Properties of single-valued neutrosophic graphs

Abstract

In this study, we introduce some new concepts related to single-valued neutrosophic graphs such as homomorphism, isomorphism, weak isomorphism and co-weak isomorphism, and define arithmetic Cartesian product, arithmetic direct product, and arithmetic lexicographical product operations on single-valued neutrosophic graphs with their examples. We also introduce intersection and union operations of single-valued neutrosophic graphs, and give some examples related to these operations. Furthermore we define two types of addition between two single-valued neutrosophic graphs based on union operation and summation operation of single valued neutrosophic numbers. Finally we present an application of single-valued neutrosophic graphs in decision making.

Keywords

Single-valued neutrosophic sets single-valued neutrosophic relation single-valued neutrosophic graphs product operations homomorphism isomorphism

1 Introduction

Graph theory, which introduced by Euler in 1736 to solve Königsberg bridge problem, has an important role in many areas such as mathematics, computer sciences, chemistry, physics, engineering, medicine and sociology. The concept of fuzzy graphs was defined by Rosenfeld [38] in 1975 based on fundamental idea proposed by Kauffman [27] in 1973. He also obtained some structures, which exist in graph theory, for fuzzy graphs. Bhattacharya [13] showed relation between fuzzy graph and fuzzy groups, and gave some remarks on fuzzy graphs. Bhutani [14] introduced some concepts such as weak isomorphism, co-weak isomorphism and isomorphism between fuzzy graphs. Mordeson and Peng [31] defined Cartesian product, composition, union and join operations of fuzzy graphs and investigated some of their properties. Mordeson and Nair [32] defined complement of fuzzy graph. In 2001, Sunitha and Kumar [49] modified definition of complement of a fuzzy graph and studied some properties of self complementary fuzzy graphs.

Intuitionistic fuzzy set theory was introduced by Atanassov [1] in 1983 as a generalization of fuzzy sets. Atanassov also defined the concept of intuitionistic fuzzy relation and introduced intuitionistic fuzzy graph theory [2]. Karunambigai and Parvathi [23] introduced intuitionistic fuzzy graph considering a special case of Atanassov’s definition of intuitionistic fuzzy graph. They also defined operations on intuitionistic fuzzy graphs [36]. Some more work on intuitionistic fuzzy relation and intuitionistic fuzzy graphs may be found on [3 , 42]. Akram and Davvaz [5] introduced the notion of intuitionistic fuzzy graphs and investigated some of their properties. Sahoo and Pal [43] defined some product operations on intuitionistic fuzzy graphs and investigated some of their properties. The concept of rough set was introduced by Pawlak [37]. Tong et al. [51] introduced rough graph structure by combining rough set theory and classical graph theory. By combining soft graph with rough set theory, concept of soft rough graph was defined by Noor et al. [34].

Fuzzy set theory and intuitionistic fuzzy set theory are very important tools for dealing with uncertainty and incomplete information. But they may not be sufficient in modeling of indeterminate and inconsistent information encountered in real world. In order to cope with this issue, neutrosophic set theory was proposed by Smarandache [44, 45] as a generalization of fuzzy sets and intuitionistic fuzzy sets. However, since neutrosophic sets are identified by three functions called truth-membership (t), indeterminacy-membership(i) and falsity-membership (f) whose values are real standard or non-standard subset of unit interval ] ^-0, 1⁺ [, there are some difficulties in modeling of some problems in engineering and sciences. To overcome these difficulties, in 2010, concept of single valued neutrosophic(SVN) sets and its operations defined by Wang [50] as a generalization of intuitionistic fuzzy sets. Yang et al. [52] introduced concept of single-valued neutrosophic relation based on single-valued neutrosophic set. They also developed kernels and closures of a single-valued neutrosophic set.

As mentioned above, many researchers have studied on fuzzy graph theory and intuitionistic fuzzy graph theory by taking their vertex sets and edge sets as fuzzy sets and intuitionistic fuzzy set, respectively. But in some applications fuzzy vertices or intuitionistic fuzz vertices may not be sufficient in modeling of problems. Because of this reason, Smarandache [20 , 46–48] proposed notion of neutrosophic graph and separated them to four main categories called I-edge neutrosophic graph, I-vertex neutrosophic graph, (T,I,F)-edge neutrosophic graph and (T,I,F)-vertex neutrosophic graph. Then Broumi et al. [17] introduced concept of single-valued neutrosophic graphs (SVN-graphs) and bipolar single valued neutrosophic graphs [18], and gave basic definitions, which are exist in graph theory, for SVN-graphs. They also investigated some properties of SVN-graphs. Naz et al. [33] defined operations of SVN-graphs based on definition of SVN-graphs given by Broumi et al. [17]. Akram and Shahzadi [8] showed that there are some flaws in Broumi et al.’s [17] and Shah et al.’s [39] definitions, and first introduced with a new approach the concept of SVN-graphs giving true results in some SVN-graph operations such as joint and complement. Further several new concepts on neutrosophic graphs with their applications were discussed in [6 , 53]. Dhavaseelan et al. [19] introduced the concept of strong neutrosophic graph, and studied on some interesting properties of strong neutrosophic graphs. Shah [40] discussed some new concepts such as homomorphism, isomorphism, weak isomorphism and co-weak isomorphism between two neutrosophic graphs. Shah and Hussain [39] defined concept of neutrosophic soft graphs and obtained some of their properties. Akram and Shahzadi [9] defined concept of neutrosophic soft graphs, and introduced some novel concepts as strong neutrosophic soft graphs, complete neutrosophic soft graphs. They also obtained some of properties related to these new concepts. Furthermore, they gave an application of neutrosophic soft graphs in decision making.

In this paper, we give some new definitions related to single-valued neutrosophic relations (SVN-relations) and obtain some properties of SVN-relations, and define composition operations of SVN-relations. We also introduce concepts of homomorphism, isomorphism, weak isomorphism and co-weak isomorphism between two SVN-graphs based on definition given by Broumi in [17]. Furthermore we define some new operations such as arithmetic Cartesian products, arithmetic direct products, arithmetic lexicographical products, intersection, join and direct join operations between SVN-graphs and give examples related these operations, and obtain some of their properties. Finally, to determine dominate element in a group we give an application of the SVN-graphs in decision making.

2 Preliminaries

In this section, we present basic definitions in single valued neutrosophic set theory.

