A study of m -polar neutrosophic graph with applications

Abstract

Information in many real life problems collect from multi agents, i.e., "multipolar information" exists. This multipolar information cannot be properly modeled by m- polar fuzzy graph or intutionistic fuzzy graph. An m-polar neutrosophic model is very much efficient for such real word problems which can construct more precise, flexible, and comparable system as compared to the classical, fuzzy and neutrosophic graph models. In this paper, we present the definition of m-polar neutrosophic graph model. Some new operations, such as union, join, composition and ring sum of two m-polar neutrosophic graph are defined here. We define six new products on m-polar neutrosophic graphs namely strong product, semi strong product, complete product, direct product, cartesian product and lexicographic product. Some idea of complement, isomorphism, weak and co weak isomorphism on m-polar neutrosophic graph are introduced here. We also present several associated properties and theorems of m-polar neutrosophic graph. We introduce a model of m-polar neutrosophic graph, which is applied in evaluating the teacher’s performance of a college. The performances of the teachers are computed based on the response score (feedback) of the students of the college. We also present a numerical example to illustrate our proposed model.

Keywords

Multipolar information neutrosophic set union direct product

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 $N = {(\frac{< T_{N} (ξ), I_{N} (ξ), F_{N} (ξ) >}{ξ}) : 0 \leq T_{N} (ξ) + I_{N} (ξ) + F_{N} (ξ) \leq 3, ξ \in X}$ .

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;

T_Γ : V_G × V_G → [0, 1], I_Γ : V_G × V_G → [0, 1] and F_Γ : V_G × V_G → [0, 1] are the truth membership function, indeterminacy membership function and falsity membership fuction of the edge (ξ, η) respectively, s.t., 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), ∀ ξη ∈ E_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), $\forall ξ η \in {\tilde{V}}_{G}^{2}$ .

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 $Γ : {\tilde{V}}_{G}^{2} \to [0, 1]^{m}$ 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)), $\forall xy \in {\tilde{V}}_{G}^{2}$ .

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

$N = {(\frac{< T_{N} (ξ), I_{N} (ξ), F_{N} (ξ) >}{ξ}) : 0 \leq ϕ_{j} (T_{N} (ξ)) + ϕ_{j} (I_{N} (ξ)) + ϕ_{j} (F_{N} (ξ)) \leq 3, ξ \in X$ 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;

$T_{Γ} : {\tilde{V}}_{G}^{2} \to [0, 1]^{m}$ , $I_{Γ} : {\tilde{V}}_{G}^{2} \to [0, 1]^{m}$ and $F_{Γ} : {\tilde{V}}_{G}^{2} \to [0, 1]^{m}$ are the truth membership function, indeterminacy membership function and falsity membership fuction of the edge ξη respectively, s.t., ϕ_j (T_Γ (ξη)) ≤ ϕ_j (T_ϒ (ξ)) ∧ ϕ_j (T_ϒ (η)), ϕ_j (I_Γ (ξη)) ≤ ϕ_j (I_ϒ (ξ)) ∧ ϕ_j (I_ϒ (η)) and ϕ_j (F_Γ (ξη)) ≤ ϕ_j (F_ϒ (ξ)) ∨ ϕ_j (F_ϒ (η)), ∀ ξη ∈ E_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));

ϕ_j ((I_{ϒ
₁} × I_{ϒ
₂}) (α, β)) = ϕ_j (I_{ϒ
₁} (α)) ∧ ϕ_j (I_{ϒ
₂} (β));

ϕ_j ((F_{ϒ
₁} × F_{ϒ
₂}) (α, β)) = ϕ_j (F_{ϒ
₁} (α)) ∨ ϕ_j (F_{ϒ
₂} (β));

∀γ ∈ V_G and ∀αβ ∈ E_H,

ϕ_j ((T_{Γ
₁} × T_{Γ
₂}) ((γ, α) (γ, β))) = ϕ_j (T_{ϒ
₁} (γ)) ∧ ϕ_j (T_{Γ
₂} (αβ));

ϕ_j ((I_{Γ
₁} × I_{Γ
₂}) ((γ, α) (γ, β))) = ϕ_j (I_{ϒ
₁} (γ)) ∧ ϕ_j (I_{Γ
₂} (αβ));

ϕ_j ((F_{Γ
₁} × F_{Γ
₂}) ((γ, α) (γ, β))) = ϕ_j (F_{ϒ
₁} (γ)) ∨ ϕ_j (F_{Γ
₂} (αβ));

∀γ ∈ V_H and ∀αβ ∈ E_G,

ϕ_j ((T_{Γ
₁} × T_{Γ
₂}) ((α, γ) (β, γ))) = ϕ_j (T_{Γ
₁} (αβ)) ∧ ϕ_j (T_{ϒ
₂} (γ));

ϕ_j ((I_{Γ
₁} × I_{Γ
₂}) ((α, γ) (β, γ))) = ϕ_j (I_{Γ
₁} (αβ)) ∧ ϕ_j (I_{ϒ
₂} (γ));

ϕ_j ((F_{Γ
₁} × F_{Γ
₂}) ((α, γ) (β, γ))) = ϕ_j (F_{Γ
₁} (αβ)) ∨ ϕ_j (F_{ϒ
₂} (γ));

$\forall (α, β) (γ, δ) \in (V_{G} \tilde{\times} V_{H})^{2} - E$ ,

ϕ_j ((T_{Γ
₁} × T_{Γ
₂}) ((α, β) (γ, δ))) =0;

ϕ_j ((I_{Γ
₁} × I_{Γ
₂}) ((α, β) (γ, δ))) =0;

ϕ_j ((F_{Γ
₁} × F_{Γ
₂}) ((α, β) (γ, δ))) =0.

Here E = {((α, γ) (β, γ)) : αβ ∈ E_G, γ ∈ V_H} ∪ {((α, γ) (α, δ)) : α ∈ V_G, γδ ∈ E_H}.

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.,

$T_{ϒ_{G}} = {\frac{< 0.3, 0.4, 0.6 >}{α}, \frac{< 0.3, 0.5, 0.7 >}{β}}$ ,

$T_{Γ_{G}} = {\frac{< 0.1, 0.2, 0.5 >}{α β}}$ ,

$I_{ϒ_{G}} = {\frac{< 0.6, 0.7, 0.4 >}{α}, \frac{< 0.4, 0.3, 0.6 >}{β}}$ ,

$I_{Γ_{G}} = {\frac{< 0.3, 0.2, 0.4 >}{α β}}$ ,

$F_{ϒ_{G}} = {\frac{< 0.5, 0.4, 0.7 >}{α}, \frac{< 0.4, 0.5, 0.4 >}{β}}$ ,

$F_{Γ_{G}} = {\frac{< 0.5, 0.5, 0.6 >}{α β}}$ ,

$T_{ϒ_{H}} = {\frac{< 0.7, 0.8, 0.9 >}{γ}, \frac{< 0.5, 0.6, 0.7 >}{δ}}$ ,

$T_{Γ_{H}} = {\frac{< 0.4, 0.5, 0.7 >}{γ δ}}$ ,

$I_{ϒ_{H}} = {\frac{< 0.4, 0.5, 0.7 >}{γ}, \frac{< 0.3, 0.4, 0.2 >}{δ}}$ ,

$I_{Γ_{H}} = {\frac{< 0.3, 0.3, 0.1 >}{γ δ}}$ ,

$F_{ϒ_{H}} = {\frac{< 0.5, 0.4, 0.2 >}{γ}, \frac{< 0.5, 0.7, 0.4 >}{δ}}$ ,

$F_{Γ_{H}} = {\frac{< 0.4, 0.5, 0.3 >}{γ δ}}$ .

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.

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,

$\begin{array}{l} ϕ_{j} ((T_{Γ_{G}} \times T_{Γ_{H}}) ((γ, α) (γ, β))) = ϕ_{j} (T_{Υ_{G}} (γ)) \land ϕ_{j} (T_{Γ_{H}} (α β)) \\ \leq ϕ_{j} (T_{Υ_{G}} (γ)) \land {ϕ_{j} (T_{Υ_{H}} (α)) \land ϕ_{j} (T_{Υ_{H}} (β))} \\ = {ϕ_{j} (T_{Υ_{G}} (γ)) \land ϕ_{j} (T_{Υ_{H}} (α))} \land {ϕ_{j} (T_{Υ_{G}} (γ)) \land ϕ_{j} (T_{Υ_{H}} (β))} \\ = ϕ_{j} ((T_{Υ_{G}} \times T_{Υ_{H}}) (γ, α)) \land ϕ_{j} ((T_{Υ_{G}} \times T_{Υ_{H}}) (γ, β)) . \end{array}$

∴ 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,

$\begin{array}{l} ϕ_{j} ((T_{Γ_{G}} \times T_{Γ_{H}}) ((α, γ) (β, γ))) = ϕ_{j} (T_{Γ_{G}} (α β)) \land ϕ_{j} (T_{Υ_{H}} (γ)) \\ \leq {ϕ_{j} (T_{Υ_{G}} (α)) \land ϕ_{j} (T_{Υ_{G}} (β))} \land ϕ_{j} (T_{Υ_{H}} (γ)) \\ = {ϕ_{j} (T_{Υ_{G}} (α)) \land ϕ_{j} (T_{Υ_{H}} (γ))} \land {ϕ_{j} (T_{Υ_{G}} (β)) \land ϕ_{j} (T_{Υ_{H}} (γ))} \\ = ϕ_{j} ((T_{Υ_{G}} \times T_{Υ_{H}}) (α, γ) \land ϕ_{j} ((T_{Υ_{G}} \times T_{Υ_{H}}) (β, γ)) . \end{array}$

∴ 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 $(α, β) (γ, δ) \in (V_{G} \tilde{\times} V_{H})^{2} - E .$ 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,

$\begin{array}{l} ϕ_{j} ((I_{Γ_{G}} \times I_{Γ_{H}}) ((γ, α) (γ, β))) = ϕ_{j} (I_{Υ_{G}} (γ)) \land ϕ_{j} (I_{Γ_{H}} (α β)) \\ \leq ϕ_{j} (I_{Υ_{G}} (γ)) \land {ϕ_{j} (I_{Υ_{H}} (α)) \land ϕ_{j} (I_{Υ_{H}} (β))} \\ = {ϕ_{j} (I_{Υ_{G}} (γ)) \land ϕ_{j} (I_{Υ_{H}} (α))} \land {ϕ_{j} (I_{Υ_{G}} (γ)) \land ϕ_{j} (I_{Υ_{H}} (β))} \\ = ϕ_{j} ((I_{Υ_{G}} \times I_{Υ_{H}}) (γ, α)) \land ϕ_{j} ((I_{Υ_{G}} \times I_{Υ_{H}}) (γ, β)) . \end{array}$

∴ 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,

$\begin{array}{l} ϕ_{j} ((I_{Γ_{G}} \times I_{Γ_{H}}) ((α, γ) (β, γ))) = ϕ_{j} (I_{Γ_{G}} (α β)) \\ \land ϕ_{j} (I_{Υ_{H}} (γ)) \\ \leq {ϕ_{j} (I_{Υ_{G}} (α)) \land ϕ_{j} (I_{Υ_{G}} (β))} \land ϕ_{j} (I_{Υ_{H}} (γ)) \\ = {ϕ_{j} (I_{Υ_{G}} (α)) \land ϕ_{j} (I_{Υ_{H}} (γ))} \land {ϕ_{j} (I_{Υ_{G}} (β)) \land ϕ_{j} (I_{Υ_{H}} (γ))} \\ = ϕ_{j} ((I_{Υ_{G}} \times I_{Υ_{H}}) (α, γ) \land ϕ_{j} ((I_{Υ_{G}} \times I_{Υ_{H}}) (β, γ)) . \end{array}$

∴ 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 $(α, β) (γ, δ) \in (V_{G} \tilde{\times} V_{H})^{2} - E .$ 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,

$\begin{array}{l} ϕ_{j} ((F_{Γ_{G}} \times F_{Γ_{H}}) ((γ, α) (γ, β))) = ϕ_{j} (F_{Υ_{G}} (γ)) \\ \lor ϕ_{j} (F_{Γ_{H}} (α β)) \\ \leq ϕ_{j} (F_{Υ_{G}} (γ)) \lor {ϕ_{j} (F_{Υ_{H}} (α)) \lor ϕ_{j} (F_{Υ_{H}} (β))} \\ = {ϕ_{j} (F_{Υ_{G}} (γ)) \lor ϕ_{j} (F_{Υ_{H}} (α))} \lor {ϕ_{j} (F_{Υ_{G}} (γ)) \lor ϕ_{j} (F_{Υ_{H}} (β))} \\ = ϕ_{j} ((F_{Υ_{G}} \times F_{Υ_{H}}) (γ, α)) \lor ϕ_{j} ((F_{Υ_{G}} \times F_{Υ_{H}}) (γ, β)) . \end{array}$

∴ 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,

$\begin{array}{l} ϕ_{j} ((F_{Γ_{G}} \times F_{Γ_{H}}) ((α, γ) (β, γ))) = ϕ_{j} (F_{Γ_{G}} (α β)) \lor \\ ϕ_{j} (F_{γ_{_{H}}} (γ)) \\ \leq ϕ_{j} (F_{Υ_{G}} (α)) \lor ϕ_{j} (F_{Υ_{G}} (β)) \land ϕ_{j} (F_{Υ_{H}} (γ)) \\ = {ϕ_{j} (F_{Υ_{G}} (α)) \lor ϕ_{j} (F_{Υ_{H}} (γ))} \lor {ϕ_{j} (F_{Υ_{G}} (β)) \lor \\ ϕ_{j} (F_{Υ_{H}} (γ))} \\ = ϕ_{j} ((F_{Υ_{G}} \times F_{Υ_{H}}) (α, γ)) \lor ϕ_{j} ((F_{Υ_{G}} \times F_{Υ_{H}}) (β, γ)) . \end{array}$

∴ 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 $(α, β) (γ, δ) \in (V_{G} \tilde{\times} V_{H})^{2} - E .$ 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));

ϕ_j ((I_{ϒ
_G} ∘ I_{ϒ
_H}) (α, β)) = ϕ_j (I_{ϒ
_G} (α)) ∧ ϕ_j (I_{ϒ
_H} (β));

ϕ_j ((F_{ϒ
_G} ∘ F_{ϒ
_H}) (α, β)) = ϕ_j (F_{ϒ
_G} (α)) ∨ ϕ_j (F_{ϒ
_H} (β));

∀γ ∈ V_G and ∀ (α, β) ∈ E_H,

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 ((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 ((F_{Γ
_G} ∘ F_{Γ
_H}) ((γ, α) (γ, β))) = ϕ_j (F_{ϒ
_G} (γ)) ∨ ϕ_j (F_{Γ
_H} (α, β));

∀γ ∈ V_H and ∀αβ ∈ E_G,

ϕ_j ((T_{Γ
_G} ∘ T_{Γ
_H}) ((α, γ) (β, γ))) = ϕ_j (T_{Γ
_G} (αβ)) ∧ ϕ_j (T_{ϒ
_H} (γ));

ϕ_j ((I_{Γ
_G} ∘ I_{Γ
_H}) ((α, γ) (β, γ))) = ϕ_j (I_{Γ
_G} (αβ)) ∧ ϕ_j (I_{ϒ
_H} (γ));

ϕ_j ((F_{Γ
_G} ∘ F_{Γ
_H}) ((α, γ) (β, γ))) = ϕ_j (F_{Γ
_G} (αβ)) ∨ ϕ_j (F_{ϒ
_H} (γ));

∀ (α, β) (γ, δ) ∈ E₀ - E,

ϕ_j ((T_{Γ
_G} ∘ T_{Γ
_H}) ((α, β) (γ, δ))) = ϕ_j (T_{Γ
_G} (αγ)) ∧ ϕ_j (T_{ϒ
_H} (β)) ∧ ϕ_j (T_{ϒ
_H} (δ));

ϕ_j ((I_{Γ
_G} ∘ I_{Γ
_H}) ((α, β) (γ, δ))) = ϕ_j (I_{Γ
_G} (αγ)) ∧ ϕ_j (I_{ϒ
_H} (β)) ∧ ϕ_j (I_{ϒ
_H} (δ));

ϕ_j ((F_{Γ
_G} ∘ F_{Γ
_H}) ((α, β) (γ, δ))) = ϕ_j (F_{Γ
_G} (αγ)) ∨ ϕ_j (F_{ϒ
_H} (β)) ∨ ϕ_j (F_{ϒ
_H} (δ));

$\forall (α, β) (γ, δ) \in (V_{G} \tilde{\times} V_{H})^{2} - E_{0}$ ,

ϕ_j ((T_{Γ
_G} ∘ T_{Γ
_H}) ((α, β) (γ, δ))) =0;

ϕ_j ((I_{Γ
_G} ∘ I_{Γ
_H}) ((α, β) (γ, δ))) =0;

ϕ_j ((F_{Γ
_G} ∘ F_{Γ
_H}) ((α, β) (γ, δ))) =0.

