ G × I Υ H ) ( α , γ ) ∧ ϕ j ( ( I Υ G × I Υ H ) ( β , γ ) ) .

∴ ϕ _j  ((I _{Γ G}  × I _{Γ H} ) ((α, γ) (β, γ))) ≤ ϕ _j  ((I _{ϒ G}  × I _{ϒ H} ) (α, γ)) ∧ ϕ _j  ((I _{ϒ G}  × I _{ϒ H} ) (β, γ)).

Let ( α , β ) ( γ , δ ) ∈ ( V G × ˜ V H ) 2 - E . Then for each j = 1, 2, . . . m,

ϕ _j  ((I _{Γ G}  × I _{Γ H} ) ((α, β) (γ, δ))) =0 ≤ ϕ _j  ((I _{ϒ G}  × I _{ϒ H} ) (α, β)) ∧ ϕ _j  ((I _{ϒ G}  × I _{ϒ H} ) (γ, δ)).

Let γ ∈ V _G and αβ ∈ E _H . Then for each j = 1, 2, 3, . . . , m,

ϕ j ( ( F Γ G × F Γ H ) ( ( γ , α ) ( γ , β ) ) ) = ϕ j ( F Υ G ( γ ) ) ∨ ϕ j ( F Γ H ( α β ) ) ≤ ϕ j ( F Υ G ( γ ) ) ∨ { ϕ j ( F Υ H ( α ) ) ∨ ϕ j ( F Υ H ( β ) ) } = { ϕ j ( F Υ G ( γ ) ) ∨ ϕ j ( F Υ H ( α ) ) } ∨ { ϕ j ( F Υ G ( γ ) ) ∨ ϕ j ( F Υ H ( β ) ) } = ϕ j ( ( F Υ G ​ × ​ F Υ H ) ( γ , α ) ) ∨ ϕ j ( ( F Υ G × F Υ H ) ( γ , β ) ) .

∴ ϕ _j  ((F _{Γ G}  × F _{Γ H} ) ((γ, α) (γ, β)))   ≤  ϕ _j  ((F _{ϒ G}   ×  F _{ϒ H} ) (γ, α)) ∨ ϕ _j  ((F _{ϒ G}  × F _{ϒ H} ) (γ, β)).

Let γ ∈ V _H and α, β ∈ E _G . Then for each j = 1, 2, 3, . . . , m,

ϕ j ( ( F Γ G × F Γ H ) ( ( α , γ ) ( β , γ ) ) ) = ϕ j ( F Γ G ( α β ) ) ∨ ϕ j ( F γ H ( γ ) ) ≤ ϕ j ( F Υ G ( α ) ) ∨ ϕ j ( F Υ G ( β ) ) ∧ ϕ j ( F Υ H ( γ ) ) = { ϕ j ( F Υ G ( α ) ) ∨ ϕ j ( F Υ H ( γ ) ) } ∨ { ϕ j ( F Υ G ( β ) ) ∨ ϕ j ( F Υ H ( γ ) ) } = ϕ j ( ( F Υ G ​ × ​ F Υ H ) ( α , γ ) ) ∨ ϕ j ( ( F Υ G × F Υ H ) ( β , γ ) ) .

∴ ϕ _j  ((F _{Γ G}   ×  F _{Γ H} ) ((α, γ) (β, γ)))   ≤  ϕ _j  ((F _{ϒ G}  × F _{ϒ H} ) (α, γ)) ∧ ϕ _j  ((F _{ϒ G}  × F _{ϒ G} ) (β, γ)).

Let ( α , β ) ( γ , δ ) ∈ ( V G × ˜ V H ) 2 - E . Then for each j = 1, 2, . . . m,

ϕ _j  ((F _{Γ G}  × F _{Γ H} ) ((α, β)  (γ, δ)))  = 0 ≤ ϕ _j  ((F _{ϒ G}  × F _{ϒ H} ) (α, β)) ∧ ϕ _j  ((F _{ϒ G}  × F _{ϒ H} ) (γ, δ)).