2.1 Single-valued neutrosophic sets

A neutrosophic set A on the universe of discourse X is defined as $A = {〈 x, T_{A} (x), I_{A} (x), F_{A} (x) 〉 : x \in X}$ where T_A, I_A, F_A : X →] ^-0, 1⁺ [ and ^-0 ≤ T_A (x) + I_A (x) + F_A (x) ≤3⁺ [44]. From philosophical point of view, the neutrosophic set takes the value from real standard or non-standard subsets of ] ^-0, 1⁺ [. But in real life application in scientific and engineering problems it is difficult to use neutrosophic set with value from real standard or non-standard subset of ] ^-0, 1⁺ [. Hence we consider the neutrosophic set which takes the value from the subset of [0, 1].

Let X be a nonempty set (initial universe), with a generic element in X denoted by x. A single-valued neutrosophic set (SVN-set) μ is characterized by three functions called truth membership function μ_t (x), indeterminacy membership function μ_i (x) and falsity membership function μ_f (x) such that μ_t (x), μ_i (x), μ_f (x) ∈ [0, 1] for all x ∈ X, as follows:

If X is continuous, a SVN-set μ can be written as follows: $μ = \int_{X} 〈 μ_{t} (x), μ_{i} (x), μ_{f} (x) 〉 / x, for all x \in X .$

If X is crisp set, a SVN-set μ can be written as follows: $μ = \sum_{x} 〈 μ_{t} (x), μ_{i} (x), μ_{f} (x) 〉 / x, for all x \in X .$

Also, finite SVN-set μ can be presented by μ = {〈x₁, μ_t (x₁), μ_i (x₁), μ_f (x₁) 〉, …, 〈x_M, μ_t (x_M), μ_i (x_M), μ_f (x_M) 〉} for all x ∈ X. Here 0 ≤ μ_t (x) + μ_i (x) + μ_f (x) ≤3 for all x ∈ X [50]. Also, we can denote SVN-set μ over X by separating three parts as follows: $\begin{matrix} μ_{t} & = & {(x, μ_{t} (x)) : x \in X}, \\ μ_{i} & = & {(x, μ_{i} (x)) : x \in X}, \\ μ_{f} & = & {(x, μ_{f} (x)) : x \in X} . \end{matrix}$

Then, we can write μ = (μ_t, μ_i, μ_f). Here μ_t, μ_i and μ_f are fuzzy sets. Throughout this paper, initial universe will be considered as a finite and crisp set, and set of all SVN-sets over X will be denoted by SVN_X. Let μ, ν ∈ SVN_X. Then,

μ ⊆ ν if and only if μ_t (x) ≤ ν_t (x), μ_i (x) ≥ ν_i (x), μ_f (x) ≥ ν_f (x) for all x ∈ X,

μ = ν if and only if μ ⊆ ν and ν ⊆ μ for all x ∈ X,

μ^c={〈x, μ_f (x), 1 - μ_i (x), μ_t (x) 〉 : x ∈ X},

μ ∪ ν= {〈x, (μ_t (x) ∨ ν_t (x)), (μ_i (x) ∧ ν_i (x)), (μ_f (x) ∧ ν_f (x)) 〉 : x ∈ X} ,

μ ∩ ν= {〈x, (μ_t (x) ∧ ν_t (x)), (μ_i (x) ∨ ν_i (x)), (μ_f (x) ∨ ν_f (x)) 〉 : x ∈ X}.

For convenience, we can use μ (x) = 〈μ_t (x), μ_i (x), μ_f (x) 〉 to represent an element in SVN-set and we will call it as a SVN-number. Operations between two SVN-numbers are defined by Liu and Wang [28] as follows. Let μ (x) = 〈μ_t (x), μ_i (x), μ_f (x) 〉 and μ (y) = 〈μ_t (y), μ_i (y), μ_f (y) 〉 be two SVN-numbers. Then,

μ (x) ⊕ μ (y) = 〈μ_t (x) + μ_t (y) - μ_t (x) μ_t (y), μ_i (x) μ_i (y), μ_f (x) μ_f (y) 〉

μ (x) ⊗ μ (y) = 〈μ_t (x) μ_t (y), μ_i (x) + μ_i (y) - μ_i (x) μ_i (y), μ_f (x) + μ_f (y) - μ_f (x) μ_f (y) 〉

λμ (x) = 〈1 - (1 - μ_t (x)) ^λ, μ_i (x) ^λ, μ_f (x) ^λ〉, λ > 0

μ (x) ^λ = 〈 (μ_t (x)) ^λ, 1 - (1 - μ_i (x)) ^λ, 1 - (1 - μ_f (x)) ^λ〉, λ > 0

Let μ and ν be two SVN-sets over X and Y, respectively. Then, the Cartesian product of μ and ν, denoted by μ × ν = γ, is defined as follows: $\begin{matrix} γ & = & {((x, y) 〈 γ_{t} (x, y), γ_{i} (x, y), γ_{f} (x, y) 〉) : \\ (x, y) \in X \times Y} \end{matrix}$ where γ_t (x, y) = min(μ_t (x), ν_t (y)), γ_i (x, y) = max(μ_i (x), ν_i (y)), and γ_f (x, y) = max(μ_f (x), ν_f (y)) . This definition can be considered as a special case of the Definition 6 given in reference [16].

We can generalize the definition of Cartesian product of SVN-sets as follows:

Let μ₁, . . . , μ_n be n SVN-sets over universal sets X₁, . . . , X_n, respectively. Then, the Cartesian product of μ₁, . . . , μ_n is denoted by μ₁ × . . . × μ_n = π, is defined by $\begin{matrix} π & = & {((x_{1}, . . ., x_{n}) 〈 π_{t} (x_{1}, . . ., x_{n}), π_{i} (x_{1}, . . ., x_{n}), \\ π_{f} (x_{1}, . . ., x_{n}) 〉) : (x_{1},, . . ., x_{n}) \in \prod_{i = 1}^{n} X_{i}} \end{matrix}$ here π_t (x₁, . . . , x_n) = min(μ₁_t (x₁), . . . , μ_n_t (x_n)), π_i (x₁, . . . , x_n) = max(μ₁_i (x₁), . . . , μ_n_i (x_n)), π_f (x₁, . . . , x_n) = max(μ₁_f (x₁), . . . , μ_n_f (x_n)) .

Let μ be a SVN-set over X. Then, level set or (α₁, α₂, α₃)-cut of SVN-set μ, denoted by μ^{(α₁,α₂,α₃)}, is defined as follows: μ^{(α₁,α₂,α₃)} = {x ∈ X : μ_t (x) ≥ α₁, μ_i (x) ≤ α₂, μ_f (x) ≤ α₃} , α₁, α₂, α₃ ∈ [0, 1] [15].