Here 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} .

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.,

$T_{ϒ_{G}} = {\frac{< 0.6, 0.5, 0.4 >}{x_{1}}, \frac{< 0.3, 0.4, 0.6 >}{x_{2}}, \frac{< 0.3, 0.5, 0.7 >}{x_{3}}, \frac{< 0.7, 0.8, 0.75 >}{x_{4}}}$ ,

$T_{Γ_{G}} = {\frac{< 0.3, 0.2, 0.4 >}{x_{2} x_{1}}, \frac{< 0.1, 0.2, 0.5 >}{x_{2} x_{3}}, \frac{< 0.3, 0.4, 0.55 >}{x_{2} x_{4}}}$ ,

$I_{ϒ_{G}} = {\frac{< 0.5, 0.4, 0.5 >}{x_{1}}, \frac{< 0.6, 0.7, 0.4 >}{x_{2}}, \frac{< 0.4, 0.3, 0.6 >}{x_{3}}, \frac{< 0.55, 0.4, 0.35 >}{x_{4}}}$ ,

$I_{Γ_{G}} = {\frac{< 0.4, 0.4, 0.3 >}{x_{2} x_{1}}, \frac{< 0.3, 0.2, 0.4 >}{x_{2} x_{3}}, \frac{< 0.4, 0.3, 0.3 >}{x_{2} x_{4}}}$ ,

$F_{ϒ_{G}} = {\frac{< 0.4, 0.3, 0.6 >}{x_{1}}, \frac{< 0.5, 0.4, 0.7 >}{x_{2}}, \frac{< 0.4, 0.5, 0.4 >}{x_{3}}, \frac{< 0.4, 0.3, 0.6 >}{x_{4}}}$ ,

$F_{Γ_{G}} = {\frac{< 0.4, 0.3, 0.5 >}{x_{2} x_{1}}, \frac{< 0.4, 0.3, 0.35 >}{x_{2} x_{3}}, \frac{< 0.4, 0.25, 0.55 >}{x_{2} x_{4}}}$

$T_{ϒ_{H}} = {\frac{< 0.7, 0.8, 0.9 >}{y_{1}}, \frac{< 0.5, 0.6, 0.7 >}{y_{2}}}$ ,

$T_{Γ_{H}} = {\frac{< 0.4, 0.5, 0.7 >}{y_{1} y_{2}}}$ ,

$I_{ϒ_{G}} = {\frac{< 0.4, 0.5, 0.7 >}{y_{1}}, \frac{< 0.3, 0.4, 0.2 >}{y_{2}}}$ ,

$I_{Γ_{H}} = {\frac{< 0.3, 0.3, 0.1 >}{y_{1} y_{2}}}$ ,

$F_{ϒ_{H}} = {\frac{< 0.5, 0.4, 0.2 >}{y_{1}}, \frac{< 0.5, 0.7, 0.4 >}{y_{2}}}$ ,

$F_{Γ_{H}} = {\frac{< 0.4, 0.5, 0.3 >}{y_{1} y_{2}}}$ .

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.

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, $\begin{matrix} ϕ_{j} ((T_{Γ_{G}} \circ T_{Γ_{H}}) ((γ, α) (γ, β))) \\ = ϕ_{j} (T_{ϒ_{G}} (γ)) \land ϕ_{j} (T_{Γ_{H}} (α β)) \\ \leq ϕ_{j} (T_{ϒ_{G}} (γ)) \land {ϕ_{j} (T_{ϒ_{H}} (α)) \land ϕ_{j} (T_{ϒ_{H}} (β))} \\ = {ϕ_{j} (T_{ϒ_{G}} (γ)) \land ϕ_{j} (T_{ϒ_{H}} (α))} \land {ϕ_{j} (T_{ϒ_{G}} (γ)) \land \\ ϕ_{j} (T_{ϒ_{H}} (β))} \\ = ϕ_{j} ((T_{ϒ_{G}} \circ T_{ϒ_{H}}) (γ, α)) \land ϕ_{j} ((T_{ϒ_{G}} \circ T_{ϒ_{H}}) (γ, β)) . \end{matrix}$ ∴ 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, $\begin{matrix} ϕ_{j} ((T_{Γ_{G}} \circ T_{Γ_{H}}) ((α, γ) (β, γ))) \\ = ϕ_{j} (T_{Γ_{G}} (α β)) \land ϕ_{j} (T_{ϒ_{H}} (γ)) \\ \leq {ϕ_{j} (T_{ϒ_{G}} (α)) \land ϕ_{j} (T_{ϒ_{G}} (β))} \land ϕ_{j} (T_{ϒ_{H}} (γ)) \\ = {ϕ_{j} (T_{ϒ_{G}} (α)) \land ϕ_{j} (T_{ϒ_{H}} (γ))} \land {ϕ_{j} (T_{ϒ_{G}} (β)) \land \\ ϕ_{j} (T_{ϒ_{H}} (γ))} \\ = ϕ_{j} ((T_{ϒ_{G}} \circ T_{ϒ_{H}}) (α, γ)) \land ϕ_{j} ((T_{ϒ_{G}} \circ T_{ϒ_{H}}) (β, γ)) . \end{matrix}$ ∴ 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, $\begin{matrix} ϕ_{j} ((T_{Γ_{G}} \circ T_{Γ_{H}}) ((α, β) (γ, δ))) \\ = ϕ_{j} (T_{Γ_{G}} (α γ)) \land ϕ_{j} (T_{ϒ_{H}} (β)) \land ϕ_{j} (T_{ϒ_{H}} (δ)) \\ \leq {ϕ_{j} (T_{ϒ_{G}} (α)) \land ϕ_{j} (T_{ϒ_{G}} (γ))} \land ϕ_{j} (T_{ϒ_{H}} (β)) \land \\ ϕ_{j} (T_{ϒ_{H}} (δ)) \\ = {ϕ_{j} (T_{ϒ_{G}} (α)) \land ϕ_{j} (T_{ϒ_{H}} (β))} \\ \land {ϕ_{j} (T_{ϒ_{G}} (γ)) \land ϕ_{j} (T_{ϒ_{H}} (δ))} \\ = ϕ_{j} ((T_{ϒ_{G}} \circ T_{ϒ_{H}}) (α, β)) \land ϕ_{j} ((T_{ϒ_{G}} \circ T_{ϒ_{H}}) (γ, δ)) \end{matrix}$ ∴ 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 $(α, β) (γ, δ) \in (V_{G} \tilde{\times} V_{H})^{2} - E .$ 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,

$\begin{matrix} ϕ_{j} ((I_{Γ_{G}} \circ I_{Γ_{H}}) ((γ, α) (γ, β))) \\ = ϕ_{j} (I_{ϒ_{G}} (γ)) \land ϕ_{j} (I_{Γ_{H}} (α β)) \\ \leq ϕ_{j} (I_{ϒ_{G}} (γ)) \land {ϕ_{j} (I_{ϒ_{H}} (α)) \land ϕ_{j} (I_{ϒ_{H}} (β))} \\ = {ϕ_{j} (I_{ϒ_{G}} (γ)) \land ϕ_{j} (I_{ϒ_{H}} (α))} \land {ϕ_{j} (I_{ϒ_{G}} (γ)) \land ϕ_{j} (I_{ϒ_{H}} (β))} \\ = ϕ_{j} ((I_{ϒ_{G}} \circ I_{ϒ_{H}}) (γ, α)) \land ϕ_{j} ((I_{ϒ_{G}} \circ I_{ϒ_{H}}) (γ, β)) . \end{matrix}$

∴ 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, $\begin{matrix} ϕ_{j} ((I_{Γ_{G}} \circ I_{Γ_{H}}) ((α, γ) (β, γ))) \\ = ϕ_{j} (I_{Γ_{G}} (α β)) \land ϕ_{j} (I_{ϒ_{H}} (γ)) \\ \leq {ϕ_{j} (I_{ϒ_{G}} (α)) \land ϕ_{j} (I_{ϒ_{G}} (β))} \land ϕ_{j} (I_{ϒ_{H}} (γ)) \\ = {ϕ_{j} (I_{ϒ_{G}} (α)) \land ϕ_{j} (I_{ϒ_{H}} (γ))} \land {ϕ_{j} (I_{ϒ_{G}} (β)) \\ \land ϕ_{j} (I_{ϒ_{H}} (γ))} \\ = ϕ_{j} ((I_{ϒ_{G}} \circ I_{ϒ_{H}}) (α, γ) \land ϕ_{j} ((I_{ϒ_{G}} \circ I_{ϒ_{H}}) (β, γ)) . \end{matrix}$ ∴ 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 $(α, β) (γ, δ) \in (V_{G} \tilde{\times} V_{H})^{2} - E .$ 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, $\begin{matrix} ϕ_{j} ((I_{Γ_{G}} \circ I_{Γ_{H}}) ((α, β) (γ, δ))) \\ = ϕ_{j} (I_{Γ_{G}} (α γ)) \land ϕ_{j} (I_{ϒ_{H}} (β)) \land ϕ_{j} (I_{ϒ_{H}} (δ)) \\ \leq {ϕ_{j} (I_{ϒ_{G}} (α)) \land ϕ_{j} (I_{ϒ_{H}} (γ))} \land ϕ_{j} (I_{ϒ_{G}} (β)) \\ \land ϕ_{j} (I_{ϒ_{H}} (δ)) \\ = {ϕ_{j} (I_{ϒ_{G}} (α)) \land ϕ_{j} (I_{ϒ_{H}} (β))} \land {ϕ_{j} (I_{ϒ_{G}} (γ)) \\ \land ϕ_{j} (I_{ϒ_{H}} (δ))} \\ = ϕ_{j} ((I_{ϒ_{G}} \circ I_{ϒ_{H}}) (α, β)) \land ϕ_{j} ((I_{ϒ_{G}} \circ I_{ϒ_{H}}) (γ, δ)) . \end{matrix}$ ∴ 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)).

Let γ ∈ V_G and αβ ∈ E_H. Then for each j = 1, 2, 3, . . . , m, $\begin{matrix} ϕ_{j} ((F_{Γ_{G}} \circ F_{Γ_{H}}) ((γ, α) (γ, β))) \\ = ϕ_{j} (F_{ϒ_{G}} (γ)) \lor ϕ_{j} (F_{Γ_{H}} (α β)) \\ \leq ϕ_{j} (F_{ϒ_{G}} (γ)) \lor {ϕ_{j} (F_{ϒ_{H}} (α)) \lor ϕ_{j} (F_{ϒ_{H}} (β))} \\ = {ϕ_{j} (F_{ϒ_{G}} (γ)) \lor ϕ_{j} (F_{ϒ_{H}} (α))} \lor {ϕ_{j} (F_{ϒ_{G}} (γ)) \\ \lor ϕ_{j} (F_{ϒ_{H}} (β))} \\ = ϕ_{j} ((F_{ϒ_{G}} \circ F_{ϒ_{H}}) (γ, α)) \lor ϕ_{j} ((F_{ϒ_{G}} \circ F_{ϒ_{H}}) (γ, β)) . \end{matrix}$ ∴ 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 αβ ∈ E_G. Then for each j = 1, 2, 3, . . . , m, $\begin{matrix} ϕ_{j} ((F_{Γ_{G}} \circ F_{Γ_{H}}) ((α, γ) (β, γ))) \\ = ϕ_{j} (F_{Γ_{G}} (α β)) \lor ϕ_{j} (F_{ϒ_{H}} (γ)) \\ \leq {ϕ_{j} (F_{ϒ_{G}} (α)) \lor ϕ_{j} (F_{ϒ_{G}} (β))} \lor ϕ_{j} (F_{ϒ_{H}} (γ)) \\ = {ϕ_{j} (F_{ϒ_{G}} (α)) \lor ϕ_{j} (F_{ϒ_{H}} (γ))} \lor {ϕ_{j} (F_{ϒ_{G}} (β)) \\ \lor ϕ_{j} (F_{ϒ_{H}} (γ))} \\ = ϕ_{j} ((F_{ϒ_{G}} \circ F_{ϒ_{H}}) (α, γ)) \lor ϕ_{j} ((F_{ϒ_{G}} \circ F_{ϒ_{H}}) (β, γ)) . \end{matrix}$ ∴ 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
_H}) (β, γ)).

Let (α, β) (γ, δ) ∈ E₀ - E. Then for each j = 1, 2, 3, . . . , m, $\begin{matrix} ϕ_{j} ((F_{Γ_{G}} \circ F_{Γ_{H}}) ((α, β) (γ, δ))) \\ = ϕ_{j} (F_{Γ_{G}} (α γ)) \lor ϕ_{j} (F_{ϒ_{H}} (β)) \lor ϕ_{j} (F_{ϒ_{H}} (δ)) \\ \leq {ϕ_{j} (F_{ϒ_{G}} (α)) \lor ϕ_{j} (F_{ϒ_{G}} (γ))} \lor ϕ_{j} (F_{ϒ_{H}} (β)) \\ \lor ϕ_{j} (F_{ϒ_{H}} (δ)) \\ = {ϕ_{j} (F_{ϒ_{G}} (α)) \lor ϕ_{j} (F_{ϒ_{H}} (β))} \lor {ϕ_{j} (F_{ϒ_{G}} (γ)) \\ \lor ϕ_{j} (F_{ϒ_{H}} (δ))} \\ = ϕ_{j} ((F_{ϒ_{G}} \circ (F_{ϒ_{H}}) (α, β)) \\ \lor ϕ_{j} ((F_{ϒ_{G}} \circ F_{ϒ_{H}}) (γ, δ)) . \end{matrix}$ ∴ ϕ_j ((F_{Γ
_G} ∘ F_{Γ
_H}) ((α, β) (γ, δ))) ≤ ϕ_j ((F_{ϒ
_G} ∘ F_{ϒ
_H}) (α, β)) ∨ ϕ_j ((F_{ϒ
_G} ∘ F_{ϒ
_H}) (γ, δ)).