This completes the proof. □

Definition 4.2. (Composition of Two m-Polar SVNG’s) The composition G ∘ H = (V _G  × V _H , ϒ _G  ∘ ϒ _H , Γ _G  ∘ Γ _H ), where (ϒ _G  ∘ ϒ _H ) = (T _{ϒ G}   ∘  T _{ϒ H} ,  I _{ϒ G}   ∘  I _{ϒ H} ,  F _{ϒ G}  ∘ F _{ϒ H} ) and (Γ _G   ∘  Γ _H ) = (T _{Γ G}  ∘ T _{Γ H} , I _{Γ G}  ∘ I _{Γ H} , F _{Γ G}  ∘ F _{Γ H} ), of two m-polar SVNG’s G = (V _G , ϒ _G , Γ _G ) and H = (V _H , ϒ _H , Γ _H ), where ϒ _G  = (T _{ϒ G} , I _{ϒ G} , F _{ϒ G} ), ϒ _H  = (T _{ϒ H} , I _{ϒ H} , F _{ϒ H} ), Γ _G  = (T _{Γ G} , I _{Γ G} , F _{Γ G} ) and Γ _H  = (T _{Γ H} , I _{Γ H} , F _{Γ H} ) of the graphs G′ = (V _G , E _G ) and H′ = (V _H , E _H ) respectively, is defined as follows: for each j = 1, 2, . . . , m,

∀ (α, β) ∈ V _G  × V _H ,

ϕ _j  ((T _{ϒ G}  ∘ T _{ϒ H} ) (α, β)) = ϕ _j  (T _{ϒ G}  (α)) ∧ ϕ _j  (T _{ϒ H}  (β));

Example 4.3. We consider two graphs G′ = (V _G , E _G ) and H = (V _H , E _H ) s.t., V _G  = {x₁, x₂, x₃, x₄}, V _H  = {y₁, y₂}, E _G  = {x₂x₁,  x₂x₃,  x₂x₄} and E _H  = {y₁y₂}. Cosider the 3-polar SVNG’s G = (V _G , ϒ _G , Γ _G ) and H = (V _H , ϒ _H , Γ _H ) of the graphs G′ = (V _G , E _G ) and H′ = (V _H , E _H ) respectively where ϒ _G  = (T _{ϒ G} , I _{ϒ G} , F _{ϒ G} ), ϒ _H  = (T _{ϒ H} , I _{ϒ H} , F _{ϒ H} ), Γ _G  = (T _{Γ G} , I _{Γ G} , F _{Γ G} ) and Γ _H  = (T _{Γ H} , I _{Γ H} , F _{Γ H} ) s.t.,

T ϒ G = { < 0.6 , 0.5 , 0.4 > x 1 , < 0.3 , 0.4 , 0.6 > x 2 , < 0.3 , 0.5 , 0.7 > x 3 , < 0.7 , 0.8 , 0.75 > x 4 } ,

T Γ G = { < 0.3 , 0.2 , 0.4 > x 2 x 1 , < 0.1 , 0.2 , 0.5 > x 2 x 3 , < 0.3 , 0.4 , 0.55 > x 2 x 4 } ,

I ϒ G = { < 0.5 , 0.4 , 0.5 > x 1 , < 0.6 , 0.7 , 0.4 > x 2 , < 0.4 , 0.3 , 0.6 > x 3 , < 0.55 , 0.4 , 0.35 > x 4 } ,

I Γ G = { < 0.4 , 0.4 , 0.3 > x 2 x 1 , < 0.3 , 0.2 , 0.4 > x 2 x 3 , < 0.4 , 0.3 , 0.3 > x 2 x 4 } ,

F ϒ G = { < 0.4 , 0.3 , 0.6 > x 1 , < 0.5 , 0.4 , 0.7 > x 2 , < 0.4 , 0.5 , 0.4 > x 3 , < 0.4 , 0.3 , 0.6 > x 4 } ,

F Γ G = { < 0.4 , 0.3 , 0.5 > x 2 x 1 , < 0.4 , 0.3 , 0.35 > x 2 x 3 , < 0.4 , 0.25 , 0.55 > x 2 x 4 }