Let I³ = [0, 1] × [0, 1] × [0, 1] and N (I³) = {(α₁, α₂, α₃) : α₁, α₂, α₃ ∈ [0, 1]}. Then, (N (I³) is a lattice together with partial ordered relation ⪯, where order relation ⪯ on N (I³) can be defined by (0, 1, 1) ⪯ (α₁, α₂, α₃) ⪯ (β₁, β₂, β₃) ⪯ (1, 0, 0) ⇔ α₁ ≤ β₁, α₂ ≥ β₂, α₃ ≥ β₃ for (α₁, α₂, α₃), (β₁, β₂, β₃) ∈ N (I³) [22].

Let ρ be a SVN-set over U × U. Then, a single valued neutrosophic relation (SVNR) over set U is a set defined by ρ = {〈 (u, v), ρ_t (u, v), ρ_i (u, v), ρ_f (u, v) 〉 : (u, v) ∈ U × U} . Here ρ_t : U × U → [0, 1], ρ_i : U × U → [0, 1] and ρ_f : U × U → [0, 1] denote the truth-membership function, indeterminacy membership function and falsity membership function of ρ, respectively [52].

2.2 Single-valued neutrosophic relations

Definition 2.1. Let X and Y be two nonempty sets and let μ and ν be two SVN-sets over X and Y, respectively. Then, a single-valued neutrosophic relation (SNR-relation) η from the SVN-set μ into SVN-set ν is a SVN-set η on X × Y such that η (x, y) ⪯ μ (x) ⋏ ν (y), for all (x, y) ∈ X × Y.

Here, η (x, y) = 〈η_t (x, y), η_i (x, y), η_f (x, y) 〉 and η_t, η_i, η_f : X × Y → [0, 1]. Also μ (x) = 〈μ_t (x), μ_i (x), μ_f (x) 〉 for all x ∈ X and ν (y) = 〈ν_t (y), ν_i (y), ν_f (y) 〉 for all y ∈ Y, and μ (x) ⋏ μ (y) = (min(μ_t (x), μ_t (y)), max(μ_i (x), μ_i (y)), max(η_f (x, y), η_f (y, x)) .

If μ = ν, then η is called a SVN-relation on SVN-set μ. Especially, if we take X = Y and μ = ν, the Definition is reduced the Definition 2.6 given in reference [17].

Corollary 2.2. Let μ be a SVN-set over X. Then, $μ^{(α_{1}, α_{2}, α_{3})} = μ_{t}^{α_{1}} \cap μ_{i}^{α_{2}} \cap μ_{f}^{α_{3}},$ where $μ_{t}^{α_{1}} = {x \in X : μ_{t} (x) \geq α_{1}}$ , $μ_{i}^{α_{2}} = {x \in X : μ_{i} (x) \leq α_{2}}$ and $μ_{f}^{α_{3}} = {x \in X : μ_{f} (x) \leq α_{3}}$ .

Proposition 2.3. Let η : X × Y → [0, 1] be a SVN-relation from SVN-set μ over X into a SVN-set μ′ over Y and η^{(α₁,α₂,α₃)}, μ^{(α₁,α₂,α₃)} and μ^{′(α₁,α₂,α₃)} be (α₁, α₂, α₃)-cuts of η, μ and μ′, respectively. Then, η^{(α₁,α₂,α₃)} ⊆ μ^{(α₁,α₂,α₃)} × μ^{′(α₁,α₂,α₃)}.

Proof. Let (x, y) ∈ η^{(α₁,α₂,α₃)}. Then, η_t (x, y) ≥ α₁, η_i (x, y) ≤ α₂, η_f (x, y) ≤ α₃ . Since η is SVN-relation, it follows that α₁ ≤ η_t (x, y) ≤ min(μ_t (x), $μ_{t}^{'} (y)) \Rightarrow α_{1} \leq μ_{t} (x)$ and $α_{1} \leq μ_{t}^{'} (y),$ $α_{2} \geq η_{i} (x, y) \geq max (μ_{i} (x), μ_{i}^{'} (y)) \Rightarrow α_{2} \geq μ_{i} (x)$ and $α_{2} \geq μ_{i}^{'} (y), α_{3} \geq η_{f} (x, y) \geq max (μ_{f} (x), μ_{f}^{'} (y)) \Rightarrow α_{3} \geq μ_{f} (x)$ and $α_{3} \geq μ_{f}^{'} (y) .$ From here, x ∈ μ^{(α₁,α₂,α₃)}, y ∈ μ^{′(α₁,α₂,α₃)} and (x, y) ∈ μ^{(α₁,α₂,α₃)} × μ^{′(α₁,α₂,α₃)}. So η^{(α₁,α₂,α₃)} ⊆ μ^{(α₁,α₂,α₃)} × μ^{′(α₁,α₂,α₃)}.

Definition 2.4. Let μ be a SVN-set on X and η be a SVN-relation on μ. Then, η is called the strongest SVN-relation on μ if and only if for all SVN-relations η′ on μ, for all x, y ∈ X η′ (x, y) ⪯ η (x, y) .

Definition 2.5. Let μ be a SVN-set on X and η be a SVN-relation on X × X. Then, μ is called the weakest SVN-set of X on which η is a SVN-relation defined by μ (x) = ⋎ {η (x, y) ⋎ η (y, x) |y ∈ X} , forall x ∈ X . Here, η (x, y) ⋎ η (y, x) = (max(η_t (x, y), η_t (y, x)), min(η_i (x, y), η_i (y, x)), min(η_f (x, y), η_f (y, x))) .

3 Single-valued neutrosophic graphs

In this section we present definitions of SVN-graph made previously. Then, we define some novel operations of SVN-graphs based on definition given by Broumi et al. [17].

Definition 3.1. [8] A single-valued neutrosophic graph is a pair G = (μ, η), where μ : V → [0, 1] is SVN-set in V and η : V × V → [0, 1] is SVN-relation on V such that η_t (xy) ≤ min(μ_t (x), μ_t (y)), η_i (xy) ≤ min(μ_i (x), μ_i (y)), η_f (xy) ≤ max(μ_f (x), μ_f (y)) for all x, y ∈ V.

Definition 3.2. [40] Let G = (V, E) be a simple graph and E ⊆ V × V. Let μ_t, μ_i, μ_f : V → [0, 1] denote the truth-membership, indeterminacy-membership and falsity-membership of an element v ∈ V and η_t, η_i, η_f : E → [0, 1] denote truth, indeterminacy and falsity- memberships of (x, y) ∈ E. The neutrosophic graph is a 3-tuple satisfied following conditions: η_t (xy) ≤ min(μ_t (x), μ_t (y)), η_i (xy) ≤ min(μ_i (x), μ_i (y)), η_f (xy) ≥ max(μ_f (x), μ_f (y)) for all x, y ∈ V.