Let $(α, β) (γ, δ) \in (V_{G} \tilde{\times} V_{H})^{2} - E .$ Then for each j = 1, 2, . . . m,

Then 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 4.3. (Union of Two m-Polar SVNG’s) The union 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,

$ϕ_{j} ((T_{ϒ_{G}} \cup T_{ϒ_{H}}) (ζ)) = {\begin{matrix} ϕ_{j} (T_{ϒ_{G}} (ζ)), if ζ \in V_{G} - V_{H} \\ ϕ_{j} (T_{ϒ_{H}} (ζ)), if ζ \in V_{G} - V_{H} \\ ϕ_{j} (T_{ϒ_{G}} (ζ)) \lor ϕ_{j} (T_{ϒ_{H}} (ζ)), if ζ \in V_{G} \cap V_{H} \end{matrix}$

$ϕ_{j} ((I_{ϒ_{G}} \cup I_{ϒ_{H}}) (ζ)) = {\begin{matrix} ϕ_{j} (I_{ϒ_{G}} (ζ)), if ζ \in V_{G} - V_{H} \\ ϕ_{j} (I_{ϒ_{H}} (ζ)), if ζ \in V_{G} - V_{H} \\ ϕ_{j} (I_{ϒ_{G}} (ζ)) \lor ϕ_{j} (I_{ϒ_{H}} (ζ)), if ζ \in V_{ϒ_{G}} \cap V_{ϒ_{H}} \end{matrix}$

$ϕ_{j} ((F_{ϒ_{G}} \cup F_{ϒ_{H}}) (ζ)) = {\begin{matrix} ϕ_{j} (F_{ϒ_{G}} (ζ)), if ζ \in V_{G} - V_{H} \\ ϕ_{j} (F_{ϒ_{H}} (ζ)), if ζ \in V_{G} - V_{H} \\ ϕ_{j} (F_{ϒ_{G}} (ζ)) \land ϕ_{j} (F_{ϒ_{H}} (ζ)), if ζ \in V_{G} \cap V_{H} \end{matrix}$

$ϕ_{j} ((T_{Γ_{G}} \cup T_{Γ_{H}}) (ζ η)) = {\begin{matrix} ϕ_{j} (T_{Γ_{G}} (ζ η)), if ζ η \in E_{G} - E_{H} \\ ϕ_{j} (T_{Γ_{H}} (ζ η)), if ζ η \in E_{G} - E_{H} \\ ϕ_{j} (T_{Γ_{G}} (ζ η)) \lor ϕ_{j} (T_{Γ_{H}} (ζ η)), if ζ η \in E_{G} \cap E_{H} \end{matrix}$

$ϕ_{j} ((I_{Γ_{G}} \cup I_{Γ_{H}}) (ζ η)) = {\begin{matrix} ϕ_{j} (I_{Γ_{G}} (ζ η)), if ζ η \in E_{G} - E_{H} \\ ϕ_{j} (I_{Γ_{H}} (ζ η)), if ζ η \in E_{G} - E_{H} \\ ϕ_{j} (I_{Γ_{G}} (ζ η)) \lor ϕ_{j} (I_{Γ_{H}} (ζ η)), if ζ η \in E_{G} \cap E_{H} \end{matrix}$

$ϕ_{j} ((F_{Γ_{G}} \cup F_{Γ_{H}}) (ζ η)) = {\begin{matrix} ϕ_{j} (F_{Γ_{G}} (ζ η)), if ζ η \in E_{G} - E_{H} \\ ϕ_{j} (F_{Γ_{G}} (ζ η)), if ζ η \in E_{G} - E_{H} \\ ϕ_{j} (F_{Γ_{G}} (ζ η)) \land ϕ_{j} (F_{Γ_{H}} (ζ η)), if ζ η \in E_{G} \cap E_{H} \end{matrix}$

$\forall ζ η \in ({\tilde{V}}_{G}^{2} \cup {\tilde{V}}_{H}^{2}) - E_{G} \cup E_{H}$ ,

ϕ_j ((T_{Γ
_G} ∪ T_{Γ
_H}) (ζη)) =0,

ϕ_j ((I_{Γ
_G} ∪ I_{Γ
_H}) (ζη)) =0,

ϕ_j ((F_{Γ
_G} ∪ F_{Γ
_H}) (ζη)) =0.

Theorem 4.5. The union (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_G, E_G) is an m-polar SVNG of G′ ∪ H′.

Definition 4.4. (Join of Two m-polar SVNG’s) The join 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}), ϒ₂ = (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,

if ξ ∈ V_G ∪ V_H,

ϕ_j ((T_{ϒ
_G} + T_{ϒ
_H}) (ξ)) = ϕ_j ((T_{ϒ
_G} ∪ T_{ϒ
_H}) (ξ));

ϕ_j ((I_{ϒ
_G} + I_{ϒ
_H}) (ξ)) = ϕ_j ((I_{ϒ
_G} ∪ I_{ϒ
_H}) (ξ));

ϕ_j ((F_{ϒ
_G}+ F_{ϒ
_H}) (ξ)) = ϕ_j ((F_{ϒ
_G} ∪ F_{ϒ
_H}) (ξ)) ;

if ξς ∈ E_G ∪ E_H,

ϕ_j ((T_{Γ
_G} + T_{Γ
_H}) (ξς)) = ϕ_j ((T_{Γ
_G} ∪ T_{Γ
_H}) (ξς));

ϕ_j ((I_{Γ
_G} + I_{Γ
_H}) (ξς)) = ϕ_j ((I_{Γ
_G} ∪ I_{Γ
_H}) (ξς));

ϕ_j ((F_{Γ
_G}+ F_{Γ
_H}) (ξς)) = ϕ_j ((F_{Γ
_G} ∪ F_{Γ
_H}) (ξς)) ;

E^a=Collection of all edges adding the elements of V_G and V_H and assuming that V_G ∩ V_H = ϕ and if ξς ∈ E^a,

ϕ_j ((T_{Γ
_G} + T_{Γ
_H}) (ξ, ς)) = ϕ_j (T_{ϒ
_G} (ξ) ∧ ϕ_j (T_{ϒ
_H} (ς));

ϕ_j ((I_{Γ
_G} + I_{Γ
_H}) (ξ, ς)) = ϕ_j (I_{ϒ
_G} (ξ) ∧ ϕ_j (I_{ϒ
_H} (ς));

ϕ_j ((F_{Γ
_G} + F_{Γ
_H}) (ξ, ς)) = ϕ_j (F_{ϒ
_G} (ξ) ∨ ϕ_j (F_{ϒ
_H} (ς));

$ξ ς \in ({\tilde{V}}_{G}^{2} \cup {\tilde{V}}_{H}^{2}) - E_{G} \cup E_{H} \cup E^{a}$ , then

ϕ_j ((T_{Γ
_G} + T_{Γ
_H}) (ξς)) =0,

ϕ_j ((I_{Γ
_G} + I_{Γ
_H}) (ξς)) =0,

ϕ_j ((F_{Γ
_G} + F_{Γ
_H}) (ξς)) =0.

Example 4.6. We consider two graphs G′ = (V_G, E_G) and H′ = (V_H, E_H) s.t., V_G = {x₁, x₂}, V_H = {y₁, y₂}, E_G = {x₂x₁} 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.,

$T_{ϒ_{G}} = {\frac{< 0.6, 0.5, 0.4 >}{x_{1}}, \frac{< 0.3, 0.4, 0.6 >}{x_{2}}}$ ,

$T_{Γ_{G}} = {\frac{< 0.3, 0.2, 0.4 >}{x_{2} x_{1}}}$ ,

$I_{ϒ_{G}} = {\frac{< 0.5, 0.4, 0.5 >}{x_{1}}, \frac{< 0.6, 0.7, 0.4 >}{x_{2}}}$ ,

$I_{Γ_{G}} = {\frac{< 0.4, 0.4, 0.3 >}{x_{2} x_{1}}}$ ,

$F_{ϒ_{G}} = {\frac{< 0.4, 0.3, 0.6 >}{x_{1}}, \frac{< 0.5, 0.4, 0.7 >}{x_{2}}}$ ,

$F_{Γ_{G}} = {\frac{< 0.4, 0.3, 0.5 >}{x_{2} x_{1}}}$

$T_{ϒ_{H}} = {\frac{< 0.7, 0.8, 0.9 >}{y_{1}}, \frac{< 0.5, 0.6, 0.7 >}{y_{2}}}$ ,

$T_{Γ_{H}} = {\frac{< 0.4, 0.5, 0.7 >}{y_{1} y_{2}}}$ ,

$I_{ϒ_{H}} = {\frac{< 0.4, 0.5, 0.7 >}{y_{1}}, \frac{< 0.3, 0.4, 0.2 >}{y_{2}}}$ ,

$I_{Γ_{H}} = {\frac{< 0.3, 0.3, 0.1 >}{y_{1} y_{2}}}$ ,

$F_{ϒ_{H}} = {\frac{< 0.5, 0.4, 0.2 >}{y_{1}}, \frac{< 0.5, 0.7, 0.4 >}{y_{2}}}$ ,

$F_{Γ_{H}} = {\frac{< 0.4, 0.5, 0.3 >}{y_{1} y_{2}}}$ .

Compute G + H. Then we have the following:

(T_{Γ
_G}+ T_{Γ
_H}) (x₁x₂) = <0.3, 0.2, 0.4 >,

(I_{Γ
_G}+ I_{Γ
_H}) (x₁x₂) = <0.4, 0.4, 0.3 >,

(F_{Γ
_G}+ F_{Γ
_H}) (x₁x₂) = <0.4, 0.3, 0.5 >.

Similarly, we get

(T_{ϒ
_G}+ T_{ϒ
_H}) (x₁) = <0.6, 0.5, 0.4 >,

(I_{ϒ
_G}+ I_{ϒ
_H}) (x₁) = <0.5, 0.4, 0.5 >,

(F_{ϒ
_G}+ F_{ϒ
_H}) (x₁) = <0.4, 0.3, 0.6 >. See Fig. 3.

Fig. 3

Join of two 3-polar SVNG’s.

Theorem 4.7. The join (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′.

Theorem 4.8. The union (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_H) and H′ = (V_H, E_H) is an m-polar SVNG of the graph G′ ∪ H′ iff G = (V_G, ϒ_G, Γ_G) and H = (V_H, ϒ_H, Γ_H) are m-polar SVNG’s of the graphs G′ = (V_G, E_G) and H′ = (V_H, E_H).

4.1 Isomorphism and homomorphism of m-polar SVNG

Definition 4.5. (Homomorphism Between Two m-Polar SVNG’s) Let G = (V_G, ϒ_G, Γ_G) and H = (V_H, ϒ_H, Γ_H) be two m-polar SVNG’s of the graphs G′ = (V_G, E_G) and H′ = (V_H, E_H) respectively, where ϒ_G = (T_G, I_G, F_G), Γ_G = (T_{Γ
_G}, I_{Γ
_G}, F_{Γ
_G}), ϒ_H = (T_{ϒ
_H}, I_{ϒ
_H}, F_{ϒ
_H}) and Γ_H = (T_{Γ
_H}, I_{Γ
_H}, F_{Γ
_H}). A homomorphism between G and H is a mapping θ : V_G → V_H s.t., for each j = 1, 2, . . . , m,

∀ϑ ∈ V_G,

ϕ_j (T_{ϒ
_G} (ϑ)) ≤ ϕ_j (T_{ϒ
_H} (θ (ϑ))),

(b) ϕ_j (I_{ϒ
_G} (ϑ)) ≤ ϕ_j (I_{ϒ
_H} (θ (ϑ)))

and (c) ϕ_j (F_{ϒ
_G} (ϑ)) ≤ ϕ_j (F_{ϒ
_H} (θ (ϑ))),

$\forall ζ η \in {\tilde{V}}_{G}^{2}$ ,

ϕ_j (T_{Γ
_G} (ζη)) ≤ ϕ_j (T_{Γ
_H} (θ (ζη))),

(b) ϕ_j (I_{Γ
_G} (ζη)) ≤ ϕ_j (I_{Γ
_H} (θ (ζη)))

and (c) ϕ_j (F_{Γ
_G} (ζη)) ≤ ϕ_j (F_{Γ
_H} (θ (ζη))).

Definition 4.6. (Isomorphism Between Two m-Polar SVNG’s) Let G = (V_G, ϒ_G, Γ_G) and H = (V_H, ϒ_H, Γ_H) be two m-polar SVNG’s 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}), Γ_G = (T_{Γ
_G}, I_{Γ
_G}, F_{Γ
_G}), ϒ_H = (T_{ϒ
_H}, I_{ϒ
_H}, F_{ϒ
_H}) and Γ_H = (T_{Γ
_H}, I_{Γ
_H}, F_{Γ
_H}). An isomorphism between G and H is a mapping θ : V_G → V_H s.t., for each j = 1, 2, . . . , m,

∀ϑ ∈ V_G, ϕ_j (T_{ϒ
_G} (ϑ)) = ϕ_j (T_{ϒ
_H} (θ (ϑ))), ϕ_j (I_{ϒ
_G} (ϑ)) = ϕ_j (I_{ϒ
_H} (θ (ϑ))) and (c) ϕ_j (F_{ϒ
_H} (ϑ)) = ϕ_j (F_{ϒ
_H} (θ (ϑ))),

$\forall ζ η \in {\tilde{V}}_{G}^{2}$ , ϕ_j (T_{Γ
_G} (ζη)) = ϕ_j (T_{Γ
_H} (θ (ζη))), ϕ_j (I_{Γ
_H} (ζη)) = ϕ_j (I_{Γ
_H} (θ (ζη))) and ϕ_j (F_{Γ
_G} (ζη)) = ϕ_j (F_{Γ
_H} (θ (ζη))).

If G is isomorphic to H, then we can write G ≅ H.

Definition 4.7. (Weak Isomorphism Between Two m-Polar SVNG’s) Let G = (V_G, ϒ_G, Γ_G) and H = (V_H, ϒ_H, Γ_H) be two m-polar SVNG’s 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}), Γ_G = (T_{Γ
_G}, I_{Γ
_G}, F_{Γ
_G}), ϒ_H = (T_{ϒ
_H}, I_{ϒ
_H}, F_{ϒ
_H}), and Γ_H = (T_{Γ
_H}, I_{Γ
_H}, F_{Γ
_H}). A weak isomorphism between G and H is a mapping θ : V_G → V_H s.t., for each j = 1, 2, . . . , m,

θ is a homomorphism and

∀ϑ ∈ V_G, ϕ_j (T_{ϒ
_G} (ϑ)) = ϕ_j (T_{ϒ
_H} (θ (ϑ))), ϕ_j (I_{ϒ
_G} (ϑ)) = ϕ_j (I_{ϒ
_H} (θ (ϑ))) and ϕ_j (F_{ϒ
_G} (ϑ)) = ϕ_j (F_{ϒ
_H} (θ (ϑ))).

A weak isomorphism allways preserve the weights of the vertices, but not necessarily the weights of the edges.

Definition 4.8. (Co-weak Isomorphism Between Two m-Polar SVNG’s) Let G = (V_G, ϒ_G, Γ_G) and H = (V_H, ϒ_H, Γ_H) be two m-polar SVNG’s 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}), Γ_G = (T_{Γ
_G}, I_{Γ
_G}, F_{Γ
_G}), ϒ_H = (T_{ϒ
_H}, I_{ϒ
_H}, F_{ϒ
_H}), and Γ_H = (T_{Γ
_H}, I_{Γ
_H}, F_{Γ
_H}). A co-weak isomorphism between G and H is a bijective mapping θ : V_G → V_H s.t., for j = 1, 2, . . . , m,

θ is a homomorphism, and

$\forall ζ η \in {\tilde{V}}_{G}^{2}$ , ϕ_j (T_{Γ
_G} (ζη)) = ϕ_j (T_{Γ
_H} (θ (ζη))), ϕ_j (I_{Γ
_G} (ζη)) = ϕ_j (I_{Γ
_H} (θ (ζη))) and ϕ_j (F_{Γ
_G} (ζη)) = ϕ_j (F_{Γ
_H} (θ (ζη))).

Definition 4.9. (Strong m-Polar SVNG) An m-polar SVNG G = (V_G, ϒ_G, Γ_G) of the graph G′ = (V_G, E_G), where ϒ_G = (T_{ϒ
_G}, I_{ϒ
_G}, F_{ϒ
_G}), Γ_G = (T_{Γ
_G}, I_{Γ
_G}, F_{Γ
_G}) is called strong m-polar SVNG if ∀αβ ∈ E_G and for each j = 1, 2, . . . , m,

ϕ_j (T_{Γ
_G} (αβ)) = ϕ_j (T_{ϒ
_G} (α)) ∧ ϕ_j (T_{ϒ
_G} (β)),

ϕ_j (I_{Γ
_G} (αβ)) = ϕ_j (I_{ϒ
_G} (α)) ∧ ϕ_j (I_{ϒ
_G} (β)),

ϕ_j (F_{Γ
_G} (αβ)) = ϕ_j (F_{ϒ
_G} (α)) ∨ ϕ_j (F_{ϒ
_G} (β)).