T ϒ H = { < 0.7 , 0.8 , 0.9 > y 1 , < 0.5 , 0.6 , 0.7 > y 2 } ,

T Γ H = { < 0.4 , 0.5 , 0.7 > y 1 y 2 } ,

I ϒ G = { < 0.4 , 0.5 , 0.7 > y 1 , < 0.3 , 0.4 , 0.2 > y 2 } ,

I Γ H = { < 0.3 , 0.3 , 0.1 > y 1 y 2 } ,

F ϒ H = { < 0.5 , 0.4 , 0.2 > y 1 , < 0.5 , 0.7 , 0.4 > y 2 } ,

F Γ H = { < 0.4 , 0.5 , 0.3 > y 1 y 2 } .

Compute G₁ ∘ G₂. Then we have the following:

(T _{Γ G} ∘ T _{Γ H} ) ((x₁, y₁) (x₁, y₂)) = <0.4, 0.5, 0.4 >,

(I _{Γ G} ∘ I _{Γ H} ) ((x₁, y₁) (x₁, y₂)) = <0.3, 0.3, 0.1 >,

(F _{Γ G} ∘ F _{Γ H} ) ((x₁, y₁) (x₁, y₂)) = <0.4, 0.5, 0.6 >.

Similarly, we get

(T _{ϒ G} ∘ T _{ϒ H} ) (x₁, y₁) = <0.6, 0.5, 0.4 >,

(I _{ϒ G} ∘ I _{ϒ H} ) (x₁, y₁) = <0.4, 0.4, 0.5 >,

(F _{ϒ G} ∘ F _{ϒ H} ) (x₁, y₁) = <0.5, 0.4, 0.5 >. See Fig. 2.

OPEN IN VIEWER

Fig. 2

Composition of two 3-polar SVNG’s.

Theorem 4.4. The composition (G ∘ H) = (V _G  × V _H , ϒ _G  ∘ ϒ _H , Γ _G  ∘ Γ _H ), where ϒ _G  ∘ ϒ _H  = (T _{ϒ G}  ∘ T _{ϒ H} , I _{ϒ G}  ∘ I _{ϒ H} , F _{ϒ G}  ∘ F _{ϒ H} ) and Γ _G  ∘ Γ _H  = (T _{Γ G}  ∘ T _{Γ H} , I _{Γ G}  ∘ I _{Γ H} , F _{Γ G}  ∘ F _{Γ H} ) of two m- polar SVNG’s G = (V _G , ϒ _G , Γ _G ) and H = (V _H , ϒ _H , Γ _H ) of the graphs G′ = (V _G , E _G ) and H′ = (V _H , E _H ) is an m-polar SVNG of G′ ∘ H′.

Proof. We consider E = {((α, γ) (β, γ)) : αβ ∈ E _G ,

  γ ∈ V _H } ∪ {((α, γ) (α, δ)) : α ∈ V _G ,   γδ ∈ E _H } and E⁰ = E ∪ {((α, γ) (β, δ)) : αγ ∈ E _G   γ ≠ δ ∈ V _H } . Let γ ∈ V _G and αβ ∈ E _H . Then for each j = 1, 2, 3, . . . , m, ϕ j ( ( T Γ G ∘ T Γ H ) ( ( γ , α ) ( γ , β ) ) ) = ϕ j ( T ϒ G ( γ ) ) ∧ ϕ j ( T Γ H ( α β ) ) ≤ ϕ j ( T ϒ G ( γ ) ) ∧ { ϕ j ( T ϒ H ( α ) ) ∧ ϕ j ( T ϒ H ( β ) ) } = { ϕ j ( T ϒ G ( γ ) ) ∧ ϕ j ( T ϒ H ( α ) ) } ∧ { ϕ j ( T ϒ G ( γ ) ) ∧ ϕ j ( T ϒ H ( β ) ) } = ϕ j ( ( T ϒ G ∘ T ϒ H ) ( γ , α ) ) ∧ ϕ j ( ( T ϒ G ∘ T ϒ H ) ( γ , β ) ) . ∴ ϕ _j  ((T _{Γ G}  ∘ T _{Γ H} ) ((γ, α) (γ, β))) ≤ ϕ _j  ((T _{ϒ G}  ∘ T _{ϒ H} ) (γ, α)) ∧ ϕ _j  ((T _{ϒ G}  ∘ T _{ϒ H} ) (γ, β)).