Definition 3.3. [17] Let G = (V, E) be a graph, μ be a SVN-set on V and η be a SVN-set on E ⊆ V × V. If η (x, y) ⪯ (μ (x) ⋏ μ (y)) for all (x, y) ∈ E, it is called a single-valued neutrosophic graph (SVN-graph) over graph G = (V, E) and denoted by $\tilde{G} = (μ, η)$ .

SVN-set μ over V is SVN-vertex set of SVN-graph and SVN-set η over E is the SVN-edge set of SVN-graph. Note that η is a symmetric SVN-relation on μ.

From now on we use the notation xy for (x, y) ∈ E. In this study, we will not appear elements of which SVN-values are (0, 1, 1) in SVN-graph. Throughout this paper, set of all SVN-graphs over vertex set V will be denoted by $SVN (V)$ .

In this paper, we use definition of SVN-graph given in [17].

Example 1. Let V = {x₁, x₂, x₃, x₄, x₅} be an initial universe, μ be a SVN-set over V and η be a SVN-set over E = {x₁x₂, x₂x₃, x₃x₁, x₂x₄} ⊆ V × V given as in Table 1.

Table 1
SVN-graph $\tilde{G}$

x ₁ x ₂ x ₃ x ₄

μ _t 0.7 0.5 0.6 0.8

μ _i 0.5 0.4 0.2 0.7

μ _f 0.6 0.2 0.5 0.4

x ₁ x ₂ x ₂ x ₃ x ₃ x ₁ x ₂ x ₄

η _t 0.3 0.4 0.6 0.2

η _i 0.6 0.5 0.7 0.8

η _f 0.9 1.0 0.9 0.6

	x ₁	x ₂	x ₃	x ₄
μ _t	0.7	0.5	0.6	0.8
μ _i	0.5	0.4	0.2	0.7
μ _f	0.6	0.2	0.5	0.4

	x ₁ x ₂	x ₂ x ₃	x ₃ x ₁	x ₂ x ₄
η _t	0.3	0.4	0.6	0.2
η _i	0.6	0.5	0.7	0.8
η _f	0.9	1.0	0.9	0.6

Then, SVN-graph $\tilde{G} = (μ, η)$ can be depicted as in Fig. 1.

Fig.1

$\tilde{G} = (μ, η)$ SVN-graph.

Definition 3.4. Let ${\tilde{G}}_{1} = (μ, η)$ and ${\tilde{G}}_{2} = (μ^{'}, η^{'})$ be two SVN-graphs over graphs G₁ = (V₁, E₁) and G₂ = (V₂, E₂), respectively. If μ ⊆ μ′ and η ⊆ η′, then SVN-graph ${\tilde{G}}_{1} = (μ, η)$ is called a SVN-subgraph of ${\tilde{G}}_{2} = (μ^{'}, η^{'})$ and denoted by ${\tilde{G}}_{1} ⊑ {\tilde{G}}_{2} .$

Definition 3.5. [17] Let ${\tilde{G}}_{1} = (μ, η)$ and ${\tilde{G}}_{2} = (μ^{'}, η^{'})$ be two SVN-graphs over graphs G₁ = (V₁, E₁) and G₂ = (V₂, E₂), respectively. If V₁ ⊆ V₂, μ (x) ⪯ μ′ (x) for all x ∈ V₁ and η (xy) ⪯ η′ (xy) for all x, y ∈ V₁, then SVN-graph ${\tilde{G}}_{1} = (μ, η)$ is called a partial SVN-subgraph of ${\tilde{G}}_{2} = (μ^{'}, η^{'})$ and denoted by ${\tilde{G}}_{1} ⊏ {\tilde{G}}_{2} .$

Example 2. Let us consider SVN-graph $\tilde{G} = (μ, η)$ given as in Example 1. Given SVN-graphs ${\tilde{G}}_{1} = (μ^{'}, η^{'})$ and ${\tilde{G}}_{2} = (μ^{″}, η^{″})$ as in Tables 2 and 3.

Table 2

SVN-subgraph of SVN-graph $\tilde{G}$

	x ₂	x ₃	x ₄
$μ_{t}^{'}$	0.5	0.6	0.8
$μ_{i}^{'}$	0.4	0.2	0.7
$μ_{f}^{'}$	0.2	0.5	0.4

	x ₂ x ₃	x ₂ x ₄
$η_{t}^{'}$	0.3	0.1
$η_{i}^{'}$	0.6	0.8
$η_{f}^{'}$	1.0	0.7

Table 3

Partial SVN-subgraph of SVN-graph $\tilde{G}$

	x ₁	x ₂	x ₃	x ₄
$μ_{t}^{″}$	0.6	0.5	0.5	0.7
$μ_{i}^{″}$	0.6	0.4	0.2	0.8
$μ_{f}^{″}$	0.7	0.3	1.0	0.4

	x ₁ x ₂	x ₂ x ₃	x ₃ x ₁	x ₂ x ₄
$η_{t}^{″}$	0.2	0.3	0.4	0.1
$η_{i}^{″}$	0.7	0.5	0.8	0.8
$η_{f}^{″}$	1.0	1.0	1.0	0.7

Then, ${\tilde{G}}_{1}$ and ${\tilde{G}}_{2}$ is partial SVN-subgraph and SVN-subgraph of SVN-graph $\tilde{G}$ , respectively.

They are depicted in Figs. 2 and 3, respectively.

Fig.2

Partial SVN-subgraph of SVN-graph $\tilde{G}$ .

Fig.3

SVN-subgraph of SVN-graph $\tilde{G}$ .

Proposition 3.6. Let $\tilde{G} = (μ, η) \in SVN (V)$ . If α and β are SVN-numbers such that α = (α_t, α_i, α_f) ⪯ β = (β_t, β_i, β_f), then (μ^β, η^β) is a subgraph of (μ^α, η^α).

Proof. The proof is obvious from Corollary 2.2.

Proposition 3.7. Let ${\tilde{G}}_{1} = (μ, η)$ be a partial SVN-graph of ${\tilde{G}}_{2} = (μ^{'}, η^{'})$ . Then, (μ^α, η^α) is a subgraph of (μ^′α, η^′α) such that α = (α_t, α_i, α_f) ∈ N (I³).