Theorem 4.9. If G and H are the strong m-polar SVNG’s of the graphs G′ = (V_G, E_G) and H′ = (V_H, E_H) respectively, then (G × H), (G ∘ H) and (G + H) are strong m-polar SVNG’s of the graphs (G × H), (G ∘ H) and (G + H).

Theorem 4.10. If G × H is strong m-polar SVNG, then at least G or H must be strong.

Proof. Let G × H is a strong m-polar SVNG. If possible, let G and H both are not strong m-polar SVNG’s. Then, ∃ at least one edge αβ ∈ E_G and at least one edge γδ ∈ E_H s.t.,

ϕ_j (T_{Γ
_G} (αβ)) < ϕ_j (T_{ϒ
_G} (α)) ∧ ϕ_j (T_{ϒ
_G} (β)) and

ϕ_j (T_{Γ
_H} (γδ)) < ϕ_j (T_{ϒ
_H} (γ)) ∧ ϕ_j (T_{ϒ
_H} (δ)).

Without loss of generality, we assume that

$\begin{array}{l} ϕ_{j} (T_{Γ_{H}} (γ δ)) \leq ϕ_{j} (T_{Γ_{G}} (α β)) \\ < ϕ_{j} (T_{Υ_{G}} (α)) \land ϕ_{j} (T_{Υ_{G}} (β)) \\ \leq ϕ_{j} (T_{Υ_{G}} (α)) . \end{array}$

Let $ξ \in V_{G}$ and $γ δ \in E_{H} .$ Then

$\begin{array}{l} ϕ_{j} ((T_{Γ}_{G} \times T_{Γ}_{H}) ((ξ, γ) (ξ, δ))) = ϕ_{j} (T_{Υ}_{G} (ξ)) \land \\ ϕ_{j} (T_{Γ}_{H} (γ δ)) \\ < ϕ_{j} (T_{Υ}_{G} (ξ)) \land ϕ_{j} (T_{Υ}_{H} (γ)) \land ϕ_{j} (T_{Υ}_{H} (δ)) . \end{array}$

Again, ϕ_j ((T_{ϒ
_G} × T_{ϒ
_H}) (ξ, γ)) = ϕ_j (T_{ϒ
_G} (ξ)) ∧ ϕ_j (T_{ϒ
_H} (γ)) and ϕ_j ((T_{ϒ
_G} × T_{ϒ
_H}) (ξ, δ)) = ϕ_j (T_{ϒ
_G} (ξ)) ∧ ϕ_j (T_{ϒ
_H} (δ))

∴ ϕ_j ((T_{ϒ
_G} × T_{ϒ
_H}) (ξ, γ)) ∧ ϕ_j ((T_{ϒ
_G} × T_{ϒ
_H}) (ξ, δ)) = ϕ_j (T_{ϒ
_G} (ξ)) ∧ ϕ_j (T_{ϒ
_H} (γ)) ∧ ϕ_j (T_{ϒ
_H} (δ))

∴ ϕ_j ((T_{Γ
_G} × T_{Γ
_H}) ((ξ, γ) (ξ, δ))) = ϕ_j (T_{ϒ
_G} (ξ)) ∧ ϕ_j (T_{Γ
_H} (γδ)) < ϕ_j ((T_{ϒ
_G} × T_{ϒ
_H}) (ξ, γ)) ∧ ϕ_j ((T_{ϒ
_G} × T_{ϒ
_H}) (ξ, δ)).

This implies that G × H is not a strong m-polar SVNG, but G × H is strong SVNG. By contrapositively, at least one of G or H must be a strong SVNG.

This completes the proof. □

Theorem 4.11. If G ∘ H is strong m-polar SVNG, then at least G or H must be strong.

4.2 Complement of m-polar SVNG

Theorem 4.12. Let G = (V_G, ϒ_G, Γ_G) be a strong m-polar SVNG of the graph G = (V_G, E_G), where ϒ_G = (T_{ϒ
_G}, F_{ϒ
_G}, I_{ϒ
_G}), Γ_G = (T_{Γ
_G}, F_{Γ
_G}, I_{Γ
_G}). If $\bar{G} = (V_{G}, ϒ_{\bar{G}}, Γ_{\bar{G}})$ satisfies $ϒ_{\bar{G}} = ϒ_{G}$ (where $ϒ_{\bar{G}} = (T_{ϒ_{\bar{G}}}, I_{ϒ_{\bar{G}}}, F_{ϒ_{\bar{G}}}))$ and $Γ_{\bar{G}} = (T_{Γ_{\bar{G}}}, I_{Γ_{\bar{G}}}, F_{Γ_{\bar{G}}})$ is defined by $α β \in {\tilde{V}}_{G}^{2}$ and for each j = 1, 2, . . . , m,

$ϕ_{j} (T_{Γ_{\bar{G}}} (α β)) = {\begin{matrix} 0, if 0 < ϕ_{j} (T_{Γ_{G}} (α β)) \leq 1 \\ ϕ_{j} (T_{ϒ_{G}} (α)) \land ϕ_{j} (T_{ϒ_{G}} (β)), if ϕ_{j} (T_{Γ_{G}} (α, β)) = 0 . \end{matrix}$

$ϕ_{j} (I_{Γ_{\bar{G}}} (α β)) = {\begin{matrix} 0, if 0 < ϕ_{j} (I_{Γ_{G}} (α β)) \leq 1 \\ ϕ_{j} (I_{ϒ_{G}} (α)) \land ϕ_{j} (I_{ϒ_{G}} (β)), if ϕ_{j} (I_{Γ_{G}} (α, β)) = 0 . \end{matrix}$

$ϕ_{j} (F_{Γ_{\bar{G}}} (α β)) = {\begin{matrix} 0, if 0 < ϕ_{j} (F_{Γ_{G}} (α β)) \leq 1 \\ ϕ_{j} (F_{ϒ_{G}} (α)) \lor ϕ_{j} (F_{ϒ_{G}} (β)), if ϕ_{j} (F_{ϒ_{G}} (α β)) = 0 . \end{matrix}$

Then, $\bar{G}$ is a strong m-polar SVNG of the crisp graph $\bar{G^{'}} = (V_{G}, {\tilde{V}}_{G}^{2} - E_{G})$ .

Definition 4.10. (Complement of A m-Polar SVNG) The graph $\bar{G} = (V_{G}, ϒ_{\bar{G}}, Γ_{\bar{G}})$ of the graph $\bar{G} = (V_{G}, E_{\bar{G}})$ , where $E_{\bar{G}} = (V_{G} \times V_{G}) - E_{G}$ is called the complement of the strong m-polar SVNG G = (V_G, ϒ_G, Γ_G).

Definition 4.11. (Self Complementary of A m-Polar SVNG) A strong m-polar SVNG G is called self comlementary iff $G ≅ \bar{G}$ .

Theorem 4.13. Let G = (V_G, ϒ_G, Γ_G) be a strong m-polar SVNG of the graph G′ = (V_G, E_G) and $\bar{G} = (V_{G}, ϒ_{\bar{G}}, Γ_{\bar{G}})$ be the complement of G. Then $\forall α β \in {\tilde{V}}_{G}^{2}$ and for each j = 1, 2, . . . , m,

$ϕ_{j} (T_{Γ_{\bar{G}}} (α β)) = ϕ_{j} (T_{ϒ_{G}} (α)) \land ϕ_{j} (T_{ϒ_{G}} (β)) - ϕ_{j} (T_{Γ_{G}} (α β))$ ,

$ϕ_{j} (I_{Γ_{\bar{G}}} (α β)) = ϕ_{j} (I_{ϒ_{G}} (α)) \land ϕ_{j} (I_{ϒ_{G}} (β)) - ϕ_{j} (I_{Γ_{G}} (α β))$ ,

$ϕ_{j} (F_{Γ_{\bar{G}}} (α β)) = ϕ_{j} (F_{ϒ_{G}} (α)) \lor ϕ_{j} (F_{ϒ_{G}} (β)) - ϕ_{j} (F_{Γ_{G}} (α β))$ .

Proof.

Let $α β \in {\tilde{V}}_{G}^{2}$ . If 0 < ϕ_j (T_{Γ
_G} (αβ)) ≤1, for each j = 1, 2, . . . , m, then αβ ∈ E_G. As G is strong, for j = 1, 2, . . . m, $ϕ_{j} (T_{ϒ_{G}} (α)) \land ϕ_{j} (T_{ϒ_{G}} (β)) - ϕ_{j} (T_{Γ_{G}} (α β)) = 0 = ϕ_{j} (T_{Γ_{\bar{G}}} (α β))$ .

If for j = 1, 2, . . . , m, ϕ_jT_{Γ
_G} (αβ)) =0,

$ϕ_{j} (T_{ϒ_{G}} (α)) \land ϕ_{j} (T_{ϒ_{G}} (β)) - ϕ_{j} (T_{Γ_{G}} (α β)) = ϕ_{j} (T_{ϒ_{G}} (α)) \land ϕ_{j} (T_{ϒ_{G}} (β)) = ϕ_{j} T_{Γ_{\bar{G}}} (α β))$ .

Let $α β \in {\tilde{V}}_{G}^{2}$ . If 0 < ϕ_j (I_{Γ
_G} (αβ)) ≤1, for each j = 1, 2, . . . , m, then αβ ∈ E_G. As G is strong, for j = 1, 2, . . . m,

$ϕ_{j} (I_{ϒ_{G}} (α)) \land ϕ_{j} (I_{ϒ_{G}} (β)) - ϕ_{j} (I_{Γ_{G}} (α β)) = 0 = ϕ_{j} (I_{Γ_{\bar{G}}} (α β))$ .

If for j = 1, 2, . . . , m, ϕ_j (I_{Γ
_G} (αβ)) =0,

$ϕ_{j} (I_{ϒ_{G}} (α)) \land ϕ_{j} (I_{ϒ_{G}} (β)) - ϕ_{j} (I_{Γ_{G}} (α β)) = ϕ_{j} (I_{ϒ_{G}} (α)) \land ϕ_{j} (I_{ϒ_{G}} (β)) = ϕ_{j} (I_{Γ_{\bar{G}}} (α β))$ .

Let $α β \in {\tilde{V}}_{G}^{2}$ . If 0 < ϕ_j (F_{Γ
_G} (αβ)) ≤1, for each j = 1, 2, . . . , m, then αβ ∈ E_G. As G is strong, for j = 1, 2, . . . m,

$ϕ_{j} (F_{ϒ_{G}} (α)) \lor ϕ_{j} (F_{ϒ_{G}} (β)) - ϕ_{j} (F_{Γ_{G}} (α β)) = 0 = ϕ_{j} (F_{Γ_{\bar{G}}} (α β))$ .

If for j = 1, 2, . . . , m, ϕ_j (F_{Γ
_G} (αβ)) =0,

$ϕ_{j} (F_{ϒ_{G}} (α)) \lor ϕ_{j} (F_{ϒ_{G}} (β)) - ϕ_{j} (F_{Γ_{G}} (α β)) = ϕ_{j} (F_{ϒ_{G}} (α)) \lor ϕ_{j} (F_{ϒ_{G}} (β)) = ϕ_{j} (F_{Γ_{\bar{G}}} (α β))$ .

This completes the proof. □

Theorem 4.14. Let G be a self complementary strong m-polar SVNG. Then $\forall α β \in {\tilde{V}}_{G}^{2}$ , j = 1, 2, . . . , m,

$\sum_{α \neq β} ϕ_{j} (T_{Γ_{G}} (α β)) = \frac{1}{2} \sum_{α \neq β} ϕ_{j} (T_{ϒ_{G}} (α)) \land ϕ_{j} (T_{ϒ_{G}} (β))$ ,

$\sum_{α \neq β} ϕ_{j} (I_{Γ_{G}} (α β)) = \frac{1}{2} \sum_{α \neq β} ϕ_{j} (I_{ϒ_{G}} (α)) \land ϕ_{j} (I_{ϒ_{G}} (β))$ ,

$\sum_{α \neq β} ϕ_{j} (F_{Γ_{G}} (α β)) = \frac{1}{2} \sum_{α \neq β} ϕ_{j} (F_{ϒ_{G}} (α)) \lor ϕ_{j} (F_{ϒ_{G}} (β))$ .

Proof. Let G be a self complementary strong m-polar SVNG. Then, ∀ αβ ∈ E_G and for each j = 1, 2, . . . , m, ϕ_j (T_{Γ
_G} (αβ) = ϕ_j (T_{ϒ
_G} (α)) ∧ ϕ_j (T_{ϒ
_G} (β)) and ∃ an isomorphism $θ : G \to \bar{G}$ s.t., $ϕ_{j} (T_{ϒ_{G}} (ξ)) = ϕ_{j} (T_{ϒ_{\bar{G}}} (θ (ξ)))$ , $(I_{ϒ_{G}} (ξ)) = ϕ_{j} (I_{ϒ_{\bar{G}}} (θ (ξ)))$ and $(F_{ϒ_{G}} (ξ)) = ϕ_{j} (F_{ϒ_{\bar{G}}} (θ (ξ)))$ , ∀ξ ∈ V_G. Again, ϕ_j (T_{Γ
_G} $(α β)) = ϕ_{j} (T_{Γ_{\bar{G}}} (θ (α) θ (β)))$ , $ϕ_{j} (I_{Γ_{G}} (α β)) = ϕ_{j} (I_{Γ_{\bar{G}}}$ (θ (α) θ (β))) and $ϕ_{j} (T_{Γ_{G}} (α β)) = ϕ_{j} (T_{Γ_{\bar{G}}} (θ (α) θ (β)))$ , $\forall α β \in {\tilde{V}}_{G}^{2}$ . But we know that $Γ_{G} = Γ_{\bar{G}}$ . So $ϕ_{j} (T_{ϒ_{G}} (ξ)) = ϕ_{j} (T_{ϒ_{\bar{G}}} (θ (ξ))) = ϕ_{j} (T_{ϒ_{G}} (θ (ξ)))$ ,

$ϕ_{j} (I_{ϒ_{G}} (ξ)) = ϕ_{j} (I_{ϒ_{\bar{G}}} (θ (ξ))) = ϕ_{j} (I_{ϒ_{G}} (θ (ξ)))$ and $ϕ_{j} (F_{ϒ_{G}} (ξ)) = ϕ_{j} (F_{ϒ_{\bar{G}}} (θ (ξ))) = ϕ_{j} (F_{ϒ_{G}} (θ (ξ)))$ . (i) Let $α β \in {\tilde{V}}_{G}^{2}$ . Then for each j = 1, 2, . . . , m,

$ϕ_{j} (T_{Γ_{\bar{G}}} (θ (α) θ (β))) = ϕ_{j} (T_{ϒ_{G}} (θ (α))) \land ϕ_{j} (T_{ϒ_{G}}$ (θ (β))) - ϕ_j (T_{Γ
_G} (θ (α) θ (β))),

or, ϕ_j (T_{Γ
_G} (αβ)) = ϕ_j (T_{ϒ
_G} (θ (α))) ∧ ϕ_j (T_{ϒ
_G} (θ (β))) - ϕ_j (T_{Γ
_G} (θ (α) θ (β))),

(By taking ∑_α≠β on both sides)

or, ∑_α≠βϕ_j (T_{Γ
_G} (αβ)) + ∑_α≠βϕ_j (T_{Γ
_G} (θ (α) θ (β))) = ∑_α≠βϕ_j (T_{ϒ
_G} (θ (α))) ∧ ϕ_j (T_{ϒ
_G} (θ (β))),

or, 2∑_α≠βϕ_j (T_{Γ
_G} (αβ)) = ∑_α≠βϕ_j (T_{ϒ
_G} (α)) ∧ ϕ_j (T_{ϒ
_G} (β)),

or, $\sum_{α \neq β} ϕ_{j} (T_{Γ_{G}} (α β)) = \frac{1}{2} \sum_{α \neq β} ϕ_{j} (T_{ϒ_{G}} (α)) \land ϕ_{j} (T_{ϒ_{G}} (β))$ .