Let γ ∈ V _H and αβ ∈ E _G . Then for each j = 1, 2, 3, . . . , m, ϕ j ( ( T Γ G ∘ T Γ H ) ( ( α , γ ) ( β , γ ) ) ) = ϕ j ( T Γ G ( α β ) ) ∧ ϕ j ( T ϒ H ( γ ) ) ≤ { ϕ j ( T ϒ G ( α ) ) ∧ ϕ j ( T ϒ G ( β ) ) } ∧ ϕ j ( T ϒ H ( γ ) ) = { ϕ j ( T ϒ G ( α ) ) ∧ ϕ j ( T ϒ H ( γ ) ) } ∧ { ϕ j ( T ϒ G ( β ) ) ∧ ϕ j ( T ϒ H ( γ ) ) } = ϕ j ( ( T ϒ G ∘ T ϒ H ) ( α , γ ) ) ∧ ϕ j ( ( T ϒ G ∘ T ϒ H ) ( β , γ ) ) . ∴ ϕ _j  ((T _{Γ G}  ∘ T _{Γ H} ) ((α, γ) (β, γ))) ≤ ϕ _j  ((T _{ϒ G}  ∘ T _{ϒ H} ) (α, γ)) ∧ ϕ _j  ((T _{ϒ G}  ∘ T _{ϒ H} ) (β, γ)).

Let (α, β) (γ, δ) ∈ E₀ - E. Then for each j = 1, 2, 3, . . . , m, ϕ j ( ( T Γ G ∘ T Γ H ) ( ( α , β ) ( γ , δ ) ) ) = ϕ j ( T Γ G ( α γ ) ) ∧ ϕ j ( T ϒ H ( β ) ) ∧ ϕ j ( T ϒ H ( δ ) ) ≤ { ϕ j ( T ϒ G ( α ) ) ∧ ϕ j ( T ϒ G ( γ ) ) } ∧ ϕ j ( T ϒ H ( β ) ) ∧ ϕ j ( T ϒ H ( δ ) ) = { ϕ j ( T ϒ G ( α ) ) ∧ ϕ j ( T ϒ H ( β ) ) } ∧ { ϕ j ( T ϒ G ( γ ) ) ∧ ϕ j ( T ϒ H ( δ ) ) } = ϕ j ( ( T ϒ G ∘ T ϒ H ) ( α , β ) ) ∧ ϕ j ( ( T ϒ G ∘ T ϒ H ) ( γ , δ ) ) ∴ ϕ _j  ((T _{Γ G}  ∘ T _{Γ H} ) ((α, β) (γ, δ))) ≤ ϕ _j  ((T _{ϒ G}  ∘ T _{ϒ H} ) (α, β)) ∧ ϕ _j  ((T _{ϒ G}  ∘ T _{ϒ H} ) (γ, δ)).

Let ( α , β ) ( γ , δ ) ∈ ( V G × ˜ V H ) 2 - E . Then for each j = 1, 2, . . . m,

ϕ _j  ((T _{Γ G}  ∘ T _{Γ H} ) ((α, β) (γ, δ))) =0 ≤ ϕ _j  ((T _{ϒ G}  ∘ T _{ϒ H} ) (α, β)) ∧ ϕ _j  ((T _{ϒ G}  ∘ T _{ϒ H} ) (γ, δ)).

Let γ ∈ V _G and αβ ∈ E _H . Then for each j = 1, 2, 3, . . . , m,

ϕ j ( ( I Γ G ∘ I Γ H ) ( ( γ , α ) ( γ , β ) ) ) = ϕ j ( I ϒ G ( γ ) ) ∧ ϕ j ( I Γ H ( α β ) ) ≤ ϕ j ( I ϒ G ( γ ) ) ∧ { ϕ j ( I ϒ H ( α ) ) ∧ ϕ j ( I ϒ H ( β ) ) } = { ϕ j ( I ϒ G ( γ ) ) ∧ ϕ j ( I ϒ H ( α ) ) } ∧ { ϕ j ( I ϒ G ( γ ) ) ∧ ϕ j ( I ϒ H ( β ) ) } = ϕ j ( ( I ϒ G ∘ I ϒ H ) ( γ , α ) ) ∧ ϕ j ( ( I ϒ G ∘ I ϒ H ) ( γ , β ) ) .