Proof. Let x ∈ μ^α. Then, μ_t (x) ≥ α_t, μ_i (x) ≤ α_i and μ_f (x) ≤ α_f. Since μ ⊆ μ′, α_t ≤ μ_t (x) ≤ μ′t₍x), $α_{i} \geq μ_{i} (x) \geq μ_{i}^{'} (x)$ and $α_{f} \geq μ_{f} (x) \geq μ_{f}^{'} (x)$ . Thus $μ_{t}^{'} (x) \geq α_{t}$ , $μ_{i}^{'} (x) \leq α_{i}$ and $μ_{f}^{'} (x) \leq α_{f}$ . Hence x ∈ μ^′α and $μ^{α} \subseteq μ^{' α}$ (1)

Let xy ∈ η^α. Then, η_t (xy) ≥ α_t, η_i (xy) ≤ α_i and η_f (xy) ≤ α_f. Since η ⊆ η′, $η_{t} (xy) \leq η_{t}^{'} (xy)$ , $η_{i} (xy) \geq η_{i}^{'} (xy)$ and $η_{f} (xy) \geq η_{f}^{'} (xy)$ . Thus $η_{t}^{'} (xy) \geq α_{t}$ , $η_{i}^{'} (xy) \leq α_{i}$ and $η_{f}^{'} (xy) \leq α_{f}$ . Hence (xy) ∈ η^′α and $η^{α} \subseteq η^{' α}$ (2) from (1) and (2) we get ${\tilde{G}}_{1} ⊑ {\tilde{G}}_{2}$ .

Concepts of homomorphism, isomorphism, weak isomorphism and co-weak isomorphism are defined by Shah [40].

Now we redefine these concepts based on definition of SVN-graphs given by Broumi [17].

Definition 3.8. Let ${\tilde{G}}_{1} = (μ, η)$ and ${\tilde{G}}_{2} = (μ^{'}, η^{'})$ be two SVN-graphs over graphs G₁ = (V₁, E₁) and G₂ = (V₂, E₂), respectively. Then, bijective mapping h : V₁ → V₂ satisfying the following conditions is called a homomorphism from ${\tilde{G}}_{1}$ to ${\tilde{G}}_{2}$

$μ_{t} (x) \leq μ_{t}^{'} (h (x)), μ_{i} (x) \geq μ_{i}^{'} (h (x)) and μ_{f} (x) \geq μ_{f}^{'} (h (x))$ ,

$η_{t} (xy) \leq η_{t}^{'} (h (x) h (y)), η_{i} (xy) \geq η_{i}^{'} (h (x) h (y))$ and $η_{f} (xy) \geq η_{f}^{'} (h (x) h (y))$

for all x ∈ V₁, xy ∈ E₁ ⊆ V₁ × V₁ .

Definition 3.9. [40] Let ${\tilde{G}}_{1} = (μ, η)$ and ${\tilde{G}}_{2} = (μ^{'}, η^{'})$ be two SVN-graphs over graphs G₁ = (V₁, E₁) and G₂ = (V₂, E₂), respectively. Then, bijective mapping h : V₁ → V₂ satisfying the following conditions is called an isomorphism from ${\tilde{G}}_{1}$ to ${\tilde{G}}_{2}$

$μ_{t} (x) = μ_{t}^{'} (h (x)), μ_{i} (x) = μ_{i}^{'} (f (x)) and μ_{f} (x) = μ_{f}^{'} (h (x))$ ,

$η_{t} (xy) = η_{t}^{'} (h (x) h (y)), η_{i} (xy) = η_{i}^{'} (h (x) h (y))$ and $η_{f} (xy) = η_{f}^{'} (h (x) h (y))$

for all x ∈ V₁, xy ∈ E₁ ⊆ V₁ × V₁ .

Definition 3.10. Let ${\tilde{G}}_{1} = (μ, η)$ and ${\tilde{G}}_{2} = (μ^{'}, η^{'})$ be two SVN-graphs over graphs G₁ = (V₁, E₁) and G₂ = (V₂, E₂), respectively. Then, bijective mapping h : V₁ → V₂ satisfying the following conditions is called a weak isomorphism from ${\tilde{G}}_{1}$ to ${\tilde{G}}_{2}$ .

h is homomorphism

$μ_{t} (x) = μ_{t}^{'} (h (x)), μ_{i} (x) = μ_{i}^{'} (h (x)) and μ_{f} (x) = μ_{f}^{'} (h (x))$ ,

for all x ∈ V₁, xy ∈ E₁ ⊆ V₁ × V₁ .

Example 3. Let V₁ = {1, 2, 3} V₂ = {4, 5, 6} be two vertex sets, f (x) = x + 3 be a bijective mapping from V₁ to V₂ and Let ${\tilde{G}}_{1} = (μ, η)$ and ${\tilde{G}}_{2} = (μ^{'}, η^{'})$ be two SVN-graphs over graphs G₁ = (V₁, E₁) and G₂ = (V₂, E₂), respectively where μ (1) = 〈0.5, 0.7, 0.6〉, μ (2) = 〈0.4, 0.3, 0.5〉, μ (3) = 〈0.7, 0.2, 0.8〉 and η (12) = 〈0.3, 0.8, 0.7〉, η (13) = 〈0.2, 0.9, 0.8〉 μ′ (4) = 〈0.5, 0.7, 0.6〉, μ′ (5) = 〈0.4, 0.3, 0.5〉, μ′ (6) = 〈0.7, 0.2, 0.8〉 and η′ (45) = 〈0.3, 0.8, 0.7〉, η′ (46) = 〈0.2, 0.9, 0.8〉. Here since μ (1) = μ′ (4), μ (2) = μ′ (5), μ (3) = μ′ (6) and η (12) = η′ (45), η (13) = η′ (46), h is a isomorphism of ${\tilde{G}}_{1}$ onto ${\tilde{G}}_{2}$ .

Remark 1. A weak isomorphism may not be an isomorphism.

In above example, if we get η (12) = 〈0.2, 0.8, 0.8〉, η (13) = 〈0.1, 0.9, 0.9〉 and η′ (45) = 〈0.2, 0.9, 0.8〉, η′ (46) = 〈0.2, 0.9, 0.8〉 then, η (12) ¬ = η′ (45) and η (13) ¬ = η′ (46), so h is weak isomorphism but h is not an isomorphism.