(ii) Let $α β \in {\tilde{V}}_{G}^{2}$ . Then for each j = 1, 2, . . . , m,

$ϕ_{j} (I_{Γ_{\bar{G}}} (θ (α) θ (β))) = ϕ_{j} (I_{ϒ_{G}} (θ (α))) \land ϕ_{j} (I_{ϒ_{G}} (θ (β))) - ϕ_{j} (I_{Γ_{G}} (θ (α) θ (β)))$

or, $ϕ_{j} (I_{Γ_{\bar{G}}} (θ (α) θ (β))) = ϕ_{j} (I_{ϒ_{G}} (θ (α))) \land ϕ_{j} (I_{ϒ_{G}} (θ (β))) - ϕ_{j} (I_{Γ_{G}} (θ (α) θ (β)))$

(By taking ∑_α≠β on both sides)

or, ∑_α≠βϕ_j (I_{Γ
_G} (αβ)) + ∑_α≠βϕ_j (I_{Γ
_G} (θ (α) θ (β))) = ∑_α≠βϕ_j (I_{ϒ
_G} (θ (α))) ∧ ϕ_j (I_{ϒ
_G} (θ (β)))

or, 2∑_α≠βϕ_j (I_{Γ
_G} (αβ)) = ∑_α≠βϕ_j (I_{ϒ
_G} (α)) ∧ ϕ_j (I_{ϒ
_G} (β))

or, $\sum_{α \neq β} ϕ_{j} (I_{Γ_{G}} (α β)) = \frac{1}{2} \sum_{α \neq β} ϕ_{j} (I_{ϒ_{G}} (α)) \land ϕ_{j} (I_{ϒ_{G}} (β))$ .

(iii) Let $α β \in {\tilde{V}}_{G}^{2}$ . Then for each j = 1, 2, . . . , m,

$ϕ_{j} (F_{Γ_{\bar{G}}} (θ (α) θ (β))) = ϕ_{j} (F_{ϒ_{G}} (θ (α))) \lor ϕ_{j} (F_{ϒ_{G}} (θ (β))) - ϕ_{j} (F_{Γ_{G}} (θ (α) θ (β)))$

or, $ϕ_{j} (F_{Γ_{\bar{G}}} (θ (α) θ (β))) = ϕ_{j} (F_{ϒ_{G}} (θ (α))) \lor ϕ_{j} (F_{ϒ_{G}} (θ (β))) - ϕ_{j} (F_{Γ_{G}} (θ (α) θ (β)))$ (By taking ∑_α≠β on both sides)

or, ∑_α≠βϕ_j (F_{Γ
_G} (αβ)) + ∑_α≠βϕ_j (F_{Γ
_G} (θ (α) θ (β))) = ∑_α≠βϕ_j (F_{ϒ
_G} (θ (α))) ∨ ϕ_j (F_{ϒ
_G} (θ (β)))

or, 2∑_α≠βϕ_j (F_{Γ
_G} (αβ)) = ∑_α≠βϕ_j (F_{ϒ
_G} (α)) ∨ ϕ_j (F_{ϒ
_G} (β))

or, $\sum_{α \neq β} ϕ_{j} (F_{Γ_{G}} (α β)) = \frac{1}{2} \sum_{α \neq β} ϕ_{j} (F_{ϒ_{G}} (α)) \land ϕ_{j} (F_{ϒ_{G}} (β))$ .

This completes the proof. □

Theorem 4.15. Let G and H be two strong m-polar SVNG’s. Then, G ≅ H iff $\bar{G} ≅ \bar{H}$ .

Proof. Assume that G ≅ H. Then ∃ a bijective mapping θ : V_G → V_H s.t., ∀ j = 1, 2, . . . m,

ϕ_j (T_{ϒ
_G} (ξ)) = ϕ_j (T_{ϒ
_H} (θ (ξ))), ∀ξ ∈ V_G and ϕ_j (T_{Γ
_G} (ξη)) = ϕ_j (T_{Γ
_H} (θ (ξ) θ (η)), $\forall ξ η \in {\tilde{V}}_{G}^{2}$ ,

ϕ_j (I_{ϒ
_G} (ξ)) = ϕ_j (I_{ϒ
_H} (θ (ξ))), ∀ξ ∈ V_G and

ϕ_j (I_{Γ
_G} (ξη)) = ϕ_j (I_{Γ
_H} (θ (ξ) , θ (η)), $\forall ξ η \in {\tilde{V}}_{G}^{2}$ ,

ϕ_j (F_{ϒ
_G} (ξ)) = ϕ_j (F_{ϒ
_H} (θ (ξ))), ∀ξ ∈ V_G and

ϕ_j (F_{Γ
_G} (ξη)) = ϕ_j (F_{Γ
_H} (θ (ξ) θ (η)), $\forall ξ η \in {\tilde{V}}_{G}^{2}$ .

Let $α β \in {\tilde{V}}_{G}^{2}$ . If for j = 1, 2, . . . , m, ϕ_j (T_{Γ
_G} (αβ)) =0, $\begin{matrix} ϕ_{j} (T_{Γ_{\bar{G}}} (α β)) \\ = ϕ_{j} (T_{ϒ_{G}} (α)) \land ϕ_{j} (T_{ϒ_{G}} (β)) \\ = ϕ_{j} (T_{ϒ_{H}} (θ (α)) \land ϕ_{j} (T_{ϒ_{H}} (θ (β)) \\ = ϕ_{j} (T_{Γ_{\bar{H}}} (θ (α) θ (β))) . \end{matrix}$ If 0 < ϕ_j (T_{Γ
_G} (αβ)) ≤1,

0 < ϕ_j (T_{Γ
_H} (θ (α) θ (β)) ≤1.

∴ $ϕ_{j} (T_{Γ_{\bar{G}}} (α β)) = 0 = ϕ_{j} (T_{Γ_{\bar{H}}} (θ (α) θ (β))) .$

Let $α β \in {\tilde{V}}_{G}^{2}$ . If for j = 1, 2, . . . , m, ϕ_j (I_{Γ
_G} (αβ)) =0, $\begin{matrix} ϕ_{j} (I_{Γ_{\bar{G}}} (α β)) \\ = ϕ_{j} (I_{ϒ_{G}} (α)) \land ϕ_{j} (I_{ϒ_{G}} (β)) \\ = ϕ_{j} (I_{ϒ_{H}} (θ (α)) \land ϕ_{j} (I_{ϒ_{H}} (θ (β)) \\ = ϕ_{j} (I_{Γ_{\bar{H}}} (θ (α) θ (β))) . \end{matrix}$ If 0 < ϕ_j (I_{Γ
_G} (αβ)) ≤1,

0 < ϕ_j (I_{Γ
_H} (θ (α) θ (β)) ≤1.

∴ $ϕ_{j} (I_{ϒ_{\bar{G}}} (α β)) = 0 = ϕ_{j} (I_{Γ_{\bar{H}}} (θ (α) θ (β)))$ .

Let $α β \in {\tilde{V}}_{G}^{2}$ . If for j = 1, 2, . . . , m, ϕ_j (F_{Γ
_G} (αβ)) =0, $\begin{matrix} ϕ_{j} (F_{Γ_{\bar{G}}} (α β)) \\ = ϕ_{j} (F_{ϒ_{G}} (α)) \lor ϕ_{j} (F_{ϒ_{G}} (β)) \\ = ϕ_{j} (F_{ϒ_{H}} (θ (α)) \lor ϕ_{j} (F_{ϒ_{G}} (θ (β)) \\ = ϕ_{j} (F_{Γ_{\bar{H}}} (θ (α) θ (β))) . \end{matrix}$ If 0 < ϕ_j (F_{Γ
_G} (αβ)) ≤1,

0 < ϕ_j (F_{Γ
_H} (θ (α) θ (β)) ≤1.

∴ $ϕ_{j} (F_{Γ_{\bar{G}}} (α β)) = 0 = ϕ_{j} (F_{Γ_{\bar{H}}} (θ (α) θ (β)))$ .

\bar{G} ≅ \bar{H}

Conversely, let $\bar{G} ≅ \bar{H}$ . Then ∃ a bijective mapping θ : V_G → V_H s.t., for each j = 1, 2, . . . m,

$ϕ_{j} (T_{ϒ_{\bar{G}}} (ξ)) = ϕ_{j} (T_{ϒ_{\bar{H}}} (θ (ξ)))$ , ∀ξ ∈ V_G and $ϕ_{j} (T_{Γ_{\bar{G}}} (ξ η)) = ϕ_{j} (T_{Γ_{\bar{H}}} (θ (ξ) θ (η))$ , ∀ ξη $\in {\tilde{V}}_{G}^{2} - E_{G}$ ,

$ϕ_{j} (I_{ϒ_{\bar{G}}} (ξ)) = ϕ_{j} (I_{ϒ_{\bar{H}}} (θ (ξ)))$ , ∀ξ ∈ V_G and $ϕ_{j} (I_{Γ_{\bar{G}}} (ξ η)) = ϕ_{j} (I_{Γ_{\bar{H}}} (θ (ξ) θ (η))$ , ∀ ξη $\in {\tilde{V}}_{G}^{2} - E_{G}$ ,

$ϕ_{j} (F_{ϒ_{\bar{G}}} (ξ)) = ϕ_{j} (F_{ϒ_{\bar{H}}} (θ (ξ)))$ , ∀ξ ∈ V_G and $ϕ_{j} (F_{Γ_{\bar{G}}} (ξ η)) = ϕ_{j} (F_{Γ_{\bar{H}}} (θ (ξ) θ (η))$ , ∀ ξη $\in {\tilde{V}}_{G}^{2} - E_{G}$ .

Let $α β \in {\tilde{V}}_{G}^{2} - E_{G}$ . If ϕ_j (T_{Γ
_G} (αβ)) =0 for each j = 0, 1, 2, . . . , m, then $\begin{matrix} ϕ_{j} (T_{Γ_{\bar{H}}} (ψ (α) ψ (β)) \\ = ϕ_{j} (T_{Γ_{\bar{G}}} (α β)) \\ = ϕ_{j} (T_{ϒ_{G}} (α)) \land ϕ_{j} (T_{ϒ_{G}} (β)) \\ = ϕ_{j} (T_{ϒ_{\bar{G}}} (α)) \land ϕ_{j} (T_{ϒ_{\bar{G}}} (β)) \\ = ϕ_{j} (T_{ϒ_{\bar{H}}} (ψ (α)) \land ϕ_{j} (T_{ϒ_{\bar{H}}} (ψ (β)) \\ = ϕ_{j} (T_{ϒ_{H}} (ψ (α)) \land ϕ_{j} (T_{ϒ_{H}} (ψ (β)) . \end{matrix}$ Now, $ϕ_{j} (T_{Γ_{\bar{H}}} (ψ (α) ψ (β))) = ϕ_{j} (T_{ϒ_{H}} (ψ (α))) \land ϕ_{j} (T_{ϒ_{H}} (ψ (β))) - ϕ_{j} (T_{Γ_{H}} (ψ (α) ψ (β)))$ ,

or, ϕ_j (T_{ϒ
_H} (ψ (α))∧ ϕ_j (T_{ϒ
_H} (ψ (β)) = ϕ_j (T_{ϒ
_H} (ψ (α))) ∧ϕ_j (T_{ϒ
_H} (ψ (β))) - ϕ_j (T_{Γ
_H} (ψ (α) ψ (β))) ,

or, ϕ_j (T_{Γ
_H} (ψ (α) ψ (β))) =0.

∴ ϕ_j (T_{Γ
_H} (ψ (α) ψ (β))) =0 = ϕ_j (T_{Γ
_G} (αβ)), for each j = 1, 2, . . . , m.

If for each j = 1, 2, . . . m, 0 < ϕ_j(T_{Γ
_G} (αβ)) ≤1, then $ϕ_{j} (T_{Γ_{\bar{H}}} (ψ (α) ψ (β))) = 0 = ϕ_{j} (T_{Γ_{\bar{G}}} (α β))$ .

So, we have,

$\begin{array}{l} ϕ_{j} (T_{Γ_{H}} (ψ (α) ψ (β))) \\ = ϕ_{j} (T_{Υ_{H}} (ψ (α))) \land ϕ_{j} (T_{Υ_{H}} (ψ (β))) - 0 \\ = ϕ_{j} (T_{Υ_{\bar{H}}} (ψ (α)) \land ϕ_{j} (T_{Υ_{\bar{H}}} (ψ (β)) \\ = ϕ_{j} (T_{Υ_{\bar{G}}} (α)) \land ϕ_{j} (T_{Υ_{\bar{G}}} (β)) \\ = ϕ_{j} (T_{Γ_{G}} (α β)) . \end{array}$

Let $α β \in {\tilde{V}}_{G}^{2} - E_{G}$ . If ϕ_j (T_{Γ
_G} (αβ)) =0 for each j = 1, 2, . . . , m, then $\begin{matrix} ϕ_{j} (I_{Γ_{\bar{H}}} (ψ (α) ψ (β)) \\ = ϕ_{j} (I_{Γ_{\bar{G}}} (α β)) \\ = ϕ_{j} (I_{ϒ_{G}} (α)) \land ϕ_{j} (I_{ϒ_{G}} (β)) \\ = ϕ_{j} (I_{ϒ_{\bar{G}}} (α)) \land ϕ_{j} (I_{ϒ_{\bar{G}}} (β)) \\ = ϕ_{j} (I_{ϒ_{\bar{H}}} (ψ (α)) \land ϕ_{j} (I_{ϒ_{\bar{H}}} (ψ (β)) \\ = ϕ_{j} (I_{ϒ_{H}} (ψ (α)) \land ϕ_{j} (I_{ϒ_{H}} (ψ (β)) . \end{matrix}$ Now, $ϕ_{j} (I_{Γ_{\bar{H}}} (ψ (α) ψ (β))) = ϕ_{j} (I_{ϒ_{H}} (ψ (α))) \land ϕ_{j} (I_{ϒ_{H}} (ψ (β))) - ϕ_{j} (I_{Γ_{H}} (ψ (α) ψ (β)))$

or, ϕ_j (I_{ϒ
_H} (ψ (α))∧ ϕ_j (I_{ϒ
_H} (ψ (β)) = ϕ_j (I_{ϒ
_H} (ψ (α))) ∧ ϕ_j (I_{ϒ
_H} (ψ (β))) - ϕ_j (I_{Γ
_H} (ψ (α) ψ (β))) ,

or, ϕ_j (T_{Γ
_H} (ψ (α) ψ (β))) =0.

∴ ϕ_j (I_{Γ
_H} (ψ (α) ψ (β))) =0 = ϕ_j (I_{Γ
_G} (αβ)), for each j = 1, 2, . . . , m.

If for each j = 1, 2, . . . m, 0 < ϕ_j (I_{Γ
_G} (αβ)) ≤1, then $ϕ_{j} (I_{Γ_{\bar{H}}} (ψ (α) ψ (β))) = 0 = ϕ_{j} (I_{Γ_{\bar{G}}} (α β))$ .