∴ ϕ _j  ((I _{Γ G}  ∘ I _{Γ H} ) ((γ, α) (γ, β))) ≤ ϕ _j  ((I _{ϒ G}  ∘ I _{ϒ H} ) (γ, α)) ∧ ϕ _j  ((I _{ϒ G}  ∘ I _{ϒ H} ) (γ, β)).

Let γ ∈ V _H and αβ ∈ E _G . Then for each j = 1, 2, 3, . . . , m, ϕ j ( ( I Γ G ∘ I Γ H ) ( ( α , γ ) ( β , γ ) ) ) = ϕ j ( I Γ G ( α β ) ) ∧ ϕ j ( I ϒ H ( γ ) ) ≤ { ϕ j ( I ϒ G ( α ) ) ∧ ϕ j ( I ϒ G ( β ) ) } ∧ ϕ j ( I ϒ H ( γ ) ) = { ϕ j ( I ϒ G ( α ) ) ∧ ϕ j ( I ϒ H ( γ ) ) } ∧ { ϕ j ( I ϒ G ( β ) ) ∧ ϕ j ( I ϒ H ( γ ) ) } = ϕ j ( ( I ϒ G ∘ I ϒ H ) ( α , γ ) ∧ ϕ j ( ( I ϒ G ∘ I ϒ H ) ( β , γ ) ) . ∴ ϕ _j  ((I _{Γ G}  ∘ I _{Γ H} ) ((α, γ) (β, γ))) ≤ ϕ _j  ((I _{ϒ G}  ∘ I _{ϒ H} ) (α, γ)) ∧ ϕ _j  ((I _{ϒ G}  ∘ I _{ϒ H} ) (β, γ)).

Let ( α , β ) ( γ , δ ) ∈ ( V G × ˜ V H ) 2 - E . Then for each j = 1, 2, . . . m,

ϕ _j  ((I _{Γ G}  ∘ I _{Γ H} ) ((α, β) (γ, δ))) =0 ≤ ϕ _j  ((I _{ϒ G}  ∘ I _{ϒ H} ) (α, β)) ∧ ϕ _j  ((I _{ϒ G}  ∘ I _{ϒ H} ) (γ, δ)).

Let (α, β) (γ, δ) ∈ E₀ - E. Then for each j = 1, 2, 3, . . . , m, ϕ j ( ( I Γ G ∘ I Γ H ) ( ( α , β ) ( γ , δ ) ) ) = ϕ j ( I Γ G ( α γ ) ) ∧ ϕ j ( I ϒ H ( β ) ) ∧ ϕ j ( I ϒ H ( δ ) ) ≤ { ϕ j ( I ϒ G ( α ) ) ∧ ϕ j ( I ϒ H ( γ ) ) } ∧ ϕ j ( I ϒ G ( β ) ) ∧ ϕ j ( I ϒ H ( δ ) ) = { ϕ j ( I ϒ G ( α ) ) ∧ ϕ j ( I ϒ H ( β ) ) } ∧ { ϕ j ( I ϒ G ( γ ) ) ∧ ϕ j ( I ϒ H ( δ ) ) } = ϕ j ( ( I ϒ G ∘ I ϒ H ) ( α , β ) ) ∧ ϕ j ( ( I ϒ G ∘ I ϒ H ) ( γ , δ ) ) . ∴ ϕ _j  ((I _{Γ G}  ∘ I _{Γ H} ) ((α, β) (γ, δ))) ≤ ϕ _j  ((I _{ϒ G}  ∘ I _{ϒ H} ) (α, β)) ∧ ϕ _j  ((I _{ϒ G}  ∘ I _{ϒ H} ) (γ, δ)).