Definition 3.11. Let ${\tilde{G}}_{1} = (μ, η)$ and ${\tilde{G}}_{2} = (μ^{'}, η^{'})$ be two SVN-graphs over graphs G₁ = (V₁, E₁) and G₂ = (V₂, E₂), respectively. Then, bijective mapping h : V₁ → V₂ satisfying following conditions is called as a co-weak isomorphism from ${\tilde{G}}_{1}$ to ${\tilde{G}}_{2}$ .

h is homomorphism

$η_{t} (xy) = η_{t}^{'} (h (x) h (y)), η_{i} (xy) = η_{i}^{'} (h (x) h (y))$ and $η_{f} (xy) = η_{f}^{'} (h (x) h (y))$

for all xy ∈ E₁ ⊆ V₁ × V₁ .

Example 4. Let us consider function h in Example 3. Let ${\tilde{G}}_{1} = (μ, η)$ and ${\tilde{G}}_{2} = (μ^{'}, η^{'})$ be two SVN-graphs over graphs G₁ = (V₁, E₁) and G₂ = (V₂, E₂), respectively where μ (1) = 〈0.6, 0.5, 0.8〉, μ (2) = 〈0.5, 0.8, 0.4〉, μ (3) = 〈0.2, 0.6, 0.7〉 and η (12) = 〈0.3, 0.8, 0.9〉, η (13) = 〈0.2, 0.7, 0.8〉 μ′ (4) = 〈0.7, 0.5, 0.8〉, μ′ (5) = 〈0.6, 0.7, 0.2〉, μ′ (6) = 〈0.4, 0.7, 0.6〉 and η′ (45) = 〈0.3, 0.8, 0.9〉, η′ (46) = 〈0.2, 0.7, 0.8〉.

Here since h is a homomorphism from (μ, η) to (μ′, η′) and η (12) = η (45), η (13) = η′ (46), h is a co-weak isomorphism from ${\tilde{G}}_{1}$ to ${\tilde{G}}_{2}$ .

Note that h is not an isomorphism from ${\tilde{G}}_{1}$ to ${\tilde{G}}_{2}$ , since μ (13) ¬ = μ′ (45) and μ (13) ¬ = μ′ (46).

Some of definitions given below are similar the definitions in [8, 40] with regard to relations between truth-memberships. But they are different with regard to relations between indeterminacy-memberships and between falsity-memberships.

Definition 3.12. [33] Let ${\tilde{G}}_{1} = (μ, η)$ and ${\tilde{G}}_{2} = (μ^{'}, η^{'})$ be two SVN-graphs over graphs G₁ = (V₁, E₁) and G₂ = (V₂, E₂), respectively. Then, Cartesian product of two SVN-graphs ${\tilde{G}}_{1} = (μ, η)$ and ${\tilde{G}}_{2} = (μ^{'}, η^{'})$ is denoted by ${\tilde{G}}_{1} \hat{\times} {\tilde{G}}_{2} = (μ \times μ^{'}, η \times η^{'})$ and defined as follows:

$(μ_{t} \times μ_{t}^{'}) (x_{1}, x_{2}) = min (μ_{t} (x_{1}), μ_{t}^{'} (x_{2}))$ , $(μ_{i} \times μ_{i}^{'}) (x_{1}, x_{2}) = max (μ_{i} (x_{1}), μ_{i}^{'} (x_{2}))$ , $(μ_{f} \times μ_{f}^{'}) (x_{1}, x_{2}) = max (μ_{f} (x_{1}), μ_{f}^{'} (x_{2}))$ , (for all (x₁, x₂) ∈ V₁ × V₂),

$(η_{t} \times η_{t}^{'}) ((x, x_{2}) (x, y_{2})) = min (μ_{t} (x), η_{t}^{'} (x_{2} y_{2})),$ $(η_{i} \times η_{i}^{'}) ((x, x_{2}) (x, y_{2})) = max (μ_{i} (x), η_{i}^{'} (x_{2} y_{2}))$ , $(η_{f} \times η_{f}^{'}) ((x, x_{2}) (x, y_{2})) = max (μ_{f} (x), η_{f}^{'} (x_{2} y_{2})),$ (for all x ∈ V₁, x₂y₂ ∈ E₂),

$(η_{t} \times η_{t}^{'}) ((x_{1}, z) (y_{1}, z)) = min (η_{t} (x_{1} y_{1}), μ_{t}^{'} (z))$ , $(η_{i} \times η_{i}^{'}) ((x_{1}, z) (y_{1}, z)) = max (η_{i} (x_{1} y_{1}), μ_{i}^{'} (z))$ , $(η_{f} \times η_{f}^{'}) ((x_{1}, z) (y_{1}, z)) = max (η_{f} (x_{1} y_{1}), μ_{f}^{'} (z)),$ (for all z ∈ V₂, x₁y₁ ∈ E₁).

Example 5. Consider two SVN-graphs ${\tilde{G}}_{1} = (μ, η)$ and ${\tilde{G}}_{2} = (μ^{'}, η^{'})$ given as in Fig. 4. Then, Cartesian product of SVN-graphs ${\tilde{G}}_{1}$ and ${\tilde{G}}_{2}$ is as in Fig. 5.

Fig.4

${\tilde{G}}_{1}$ and ${\tilde{G}}_{2}$ SVN-graphs, respectively.

Fig.5

SVN-graph ${\tilde{G}}_{1} \hat{\times} {\tilde{G}}_{2}$ .

Proposition 3.13. Let ${\tilde{G}}_{1} = (μ, η)$ , ${\tilde{G}}_{2} = (μ^{'}, η^{'})$ and ${\tilde{G}}_{3} = (μ^{″}, η^{″})$ be three SVN-graphs. Then, $({\tilde{G}}_{1} \hat{\times} {\tilde{G}}_{2}) \hat{\times} {\tilde{G}}_{3} = {\tilde{G}}_{1} \hat{\times} ({\tilde{G}}_{2} \hat{\times} {\tilde{G}}_{3})$ .