So, we have $\begin{matrix} ϕ_{j} (I_{Γ_{H}} (ψ (α) ψ (β))) \\ = ϕ_{j} (I_{ϒ_{H}} (ψ (α))) \land ϕ_{j} (I_{ϒ_{H}} (ψ (β))) - 0 \\ = ϕ_{j} (I_{Γ_{\bar{H}}} (ψ (α)) \land ϕ_{j} (I_{Γ_{\bar{H}}} (ψ (β)) \\ = ϕ_{j} (I_{Γ_{\bar{G}}} (α)) \land ϕ_{j} (I_{Γ_{\bar{G}}} (β)) \\ = ϕ_{j} (I_{Γ_{G}} (α β)) . \end{matrix}$

Let $α β \in {\tilde{V}}_{G}^{2} - E_{G}$ . If ϕ_j (F_{Γ
_G} (αβ)) =0, for each j = 0, 1, 2, . . . , m,then $\begin{matrix} ϕ_{j} (F_{Γ_{\bar{H}}} (ψ (α) ψ (β)) \\ = ϕ_{j} (F_{Γ_{\bar{G}}} (α β)) \\ = ϕ_{j} (F_{ϒ_{G}} (α)) \lor ϕ_{j} (F_{ϒ_{G}} (β)) \\ = ϕ_{j} (F_{ϒ_{\bar{G}}} (α)) \lor ϕ_{j} (F_{ϒ_{\bar{G}}} (β)) \\ = ϕ_{j} (F_{ϒ_{\bar{H}}} (ψ (α)) \lor ϕ_{j} (F_{ϒ_{\bar{H}}} (ψ (β)) \\ = ϕ_{j} (F_{ϒ_{H}} (ψ (α)) \lor ϕ_{j} (F_{ϒ_{H}} (ψ (β)) . \end{matrix}$ Now, $ϕ_{j} (F_{Γ_{\bar{H}}} (ψ (α) ψ (β))) = ϕ_{j} (F_{ϒ_{H}} (ψ (α))) \lor ϕ_{j} (F_{ϒ_{H}} (ψ (β))) - ϕ_{j} (F_{Γ_{H}} (ψ (α) ψ (β)))$ .

or, ϕ_j (F_{ϒ
_H} (ψ (α))∨ ϕ_j (F_{ϒ
_H} (ψ (β)) = ϕ_j (F_{ϒ
_H} (ψ (α))) ∨ϕ_j (F_{ϒ
_H} (ψ (β))) - ϕ_j (F_{Γ
_H} (ψ (α) ψ (β))) ,

or, ϕ_j (F_{Γ
_H} (ψ (α) ψ (β))) =0.

∴ ϕ_j (F_{Γ
_H} (ψ (α) ψ (β))) =0 = ϕ_j (F_{Γ
_G} (αβ)), for each j = 1, 2, . . . , m.

If for each j = 1, 2, . . . , m, 0 < ϕ_j (F_{Γ
_G} (αβ)) ≤1, then $ϕ_{j} (F_{Γ_{\bar{H}}} (ψ (α) ψ (β))) = 0 = ϕ_{j} (F_{Γ_{\bar{G}}} (α β))$ .

So, we have

$\begin{array}{l} ϕ_{j} (F_{Γ}_{H} (ψ (α) ψ (β))) = ϕ_{j} (F_{Υ}_{H} (ψ (α))) \lor \\ ϕ_{j} (F_{Υ}_{H} (ψ (β))) - 0 \\ = ϕ_{j} (F_{Υ}_{\bar{H}} (ψ (α)) \lor ϕ_{j} (F_{Υ}_{\bar{H}} (ψ (β)) \\ = ϕ_{j} (F_{Υ}_{\bar{G}} (α)) \lor ϕ_{j} (F_{Υ}_{\bar{G}} (β)) = ϕ_{j} (F_{Γ}_{\bar{G}} (α β)) . \end{array}$

∴ G ≅ H.

This completes the proof. □

5 Different type of product of m-polar SVNG’s

Definition 5.1. (Direct Product of Two m-Polar SVNG’s) The direct product G ⊓ H = (V_G × V_H, ϒ_G ⊓ ϒ_H, Γ_G ⊓ Γ_H), of the graph G′ ⊓ H′ = (V_G × V_H, E²) where ϒ_G ⊓ ϒ_H = (T_{ϒ
_G} ⊓ T_{ϒ
_H}, I_{ϒ
_G} ⊓ I_{ϒ
_H}, F_{ϒ
_G} ⊓ F_{ϒ
_H}), Γ_G ⊓ Γ_H = (T_{Γ
_G} ⊓ T_{Γ
_G}, I_{Γ
_G} ⊓ I_{Γ
_H}, F_{Γ
_G} ⊓ F_{Γ
_H}), E² = {((α, γ) (β, δ)) : αβ ∈ E_G, γδ ∈ E_H} and V_G ∩ V_H = ϕ, of two m-polar SVNG’s G = (V_G, ϒ_G, Γ_G), 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}), Γ_H = (T_{Γ
_H}, I_{Γ
_H}, F_{Γ
_H}) of the graph G′ = (V_G, E_G) and H = (V_H, E_H) respectively is defined as follows:

∀ (ξ, η) ∈ V_G × V_H and for each j = 1, 2, . . . , m,

ϕ_j ((T_{ϒ
_G} ⊓ T_{ϒ
_H}) (ξ, η)) = ϕ_j (T_{ϒ
_G} (ξ)) ∧ ϕ_j (T_{ϒ
_H} (η)),

ϕ_j ((I_{ϒ
_G} ⊓ I_{ϒ
_H}) (ξ, η)) = ϕ_j (I_{ϒ
_G} (ξ)) ∧ ϕ_j (I_{ϒ
_H} (η)),

ϕ_j ((F_{ϒ
_G} ⊓ F_{ϒ
_G}) (ξ, η)) = ϕ_j (F_{ϒ
_G} (ξ)) ∨ ϕ_j (F_{ϒ
_H} (η)),

∀ ((α, γ) (β, δ) ∈ E² and for each j = 1, 2, . . . , m,

ϕ_j ((T_{Γ
_G} ⊓ T_{Γ
_H}) ((α, γ) (β, δ)) = ϕ_j (T_{Γ
_G} (αβ)) ∧ ϕ_j (T_{Γ
_H} (γδ)),

ϕ_j ((I_{Γ
_G} ⊓ I_{Γ
_H}) ((α, γ) (β, δ)) = ϕ_j (I_{Γ
_G} (αβ)) ∧ ϕ_j (I_{Γ
_H} (γδ)),

ϕ_j ((F_{Γ
_G} ⊓ F_{Γ
_H}) ((α, γ) (β, δ)) = ϕ_j (F_{Γ
_G} (αβ)) ∨ ϕ_j (F_{Γ
_H} (γδ)).

Theorem 5.1. The direct product G ⊓ H = (V_G × V_H, ϒ_G ⊓ ϒ_H, Γ_G ⊓ Γ_H), of the graph G′ ⊓ H′ = (V_G × V_H, E²), where ϒ_G ⊓ ϒ_H = (T_{ϒ
_G} ⊓ T_{ϒ
_H}, I_{ϒ
_G} ⊓ I_{ϒ
_H}, F_{ϒ
_G} ⊓ F_{ϒ
_H}), Γ_G ⊓ Γ_H = (T_{Γ
_G} ⊓ T_{Γ
_H}, I_{Γ
_G} ⊓ I_{Γ
_H}, F_{Γ
_G} ⊓ F_{Γ
_H}), E² = {((α, γ) (β, δ)) : αβ ∈ E_G, γδ ∈ E_H}, of two m-polar SVNG’s G = (V_G, ϒ_G, Γ_H), 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}), Γ_H = (T_{Γ
_H}, I_{Γ
_H}, F_{Γ
_H}) of the graph G′ = (V_G, E_G) and H′ = (V_H, E_H) respectively, is a m-polar SVNG.

Theorem 5.2. If G = (V_G, ϒ_G, Γ_G) and H = (V_H, ϒ_H, Γ_H) are strong m-polar SVNG’s of the graphs G′ = (V_G, E_G) and H = (V_H, E_H), then G ⊓ H = (V_G × V_H, ϒ_G ⊓ ϒ_H, Γ_G ⊓ Γ_H) of the graph G′ ⊓ H′ = (V_G × V_H, E²) is strong m-polar SVNG.

Proof. Since G = (V_G, ϒ_G, Γ_G) and H = (V_H, ϒ_H, Γ_H) are strong m-polar SVNG’s, ∀ αβ ∈ E_G, ∀ γδ ∈ E_H and for each j = 1, 2, . . . , m,

ϕ_j (T_{Γ
_G} (αβ)) = ϕ_j (T_{ϒ
_G} (α)) ∧ ϕ_j (T_{ϒ
_G} (β)) and ϕ_j (T_{Γ
_H} (γδ)) = ϕ_j (T_{ϒ
_H} (γ)) ∧ ϕ_j (T_{ϒ
_H} (δ)) ,

ϕ_j (I_{Γ
_G} (αβ)) = ϕ_j (I_{ϒ
_G} (α)) ∧ ϕ_j (I_{ϒ
_G} (β)) and

ϕ_j (I_{Γ
_H} (γδ)) = ϕ_j (I_{ϒ
_H} (γ)) ∧ ϕ_j (I_{ϒ
_H} (δ)),

ϕ_j (F_{Γ
_G} (αβ)) = ϕ_j (F_{ϒ
_G} (α)) ∨ ϕ_j (F_{ϒ
_G} (β)) and ϕ_j (F_{Γ
_H} (γδ)) = ϕ_j (F_{ϒ
_H} (γ)) ∨ ϕ_j (F_{ϒ
_H} (δ)).

Now, ∀ ((ξ, η) (ς, υ)) ∈ E² and for each j = 1, 2, . . . , m,

$\begin{matrix} ϕ_{j} ((T_{Γ_{G}} ⊓ T_{Γ_{H}}) ((ξ, η) (ς, υ)) \\ = ϕ_{j} (T_{Γ_{G}} (ξ ς)) \land ϕ_{j} (T_{Γ_{H}} (η υ)) \\ = ϕ_{j} (T_{ϒ_{G}} (ξ)) \land ϕ_{j} (T_{ϒ_{G}} (ς)) \land ϕ_{j} (T_{ϒ_{H}} (η)) \\ \land ϕ_{j} (T_{ϒ_{H}} (υ)) \\ = (ϕ_{j} (T_{ϒ_{G}} (ξ)) \land ϕ_{j} (T_{ϒ_{H}} (η))) \land (ϕ_{j} (T_{ϒ_{G}} (ς)) \\ \land ϕ_{j} (T_{ϒ_{H}} (υ))) \\ = ϕ_{j} ((T_{ϒ_{G}} ⊓ T_{ϒ_{H}}) (ξ, η)) \land ϕ_{j} ((T_{ϒ_{G}} ⊓ T_{ϒ_{H}}) (ς, υ)), \end{matrix}$

$\begin{matrix} ϕ_{j} ((I_{Γ_{G}} ⊓ I_{Γ_{H}}) ((ξ, η) (ς, υ)) \\ = ϕ_{j} (I_{Γ_{G}} (ξ ς)) \land ϕ_{j} (I_{Γ_{H}} (η υ)) \\ = ϕ_{j} (I_{ϒ_{G}} (ξ)) \land ϕ_{j} (I_{ϒ_{G}} (ς)) \land ϕ_{j} (I_{ϒ_{H}} (η)) \\ \land ϕ_{j} (I_{ϒ_{H}} (υ)) \\ = (ϕ_{j} (I_{ϒ_{G}} (ξ)) \land ϕ_{j} (I_{ϒ_{H}} (η))) \land (ϕ_{j} (I_{ϒ_{G}} (ς)) \\ \land ϕ_{j} (I_{ϒ_{H}} (υ))) \\ = ϕ_{j} ((I_{ϒ_{G}} ⊓ I_{ϒ_{H}}) (ξ, η)) \land ϕ_{j} ((I_{ϒ_{G}} ⊓ I_{ϒ_{H}}) (ς, υ)), \end{matrix}$

$\begin{matrix} ϕ_{j} ((F_{Γ_{G}} ⊓ F_{Γ_{H}}) ((ξ, η) (ς, υ)) \\ = ϕ_{j} (F_{Γ_{G}} (ξ ς)) \lor ϕ_{j} (F_{Γ_{H}} (η υ)) \\ = ϕ_{j} (F_{ϒ_{G}} (ξ)) \lor ϕ_{j} (F_{ϒ_{G}} (ς)) \lor ϕ_{j} (F_{ϒ_{H}} (η)) \\ \lor ϕ_{j} (F_{ϒ_{H}} (υ)) \\ = (ϕ_{j} (F_{ϒ_{G}} (ξ)) \lor ϕ_{j} (F_{ϒ_{H}} (η))) \lor (ϕ_{j} (F_{ϒ_{G}} (ς)) \\ \lor ϕ_{j} (F_{ϒ_{H}} (υ))) \\ = ϕ_{j} ((F_{ϒ_{G}} ⊓ F_{ϒ_{H}}) (ξ, η)) \lor ϕ_{j} ((F_{ϒ_{G}} ⊓ F_{ϒ_{H}}) (ς, υ)) . \end{matrix}$

Hence G ⊓ H is a strong m-polar SVNG.

This completes the proof.□

Definition 5.2. (Semi-strong Product of Two m-Polar SVNG’s) The semi-strong product G • H = (V_G × V_H, ϒ_G • ϒ_H, Γ_G • Γ_H), of the graph G • H = (V_G × V_H, E³) where ϒ_G • ϒ_H = (T_{ϒ
_G} • T_{ϒ
_H}, I_{ϒ
_G} • I_{ϒ
_H}, F_{ϒ
_G} • F_{ϒ
_H}), Γ_G • Γ_H = (T_{Γ
_G} • T_{Γ
_H}, I_{Γ
_G} • I_{Γ
_H}, F_{Γ
_G} • F_{Γ
_H}), E³ = {((ξ, η) (ξ, υ)) : ξ ∈ E_G, ηυ ∈ E_H} ∪ E² and V_G ∩ V_H = ϕ, of two m-polar SVNG’s G = (V_G, ϒ_G, Γ_G), 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}) 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:

∀ (ξ, η) ∈ V_G × V_H and for each j = 1, 2, . . . , m,

ϕ_j ((T_{ϒ
_G} • T_{ϒ
_H}) (ξ, η)) = ϕ_j (T_{ϒ
_G} (ξ)) ∧ ϕ_j (T_{ϒ
_H} (η)),

ϕ_j ((I_{ϒ
_G} • I_{ϒ
_H}) (ξ, η)) = ϕ_j (I_{ϒ
_G} (ξ)) ∧ ϕ_j (I_{ϒ
_H} (η)),

ϕ_j ((F_{ϒ
_G} • F_{ϒ
_H}) (ξ, η)) = ϕ_j (F_{ϒ
_G} (ξ)) ∨ ϕ_j (F_{ϒ
_H} (η)),

∀ ((α, γ) (β, δ) ∈ E², ∀ ((ξ, η) (ξ, υ)) ∈ (E³ - E²) and for each j = 1, 2, . . . , m,

ϕ_j ((T_{Γ
_G} • T_{Γ
_H}) ((ξ, η) (ξ, υ)) = ϕ_j (T_{ϒ
_G} (ξ)) ∧ ϕ_j (T_{Γ
_H} (ηυ)) and ϕ_j ((T_{Γ
_G} • T_{Γ
_H}) ((α, γ) (β, δ)) = ϕ_j (T_{Γ
_G} (αβ)) ∧ ϕ_j (T_{Γ
_H} (γδ)),

ϕ_j ((I_{Γ
_G} • I_{Γ
_H}) ((ξ, η) (ξ, υ)) = ϕ_j (I_{ϒ
_G} (ξ)) ∧ ϕ_j (I_{Γ
_H} (ηυ)) and ϕ_j ((I_{Γ
_G} • I_{Γ
_H}) ((α, γ) (β, δ)) = ϕ_j (I_{Γ
_G} (αβ)) ∧ ϕ_j (I_{Γ
_H} (γδ)),

ϕ_j ((F_{Γ
_G} • F_{Γ
_H}) ((ξ, η) (ξ, υ)) = ϕ_j (F_{ϒ
_G} (ξ)) ∨ ϕ_j (F_{Γ
_H} (ηυ)) and ϕ_j ((F_{Γ
_G} • F_{Γ
_H}) ((α, γ) (β, δ)) = ϕ_j (F_{Γ
_G} (αβ)) ∨ ϕ_j (F_{Γ
_H} (γδ)).

Theorem 5.3. If G = (V_G, ϒ_G, Γ_G) and H = (V_H, ϒ_H, Γ_H) are two semi strong m-polar SVNG’s of the graphs G′ = (V_G, E_G) and H = (V_H, E_H), then G • H = (V_G × V_H, ϒ_G • ϒ_H, Γ_G • Γ_H) of the graph G′ • H′ = (V_H × V_H, E³) is a semi strong m-polar SVNG.