Proof. Let (x₁, x₂) ∈ V₁ × V₂, y₁, y₂ ∈ V₃ and y₁y₂ ∈ E₃. Then, we have $\begin{matrix} ((η_{t} \times η_{t}^{'}) \times η_{t}^{″}) ((x_{1}, x_{2}), y_{1}), ((x_{1}, x_{2}), y_{2})) \\ = min ((μ_{t} \times μ_{t}^{'}) (x_{1}, x_{2}), η_{t}^{″} (y_{1} y_{2})) \\ \leq min (min (μ_{t} (x_{1}), μ_{t}^{'} (x_{2})), min (μ_{t}^{″} (y_{1}), μ_{t}^{″} (y_{2}))) \\ = min (μ_{t} (x_{1}), (μ_{t}^{'} (x_{2}), min (μ_{t}^{″} (y_{1}), μ_{t}^{″} (y_{1}))) \\ = min (μ_{t} (x_{1}), (μ_{t}^{'} (x_{2}), η_{t}^{″} (y_{1} y_{2})) \\ = min (μ_{t} (x_{1}), (η_{t}^{'} \times η_{t}^{″}) ((x_{2}, y_{1}) (x_{2}, y_{2}))) \\ = ((η_{t} \times η_{t}^{'}) \times η_{t}^{″}) ((x_{1}, (x_{2}, y_{1})) (x_{1}, (x_{2}, y_{2}))), \\ ((η_{i} \times η_{i}^{'}) \times η_{i}^{″}) (((x_{1}, x_{2}), y_{1}), ((x_{1}, x_{2}), y_{2})) \\ = max ((μ_{i} \times μ_{i}^{'}) (x_{1}, x_{2}), η_{i}^{″} (y_{1} y_{2})) \\ \geq max (max (μ_{i} (x_{1}), μ_{i}^{'} (x_{2})), \\ max (μ_{i}^{″} (y_{1}), μ_{i}^{″} (y_{2}))) \\ = max (μ_{i} (x_{1}), (μ_{i}^{'} (x_{2}), max (μ_{i}^{″} (y_{1}), μ_{i}^{″} (y_{1}))) \\ = max (μ_{i} (x_{1}), max (μ_{i}^{'} (x_{2}), η_{i}^{″} (y_{1} y_{2})) \\ = max (μ_{i} (x_{1}), (η_{i}^{'} \times η_{i}^{″}) ((x_{2}, y_{1}) (x_{2}, y_{2}))) \\ = ((η_{i} \times η_{i}^{'}) \times η_{i}^{″}) ((x_{1}, (x_{2}, y_{1})) (x_{1}, (x_{2}, y_{2}))), \\ ((η_{f} \times η_{f}^{'}) \times η_{f}^{″}) (((x_{1}, x_{2}), y_{1}), ((x_{1}, x_{2}), y_{2})) \\ = max ((μ_{f} \times μ_{f}^{'}) (x_{1}, x_{2}), η_{f}^{″} (y_{1} y_{2})) \\ \geq max (max (μ_{f} (x_{1}), μ_{f}^{'} (x_{2})), \\ max (μ_{f}^{″} (y_{1}), μ_{f}^{″} (y_{2}))) \\ = max (μ_{f} (x_{1}), (μ_{f}^{'} (x_{2}), max (μ_{f}^{″} (y_{1}), μ_{f}^{″} (y_{1}))) \\ = max (μ_{f} (x_{1}), max (μ_{f}^{'} (x_{2}), η_{f}^{″} (y_{1} y_{2})) \\ = max (μ_{f} (x_{1}), (η_{f}^{'} \times η_{f}^{″}) ((x_{2}, y_{1}) (x_{2}, y_{2}))) \\ = ((η_{f} \times η_{f}^{'}) \times η_{f}^{″}) ((x_{1}, (x_{2}, y_{1})) (x_{1}, (x_{2}, y_{2}))) . \end{matrix}$

Therefore, we concluded that $({\tilde{G}}_{1} \hat{\times} {\tilde{G}}_{2}) \hat{\times} {\tilde{G}}_{3} = {\tilde{G}}_{1} \hat{\times} ({\tilde{G}}_{2} \hat{\times} {\tilde{G}}_{3})$ .

Proposition 3.14. Let ${{\tilde{G}}_{i} = (μ^{i}, η^{i}) : i = 1, . . ., n, n \in N}$ be a family of SVN-graphs. Then, $\prod_{i = 1}^{n} {\tilde{G}}_{i}$ is a SVN-graph.

Proof. The proof can be made by using the induction law.

Now we define a new Cartesian product two SVN-sets based on product operation defined by Liu and Wang [28].

Definition 3.15. Let μ and μ′ be two SVN-set over universal sets X and Y, respectively. Then arithmetic Cartesian product of μ and μ′, denoted by μ ⊗ μ′, is defined by $\begin{matrix} μ \otimes μ^{'} & = & {(μ_{t} (x) {μ^{'}}_{t} (y), μ_{i} (x) + {μ^{'}}_{i} (y) \\ - μ_{i} (x) {μ^{'}}_{i} (y), μ_{f} (x) + {μ^{'}}_{f} (y) \\ - μ_{f} (x) {μ^{'}}_{f} (y)) : (x, y) \in X \times Y} . \end{matrix}$

Here, μ_t (x) μ′_t (y), μ_i (x) + μ′_i (y) - μ_i (x) μ′_i (y) and μ_f (x) + μ′_f (y) - μ_f (x) μ′_f (y) will be denoted by (μ_t ⊗ μ′_t) (x, y), (μ_i ⊗ μ′_i) (x, y) and (μ_f ⊗ μ′_f) (x, y)

Definition 3.16. Let ${\tilde{G}}_{1} = (μ, η)$ and ${\tilde{G}}_{2} = (μ^{'}, η^{'})$ be two SVN-graphs over graphs G₁ = (V₁, E₁) and G₂ = (V₂, E₂), respectively. Then, arithmetic Cartesian product of two SVN-graphs ${\tilde{G}}_{1} = (μ, η)$ and ${\tilde{G}}_{2} = (μ^{'}, η^{'})$ is denoted by ${\tilde{G}}_{1} \hat{\otimes} {\tilde{G}}_{2} = (μ \otimes μ^{'}, η \otimes η^{'})$ and defined as follows:

$(μ_{t} \otimes μ_{t}^{'}) (x_{1}, x_{2}) = μ_{t} (x_{1}) μ_{t}^{'} (x_{2})$ , $(μ_{i} \otimes μ_{i}^{'}) (x_{1}, x_{2}) = μ_{i} (x_{1}) + μ_{i}^{'} (x_{2}) - (μ_{i} (x_{1}) μ_{i}^{'} (x_{2}))$ , $(μ_{f} \otimes μ_{f}^{'}) (x_{1}, x_{2}) = μ_{f} (x_{1}) + μ_{f}^{'} (x_{2}) - (μ_{f} (x_{1}) μ_{f}^{'} (x_{2}))$ , (for all (x₁, x₂) ∈ V₁ × V₂),