Proof. Since G = (V_G, ϒ_G, Γ_G) and H = (V_H, ϒ_H, Γ_H) are two semi strong m-polar SVNG’s, ∀ αβ ∈ E_G, ∀ γδ ∈ E_H and for each j = 1, 2, . . . , m,

ϕ_j (T_{Γ
_G} (αβ)) = ϕ_j (T_{ϒ
_G} (α)) ∧ ϕ_j (T_{ϒ
_G} (β)) and ϕ_j (T_{Γ
_H} (γδ)) = ϕ_j (T_{ϒ
_H} (γ)) ∧ ϕ_j (T_{ϒ
_H} (δ)),

ϕ_j (I_{Γ
_G} (αβ)) = ϕ_j (I_{ϒ
_G} (α)) ∧ ϕ_j (I_{ϒ
_H} (β)) and

ϕ_j (I_{Γ
_H} (γδ)) = ϕ_j (I_{ϒ
_G} (γ)) ∧ ϕ_j (I_{ϒ
_H} (δ)),

ϕ_j (F_{Γ
_G} (αβ)) = ϕ_j (F_{ϒ
_G} (α)) ∨ ϕ_j (F_{ϒ
_G} (β)) and ϕ_j (F_{Γ
_H} (γδ)) = ϕ_j (F_{ϒ
_H} (γ)) ∨ ϕ_j (F_{ϒ
_H} (δ)).

Now, ∀ ((ξ, η) (ς, υ)) ∈ E², ∀ ((α, β) (α, γ)) ∈ E³ - E² and for each j = 1, 2, . . . , m

$\begin{matrix} ϕ_{j} ((T_{Γ_{G}} • T_{Γ_{H}}) ((α, β) (α, γ)) \\ = ϕ_{j} (T_{ϒ_{G}} (α)) \land ϕ_{j} (T_{Γ_{H}} (β γ)) \\ = ϕ_{j} (T_{ϒ_{G}} (α)) \land ϕ_{j} (T_{ϒ_{H}} (β)) \land ϕ_{j} (T_{ϒ_{H}} (γ)) \\ = ϕ_{j} (T_{ϒ_{G}} (α)) \land ϕ_{j} (T_{ϒ_{H}} (β)) \land ϕ_{j} (T_{ϒ_{G}} (α)) \\ \land ϕ_{j} (T_{ϒ_{H}} (γ)) \\ = ϕ_{j} ((T_{ϒ_{G}} • T_{ϒ_{H}}) (α, β)) \land ϕ_{j} ((T_{ϒ_{G}} • T_{ϒ_{H}}) (α, γ)), \\ ϕ_{j} ((T_{Γ_{G}} • T_{Γ_{H}}) ((ξ, η) (ς, υ)) \\ = ϕ_{j} (T_{Γ_{G}} (ξ ς)) \land ϕ_{j} (T_{Γ_{H}} (η υ)) \\ = ϕ_{j} (T_{ϒ_{G}} (ξ)) \land ϕ_{j} (T_{ϒ_{G}} (ς)) \land ϕ_{j} (T_{ϒ_{H}} (η)) \\ \land ϕ_{j} (T_{ϒ_{H}} (υ)) \\ = (ϕ_{j} (T_{ϒ_{G}} (ξ)) \land ϕ_{j} (T_{ϒ_{H}} (η))) \land (ϕ_{j} (T_{ϒ_{G}} (ς)) \\ \land ϕ_{j} (T_{ϒ_{H}} (υ))) \\ = ϕ_{j} ((T_{ϒ_{G}} • T_{ϒ_{H}}) (ξ, η)) \land ϕ_{j} ((T_{ϒ_{G}} • T_{ϒ_{H}}) (ς, υ)), \end{matrix}$

$\begin{matrix} ϕ_{j} ((I_{Γ_{G}} • I_{Γ_{H}}) ((α, β) (α, γ)) \\ = ϕ_{j} (I_{ϒ_{G}} (α)) \land ϕ_{j} (I_{Γ_{H}} (β γ)) \\ = ϕ_{j} (I_{ϒ_{G}} (α)) \land ϕ_{j} (I_{ϒ_{H}} (β)) \land ϕ_{j} (I_{ϒ_{H}} (γ)) \\ = ϕ_{j} (I_{ϒ_{G}} (α)) \land ϕ_{j} (I_{ϒ_{H}} (β)) \land ϕ_{j} (I_{ϒ_{G}} (α)) \\ \land ϕ_{j} (I_{ϒ_{H}} (γ)) \\ = ϕ_{j} ((I_{ϒ_{G}} • I_{ϒ_{H}}) (α, β)) \land ϕ_{j} ((I_{ϒ_{G}} • I_{ϒ_{H}}) (α, γ)), \\ ϕ_{j} ((I_{Γ_{G}} • I_{Γ_{H}}) ((ξ, η) (ς, υ)) \\ = ϕ_{j} (I_{Γ_{G}} (ξ ς)) \land ϕ_{j} (I_{Γ_{H}} (η υ)) \\ = ϕ_{j} (I_{ϒ_{G}} (ξ)) \land ϕ_{j} (I_{ϒ_{G}} (ς)) \land ϕ_{j} (I_{ϒ_{H}} (η)) \\ \land ϕ_{j} (I_{ϒ_{H}} (υ)) \\ = (ϕ_{j} (I_{ϒ_{G}} (ξ)) \land ϕ_{j} (I_{ϒ_{H}} (η))) \land (ϕ_{j} (I_{ϒ_{G}} (ς)) \\ \land ϕ_{j} (I_{ϒ_{H}} (υ))) \\ = ϕ_{j} ((I_{ϒ_{G}} • I_{ϒ_{H}}) (ξ, η)) \land ϕ_{j} ((I_{ϒ_{G}} • I_{ϒ_{H}}) (ς, υ)) \end{matrix}$

$\begin{matrix} ϕ_{j} ((F_{Γ_{G}} • F_{Γ_{H}}) ((α, β) (α, γ)) \\ = ϕ_{j} (F_{ϒ_{G}} (α)) \lor ϕ_{j} (F_{Γ_{H}} (β γ)) \\ = ϕ_{j} (F_{ϒ_{G}} (α)) \lor ϕ_{j} (F_{ϒ_{H}} (β)) \lor ϕ_{j} (F_{ϒ_{H}} (γ)) \\ = ϕ_{j} (F_{ϒ_{G}} (α)) \lor ϕ_{j} (F_{ϒ_{H}} (β)) \lor ϕ_{j} (F_{ϒ_{H}} (α)) \\ \lor ϕ_{j} (F_{ϒ_{H}} (γ)) \\ = ϕ_{j} ((F_{ϒ_{G}} • F_{ϒ_{H}}) (α, β)) \lor ϕ_{j} ((F_{ϒ_{G}} • F_{ϒ_{H}}) (α, γ)), \\ ϕ_{j} ((F_{Γ_{G}} • F_{Γ_{H}}) ((ξ, η) (ς, υ)) \\ = ϕ_{j} (F_{Γ_{G}} (ξ ς)) \lor ϕ_{j} (F_{Γ_{H}} (η υ)) \\ = ϕ_{j} (F_{ϒ_{G}} (ξ)) \land ϕ_{j} (F_{ϒ_{G}} (ς)) \land ϕ_{j} (F_{ϒ_{H}} (η)) \\ \land ϕ_{j} (F_{ϒ_{H}} (υ)) \\ = (ϕ_{j} (F_{ϒ_{G}} (ξ)) \land ϕ_{j} (F_{ϒ_{H}} (η))) \land (ϕ_{j} (F_{ϒ_{G}} (ς)) \\ \land ϕ_{j} (F_{ϒ_{H}} (υ))) \\ = ϕ_{j} ((F_{ϒ_{G}} • F_{ϒ_{H}}) (ξ, η)) \land ϕ_{j} ((F_{ϒ_{G}} • F_{ϒ_{H}}) (ς, υ)) . \end{matrix}$

Hence G • H is a semi strong m-polar SVNG.

This completes the proof. □

Definition 5.3. (Strong Product of Two m-Polar SVNG’s) The strong product G ⨂ H = (V_G × V_H, ϒ_G • ϒ_H, Γ_G • Γ_H) of the graph G′ ⨂ H′ = (V_G × V_H, E), where ϒ_G ⨂ ϒ_H = (T_{ϒ
_G} ⨂ T_{ϒ
_H}, I_{ϒ
_G} • I_{ϒ
_H}, F_{ϒ
_G} • F_{ϒ
_H}), Γ_G ⨂ Γ_H = (T_{Γ
_G} ⨂ T_{Γ
_H}, I_{Γ
_G} ⨂ I_{Γ
_H}, F_{Γ
_G} ⨂ F_{Γ
_H}), E⁴ = E ∪ E² and V_G ∩ V_H = ϕ, of two m-polar SVNG’s G = (V_G, ϒ_G, Γ_G), 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}), Γ_H = (T_{Γ
_H}, I_{Γ
_H}, F_{Γ
_H}) of the graph G′ = (V_G, E_G) and H = (V_H, E_H) respectively is defined as follows:

∀ (ξ, η) ∈ V_G × V_H and for each j = 1, 2, . . . , m,

ϕ_j ((T_{ϒ
_G} ⨂ T_{ϒ
_H}) (ξ, η)) = ϕ_j (T_{ϒ
_G} (ξ)) ∧ ϕ_j (T_{ϒ
_H} (η))

ϕ_j ((I_{ϒ
_G} ⨂ I_{ϒ
_H}) (ξ, η)) = ϕ_j (I_{ϒ
_G} (ξ)) ∧ ϕ_j (I_{ϒ
_H} (η))

ϕ_j ((F_{ϒ
_G} ⨂ F_{ϒ
_H}) (ξ, η)) = ϕ_j (F_{ϒ
_G} (ξ)) ∨ ϕ_j (F_{ϒ
_H} (η)).

∀ ((α, γ) (β, δ) ∈ E², ∀ ((ξ, η) (ξ, υ)) ∈ E, ((α, β) (γ, β)) ∈ E and for each j = 1, 2, . . . , m,

ϕ_j ((T_{Γ
_G} ⨂ T_{Γ
_H}) ((ξ, η) (ξ, υ))) = ϕ_j (T_{ϒ
_G} (ξ)) ∧ ϕ_j (T_{Γ
_H} (ηυ)), ϕ_j ((T_{Γ
_G} ⨂ T_{Γ
_H}) ((α, β) (γ, β))) = ϕ_j (T_{Γ
_G} (αγ)) ∧ ϕ_j (T_{ϒ
_H} (β)) and ϕ_j ((T_{Γ
_G} ⨂ T_{Γ
_H}) ((α, γ) (β, δ))) = ϕ_j (T_{Γ
_G} (αβ)) ∧ ϕ_j (T_{Γ
_H} (γδ)),

ϕ_j ((I_{Γ
_G} ⨂ I_{Γ
_H}) ((ξ, η) (ξ, υ))) = ϕ_j (I_{ϒ
_G} (ξ)) ∧ ϕ_j (I_{Γ
_H} (ηυ)), ϕ_j ((I_{Γ
_G} ⨂ I_{Γ
_H}) ((α, β) (γ, β))) = ϕ_j (I_{Γ
_G} (αγ)) ∧ ϕ_j (I_{ϒ
_H} (β)) and ϕ_j ((I_{Γ
_G} ⨂ I_{Γ
_H}) ((α, γ) (β, δ))) = ϕ_j (I_{Γ
_G} (αβ)) ∧ ϕ_j (I_{Γ
_H} (γδ)),

ϕ_j ((F_{Γ
_G} ⨂ F_{Γ
_H}) ((ξ, η) (ξ, υ))) = ϕ_j (F_{ϒ
_G} (ξ)) ∨ ϕ_j (F_{Γ
_H} (ηυ)), ϕ_j ((F_{Γ
_G} ⨂ F_{Γ
_H}) ((α, β) (γ, β))) = ϕ_j (F_{Γ
_G} (αγ)) ∨ ϕ_j (F_{ϒ
_H} (β)) and ϕ_j ((F_{Γ
_G} ⨂ F_{Γ
_H}) ((α, γ) (β, δ))) = ϕ_j (F_{Γ
_G} (αβ)) ∨ ϕ_j (F_{Γ
_H} (γδ)).

Theorem 5.4. If G = (V_G, ϒ_G, Γ_H) and H = (V_H, ϒ_H, Γ_H) are two complete m-polar SVNG’s of the graphs H = (V_H, E_H) and H = (V_H, E_H), then G ⨂ H = (V_G × V_H, ϒ_G ⨂ ϒ_H, Γ_G ⨂ Γ_H) of the graph G′ ⨂ H′ = (V_G × V_H, E⁴) is a complete m-polar SVNG.

Definition 5.4. (Product m-Polar SVNG) Let G = (V_G, ϒ_G, Γ_G) be an m-polar SVNG of the graph G′ = (V_G, E_G), where ϒ_G = (T_{ϒ
_G}, I_{ϒ
_G}, F_{ϒ
_G}) and Γ_G = (T_{Γ
_G}, I_{Γ
_G}, F_{Γ
_G}). If T_{Γ
_G} (αβ) ≤ T_{ϒ
_G} (α) × T_{ϒ
_G} (β), I_{Γ
_G} (αβ) ≤ I_{ϒ
_G} (α) × I_{ϒ
_G} (β) and F_{Γ
_G} (αβ) ≤ F_{ϒ
_G} (α) × F_{ϒ
_G} (β), $\forall α β \in {\tilde{V}}_{G}^{2}$ , where × denote the simple multiplication, and therefore the graph is called product m-polar SVNG.

Theorem 5.5. Every product m-polar SVNG is an m-polar SVNG.

Definition 5.5. (Complete Product m-Polar SVNG) A product m-polar single-valued graph G = (V_G, ϒ_G, Γ_G) of the graph G′ = (V_G, E_G) is said to be complete if T_{Γ
_G} (αβ) = T_{ϒ
_G} (α) × T_{ϒ
_G} (β), I_{Γ
_G} (αβ) = I_{ϒ
_G} (α) × I_{ϒ
_G} (β) and F_{Γ
_G} (αβ) = F_{ϒ
_G} (α) × F_{ϒ
_G} (β), $\forall α β \in {\tilde{V}}_{G}^{2}$ .

Definition 5.6. (Ring Sum of Two Product m-Polar SVNG’s) The ring sum G oplus H = (V_G ∪ V_H, ϒ_G oplus ϒ_H, Γ_G oplus Γ_H) of the graph G′ oplus H " = (V_G ∪ V_H, E_GoplusH), where (ϒ_G oplus ϒ_H) = (T_{ϒ
_G} oplus T_{ϒ
_H}, I_{ϒ
_G} oplus I_{ϒ
_H}, F_{ϒ
_G} oplus F_{ϒ
_H}) and (Γ_G oplus Γ_H) = (T_{Γ
_G} oplus T_{Γ
_H}, I_{Γ
_G} oplus I_{Γ
_H}, F_{Γ
_G} oplus F_{Γ
_H}) , of two product m-polar SVNG’s, G = (V_G, ϒ_G, Γ_G), 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 j = 1, 2, . . . , m,

$ϕ_{j} ((T_{ϒ_{G}} oplus T_{ϒ_{H}}) (ζ)) = {\begin{matrix} ϕ_{j} (T_{ϒ_{G}} (ζ)), if ζ \in V_{G} - V_{H} \\ ϕ_{j} (T_{ϒ_{H}} (ζ)), if ζ \in V_{H} - V_{G} \\ ϕ_{j} (T_{ϒ_{G}} (ζ)) \lor ϕ_{j} (T_{ϒ_{H}} (ζ)), if ζ \in V_{G} \cap V_{H} . \end{matrix}$

$ϕ_{j} ((I_{ϒ_{G}} oplus I_{ϒ_{H}}) (ζ)) = {\begin{matrix} ϕ_{j} (I_{ϒ_{G}} (ζ)), if ζ \in V_{G} - V_{H} \\ ϕ_{j} (I_{ϒ_{H}} (ζ)), if ζ \in V_{H} - V_{G} \\ ϕ_{j} (I_{G} (ζ)) \lor ϕ_{j} (I_{H} (ζ)), if ζ \in V_{G} \cap V_{H} . \end{matrix}$

$ϕ_{j} ((F_{ϒ_{G}} oplus F_{ϒ_{H}}) (ζ)) = {\begin{matrix} ϕ_{j} (F_{ϒ_{G}} (ζ)), if ζ \in V_{G} - V_{H} \\ ϕ_{j} (F_{ϒ_{H}} (ζ)), if ζ \in V_{H} - V_{G} \\ ϕ_{j} (F_{ϒ_{G}} (ζ)) \land ϕ_{j} (F_{ϒ_{H}} (ζ)), if ζ \in V_{G} \cap V_{H} . \end{matrix}$

$ϕ_{j} ((T_{Γ_{G}} oplus T_{Γ_{H}}) (ζ η)) = {\begin{matrix} ϕ_{j} (T_{Γ_{G}} (ζ η)), if ζ η \in E_{G} - E_{H} \\ ϕ_{j} (T_{Γ_{H}} (ζ η)), if ζ η \in E_{H} - E_{G} \\ 0, otherwise . \end{matrix}$

$ϕ_{j} ((I_{Γ_{G}} oplus I_{Γ_{H}}) (ζ η)) = {\begin{matrix} ϕ_{j} (I_{Γ_{G}} (ζ η)), if ζ η \in E_{G} - E_{H} \\ ϕ_{j} (I_{Γ_{H}} (ζ η)), if ζ η \in E_{H} - E_{G} \\ 0, otherwise . \end{matrix}$

$ϕ_{j} ((F_{Γ_{G}} oplus F_{Γ_{H}}) (ζ η)) = {\begin{matrix} ϕ_{j} (F_{Γ_{G}} (ζ η)), if ζ η \in E_{G} - E_{H} \\ ϕ_{j} (F_{Γ_{H}} (ζ η)), if ζ η \in E_{H} - E_{G} \\ 0, otherwise . \end{matrix}$

Theorem 5.6. If G = (V_G, ϒ_G, Γ_G) and H = (V_H, ϒ_H, Γ_H) are product m-polar SVNG’s of the graphs G′ = (V_G, E_G) and H′ = (V_H, E_H),then G oplus H = (V_G ∪ V_H, ϒ_G oplus ϒ_H, Γ_G oplus Γ_H) of the graph G′ oplus H′ = (V_G ∪ V_H, E) is product m-polar SVNG.

6 Application

The 1-polar SVNG is a special type of SVNG. The SVNG is used for modeling several real life applications, e.g., telephone network planning [12], safest path finding in an airline travel[5], multi-criteria decision making, choosing of location and gas terminal, Baltic sea [9], cellular network provider Companies [26], and so on. The bipolar SVNG (2-polar SVNG) is an example of m-polar SVNG where the value of m = 2. Bipolar SVNG has used in many real life applications such as organizational designations, radio channels [2], decision making problems [6], and brands competition [1] etc. Additionally, m-polar SVNG’s (m > 2) are very efficient to handle the uncertainties in many real life situations, e.g., when a country like India elects its prime minister or If a set of persons decides which film to watch. For e.g., we can consider a real life application of a mobile company. In a mobile company, a group of managing director decides which mobile to manufacture. This type of real life application can be modeled by m-polar SVNG. Here, different mobile can be considered as vertices and two nodes are connected by an arc if there exists any relationship between the two vertices.

The membership degree of each vertex/mobile describes the preference of the mobile which collects from a group of mobile users. The preference degree of a mobile (within [0, 1]) is expressed by a single user. Thus, each vertex has multi-preference degrees corresponding to a mobile. Similarly, the relationship degree between two nodes determines the relationship value of the arc. Between two mobiles, one mobile may have a better camera, may be in very high RAM and processor, may be battery, and so on. In such real life scenarios, there exists multipolar information between two mobiles. This network is a classical case of m-polar SVNG’s. For any mobile company, it is very significant to decide which mobile to produce so that the company can make the profit and band as much as possible. A banded and good mobile is readily purchased by several customers if the price of the mobile is also very much inexpensive. The decision of which mobile to produce is a decision making problem. By making the proper decision, a mobile company can sell their mobile throughout the world. The company can keep in their mind that the mobile is excellent quality, in very demand, easily accessible, and so on. Therefore, we can use m-polar SVNG’s to model many real life application in many field of engineering and science, electronic engineering, computer science, mathematics, artificial intelligence and artificial neural networks.

Here, we introduce a model of m-polar SVNG graph G = (V_G, ϒ_G, Γ_G) to evaluate the performance of professors by the feedback of the students of the computer science department in a college during the session 2018-2019. Here, the vertices are used to represent the professors of the computer science department and arcs are used to represent the relationship among the two professors. Suppose the computer science department has six professors represented as V = {t₁, t₂, t₃, t₄, t₅}. The value of membership of each vertex presents the corresponding professors feedback response of the students based regularity, updated information, presentation style, generation of interest, lecture quality, and promoting future study among students. All the features of a professor according to the student is uncertain, therefore 4-polar single-valued neutrosophic set is used to represent the node set V.

In the Table 1, the truth-membership values, indeterminatistic membership values of the professor’s are described which is based on the rating of the students.

Table 1
4-Polar SVNS A of V_G

t ₁ t ₂ t ₃ t ₄ t ₅

ϕ₁ (A) <.6, . 5, . 3> <.7, . 6, . 3> <.8, . 4, . 5> <.8, . 6, . 2> <.8, . 2, . 6>

ϕ₂ (A) <.7, . 4, . 4> <.8, . 2, . 3> <.6, . 4, . 2> <.9, . 4, . 3> <.8, . 5, . 5>

ϕ₃ (A) <.8, . 3, . 6> <.8, . 3, . 5> <.9, . 2, . 4> <.7, . 3, . 4> <.7, . 4, . 5>

ϕ₄ (A) <.7, . 2, . 5> <.7, . 2, . 4> <.7, . 8, . 1> <.8, . 2, . 3> <.9, . 2, . 5>

	t ₁	t ₂	t ₃	t ₄	t ₅
ϕ₁ (A)	<.6, . 5, . 3>	<.7, . 6, . 3>	<.8, . 4, . 5>	<.8, . 6, . 2>	<.8, . 2, . 6>
ϕ₂ (A)	<.7, . 4, . 4>	<.8, . 2, . 3>	<.6, . 4, . 2>	<.9, . 4, . 3>	<.8, . 5, . 5>
ϕ₃ (A)	<.8, . 3, . 6>	<.8, . 3, . 5>	<.9, . 2, . 4>	<.7, . 3, . 4>	<.7, . 4, . 5>
ϕ₄ (A)	<.7, . 2, . 5>	<.7, . 2, . 4>	<.7, . 8, . 1>	<.8, . 2, . 3>	<.9, . 2, . 5>

Arc cost is used to represent the relationship among the professor’s which can be computed as follows.

ϕ_j (T_{Γ
_G} (xy))≤ ϕ_j (T_{ϒ
_G} (x)) ∧ ϕ_j (T_{ϒ
_G} (y)) ;

ϕ_j (I_{Γ
_G} (xy))≤ ϕ_j (I_{ϒ
_G} (x)) ∧ ϕ_j (I_{ϒ
_G} (y)) ;

ϕ_j (F_{Γ
_G} (xy)) ≥ ϕ_j (F_{ϒ
_G} (x)) ∨ ϕ_j (F_{ϒ
_G} (y)), $\forall xy \in {\tilde{V}}_{G}^{2}$ and j = 1, 2, 3, 4, 5 .

The membership degrees are shown in the Table 2.

Table 2

4-Polar SVNR on V_G

	t ₁ t ₂	t ₁ t ₅	t ₂ t ₃	t ₂ t ₄
ϕ₁ (A)	<.6, . 5, . 3>	<.6, . 2, . 5>	<.7, . 4, . 5>	<.7, . 6, . 3>
ϕ₂ (A)	<.7, . 2, . 4>	<.7, . 3, . 6>	<.7, . 4, . 4>	<.6, . 2, . 1>
ϕ₃ (A)	<.8, . 3, . 4>	<.7, . 3, . 5>	<.7, . 2, . 5>	<.7, . 3, . 5>
ϕ₄ (A)	<.7, . 2, . 3>	<.7, . 2, . 5>	<.7, . 2, . 4>	<.7, . 2, . 4>
	t ₂ t ₅	t ₃ t ₄	t ₃ t ₅	t ₄ t ₅
ϕ₁ (A)	<.7, . 2, . 6>	<.7, . 4, . 4>	<.8, . 2, . 6>	<.8, . 2, . 5>
ϕ₂ (A)	<.7, . 1, . 4>	<.6, . 4, . 3>	<.6, . 2, . 2>	<.7, . 2, . 5>
ϕ₃ (A)	<.7, . 2, . 5>	<.7, . 2, . 4>	<.7, . 2, . 4>	<.7, . 2, . 5>
ϕ₄ (A)	<.7, . 2, . 5>	<.7, . 2, . 3>	<.7, . 1, . 4>	<.8, . 2, . 5>

Table 3

Score Value of The Teacher’s

	t ₁	t ₂	t ₃	t ₄	t ₅
score	. 630	. 683	. 692	. 708	. 658

The professor’s performance are ranked according to the following characteristics:

Professor’s performance value is <60%, professor’s performance is Average.

Professor’s performance value is ≥60% and <65%, professor’s performance is Good.

Professor’s performance value is ≥65% and <70%, Professor’s performance value is Very Good.

Professor’s performance value is ≥70%, Professor’s performance value is Excellent.

In the Table 1, the performance of t₂, t₃, t₅ are very good and the performance of the professor t₄ is excellent. Among all the professors, professor t₄ is the best professor according the response value of the students of the computer science department. We compute the score value of each professor according to the formula

$s (t_{i}) = \frac{2 + T - I - F}{3}$ ,

where T =average of truth membership,

I =average of indeterminastic membership and

F = average of truth of falsity membership.

7 Conclusions

Neutrosophic graph is an efficient method for modeling many uncertain decision optimization problems. Multi-polar information generally exists in almost every decision making making. This type of information cannot be properly modeled using simple neutrosophic graph. We can use the m- polar neutrosophic set/graph to deal this type of uncertainty. The main contribution of this study is to define the m- polar neutrosophic graph model and also present several operations, such as union, join, composition and ring sum on m-polar neutrosophic graph. We describe some new products (strong product, semi strong product, complete product, direct product, Cartesian product and lexicographic product) for m-polar neutrosophic graph. Finally, an application of m-polar neutrosophic graph is describe the evaluation of the teacher’s performance of a college.

In our future work, we will focus on transportation networks, production management, artificial neural networks and database theory of m-polar neutrosophic graph which are very much efficient in many branches of engineering, science and medicine and we will also try is to extend this work to m-polar fuzzy interval graphs, m-polar fuzzy intersection graphs and m-polar fuzzy hypergraphs and so on.

References

Akram

, Nasir

and Shum

K.P.

, Novel applications of bipolar single-valued neutrosophic competition graphs, Applied Mathematics-A Journal of Chinese Universities 33(4) (2018), 436–467.

Akram

, Sarwar

and Smarandache

, Bipolar neutrosophic graphs with applications.

Akram

and Shahzadi

, Operations on single-valued neutrosophic graphs. Infinite Study, 2017.

Akram

and Shahzadi

, Neutrosophic soft graphs with application, Journal of Intelligent & Fuzzy Systems 32(1) (2017), 841–858.

Akram

, Siddique

and Davvaz

, New concepts in neutrosophic graphs with application, Journal of Applied Mathematics and Computing 57(1-2) (2018), 279–302.

Akram

and Sitara

, Bipolar neutrosophic graph structures. Infinite Study, 2017.

Akram

and Raheela Younas

, Certain types of irregular-polar fuzzy graphs, Journal of Applied Mathematics and Computing 53(1-2) (2017), 365–382.

Atanassov

K.T.

, Intuitionistic fuzzy sets. In Intuitionistic fuzzy sets, (1999), pp. 1–137. Springer.

Bausys

, Zavadskas

E.K.

and Kaklauskas

, Application of neutrosophic set to multicriteria decision making by COPRAS. Infinite Study, 2015.

10.

Broumi

, Bakali

, Talea

, Smarandache

, Dey

, et al. Spanning tree problem with neutrosophic edge weights, Procedia Computer Science 127 (2018), 190–199.

11.

Broumi

and Smarandache

, New distance and similarity measures of interval neutrosophic sets. Infinite Study, 2014.

12.

Broumi

, Ullah

, Bakali

and Talea

, Novel system and method for telephone network planing based on neutrosophic graph. Infinite Study, 2018.

13.

Chen

, Li

, Ma

and Wang

, m-polar fuzzy sets: An extension of bipolar fuzzy sets. The ScientificWorld Journal, 2014, 2014.

14.

Deli

, Ali

and Smarandache

, Bipolar neutrosophic sets and their application based on multi-criteria decision making problems. In 2015 International Conference on Advanced Mechatronic Systems (ICAMechS), (2015), pp. 249–254. IEEE.

15.

Dey

, Pal

, Viet Long

, et al., Fuzzy minimum spanning tree with interval type 2 fuzzy arc length: Formulation and a new genetic algorithm, Soft Computing 1–12.

16.

Dey

, Pal

and Pal

, Interval type 2 fuzzy set in fuzzy shortest path problem, Mathematics 4(4) (2016), 62.

17.

Dey

, Pradhan

, Pal

and Pal

, A genetic algorithm for solving fuzzy shortest path problems with interval type-2 fuzzy arc lengths, Malaysian Journal of Computer Science 31(4) (2018), 255–270.

18.

Ghorai

and Pal

, On some operations and density of-polar fuzzy graphs, Pacific Science Review A: Natural Science and Engineering 17(1) (2015), 14–22.

19.

Ghorai

and Pal

, Some properties of-polar fuzzy graphs, Pacific Science Review A: Natural Science and Engineering 18(1) (2016), 38–46.

20.

Haibin

, Smarandache

, Zhang

and Sunderraman

, Single valued neutrosophic sets. Infinite Study, 2010.

21.

Liu

, Chen

and Peng

, Cosine distance measure between neutrosophic hesitant fuzzy linguistic sets and its application in multiple criteria decision making, Symmetry 10(11) (2018), 602.

22.

Liu

, Liu

and Liu

, Some similarity measures of neutrosophic sets based on the euclidean distance and thei application in medical diagnosis. Computational and mathematical methods in medicine, 2018, 2018.

23.

Rashmanlou

, Samanta

, Pal

and Ali Borzooei

, A study on bipolar fuzzy graphs, Journal of Intelligent & Fuzzy Systems 28(2) (2015), 571–580.

24.

Smarandache.

, Neutrosophic set-a generalization of the intuitionistic fuzzy set, International Journal of Pure and Applied Mathematics 24(3) (2005), 287.

25.

Wang

, Zhang

Y-Q.

and Sunderraman

, Truth-value based interval neutrosophic sets, In 2005 IEEE International Conference on Granular Computing 1 (2005), pp. 274–277. IEEE.

26.

Yaqoob

, Gulistan

, Kadry

and Wahab

, Complex intuitionistic fuzzy graphs with application in cellular network provider companies, Mathematics 7(1) (2019), 35.

27.

, Single-valued neutrosophic minimum spanning tree and its clustering method, Journal of Intelligent Systems 23(3) (2014), 311–324.

28.

Zadeh

L.A.

, Fuzzy sets, Information and Control 8(3) (1965), 338–353.

29.

Zhang

W-R.

, (yin)(yang) bipolar fuzzy sets, In 1998 IEEE International Conference on Fuzzy Systems Proceedings. IEEEWorld Congress on Computational Intelligence (Cat. No. 98CH36228) 1 (1998), pp. 835–840. IEEE.