$(η_{t} \otimes η_{t}^{'}) ((x, x_{2}) (x, y_{2})) = μ_{t} (x) η_{t}^{'} (x_{2} y_{2})$ , $(η_{i} \otimes η_{i}^{'}) ((x, x_{2}) (x, y_{2})) = μ_{i} (x) + η_{i}^{'} (x_{2} y_{2}) - (μ_{i} (x) η_{i}^{'} (x_{2} y_{2}))$ , $(η_{f} \otimes η_{f}^{'}) ((x, x_{2}) (x, y_{2})) = μ_{f} (x) + η_{f}^{'} (x_{2} y_{2}) - (μ_{f} (x) η_{f}^{'} (x_{2} y_{2}))$ , (for all x ∈ V₁, x₂y₂ ∈ E₂),

$(η_{t} \otimes η_{t}^{'}) ((x_{1}, z) (y_{1}, z)) = η_{t} (x_{1} y_{1}) μ_{t}^{'} (z)$ , $(η_{i} \otimes η_{i}^{'}) ((x_{1}, z) (y_{1}, z)) = η_{i} (x_{1} y_{1}) + μ_{i}^{'} (z) - (η_{i} (x_{1} y_{1}) μ_{i}^{'} (z))$ , $(η_{f} \otimes η_{f}^{'}) ((x_{1}, z) (y_{1}, z)) = η_{f} (x_{1} y_{1}) + μ_{f}^{'} (z) - (η_{f} (x_{1} y_{1}) μ_{f}^{'} (z))$ , (for all z ∈ V₂, x₁y₁ ∈ E₁).

Example 6. Consider two SVN-graphs ${\tilde{G}}_{1} = (μ, η)$ and ${\tilde{G}}_{2} = (μ^{'}, η^{'})$ given as in Example 5 Then, arithmetic Cartesian product of SVN-graphs ${\tilde{G}}_{1}$ and ${\tilde{G}}_{2}$ is as in Fig. 6.

Fig.6

SVN-graph ${\tilde{G}}_{1} \hat{\otimes} {\tilde{G}}_{2}$ .

Proposition 3.17. Let ${\tilde{G}}_{1} = (μ, η)$ and ${\tilde{G}}_{2} = (μ^{'}, η^{'})$ be two SVN-graphs over graphs G₁ = (V₁, E₁) and G₂ = (V₂, E₂), respectively. Then, ${\tilde{G}}_{1} \hat{\otimes} {\tilde{G}}_{2}$ is a SVN-graph.

Proof. In Definition 3.12 Case 1 is clear form definition of Cartesian product of two SVN-sets. Let x ∈ V₁, x₂y₂ ∈ E₂. Then, we have $\begin{matrix} (η_{t} \otimes η_{t}^{'}) ((x, x_{2}) (x, y_{2})) \\ = μ_{t} (x) η_{t}^{'} (x_{2} y_{2}) \\ \leq μ_{t} (x) min (μ_{t}^{'} (x_{2}), μ_{t}^{'} (y_{2})) \\ = min (μ_{t} (x) μ_{t}^{'} (x_{2}), μ_{t} (x) μ_{t}^{'} (y_{2})) \\ = min ((μ_{t} \otimes μ_{t}^{'}) (x, x_{2}), (μ_{t} \otimes μ_{t}^{'}) (x, y_{2})), \\ (η_{i} \otimes η_{i}^{'}) ((x, x_{2}) (x, y_{2})) \\ = μ_{i} (x) + η_{i}^{'} (x_{2} y_{2}) - μ_{i} (x) η_{i}^{'} (x_{2} y_{2}) \\ \geq μ_{i} (x) + max (μ_{i}^{'} (x_{2}), μ_{i}^{'} (y_{2})) \\ - μ_{i} (x) max (μ_{i}^{'} (x_{2}), μ_{i}^{'} (y_{2})) \\ = max (μ_{i} (x) + μ_{i}^{'} (x_{2}), μ_{i} (x) + μ_{i}^{'} (y_{2})) \\ - max (μ_{i} (x) μ_{i}^{'} (x_{2}), μ_{i} (x) μ_{i}^{'} (y_{2})) \\ = max (μ_{i} (x) + μ_{i}^{'} (x_{2}) - μ_{i} (x) μ_{i}^{'} (x_{2}), \\ μ_{i} (x) + μ_{i}^{'} (y_{2}) - μ_{i} (x) μ_{i}^{'} (y_{2})) \\ = max ((μ_{i} \otimes μ_{i}^{'}) (x, x_{2}), (μ_{i} \otimes μ_{i}^{'}) (x, y_{2})), \\ (η_{f} \otimes η_{f}^{'}) ((x, x_{2}) (x, y_{2})) \\ = μ_{f} (x) + η_{f}^{'} (x_{2} y_{2}) - μ_{f} (x) η_{f}^{'} (x_{2} y_{2}) \\ \geq μ_{f} (x) + max (μ_{f}^{'} (x_{2}), μ_{f}^{'} (y_{2})) \\ - μ_{f} (x) max (μ_{f}^{'} (x_{2}), μ_{f}^{'} (y_{2})) \\ = max (μ_{f} (x) + μ_{f}^{'} (x_{2}), μ_{f} (x) + μ_{f}^{'} (y_{2})) \\ - max (μ_{f} (x) μ_{f}^{'} (x_{2}), μ_{f} (x) μ_{f}^{'} (y_{2})) \\ = max (μ_{f} (x) + μ_{f}^{'} (x_{2}) - μ_{f} (x) μ_{f}^{'} (x_{2}), \\ μ_{f} (x) + μ_{f}^{'} (y_{2}) - μ_{f} (x) μ_{f}^{'} (y_{2})) \\ = max ((μ_{f} \otimes μ_{f}^{'}) (x, x_{2}), (μ_{f} \otimes μ_{f}^{'}) (x, y_{2})) . \end{matrix}$

Let z ∈ V₂, x₁y₁ ∈ E₁. Then, $\begin{matrix} (η_{t} \otimes η_{t}^{'}) ((x_{1}, z) (y_{1}, z)) \\ = η_{t} (x_{1} y_{1}) μ_{t}^{'} (z) \\ \leq min (μ_{t} (x_{1}), μ_{t} (y_{1})) μ_{t}^{'} (z) \\ = min (μ_{t} (x_{1}) μ_{t}^{'} (z), μ_{t} (y_{1}) μ_{t}^{'} (z)) \\ = min ((μ_{t} \otimes μ_{t}^{'}) (x_{1}, z), (μ_{t} \otimes μ_{t}^{'}) (y_{1}, z)), \\ (η_{i} \otimes η_{i}^{'}) ((x_{1}, z) (y_{1}, z)) \\ = η_{i} (x_{1} y_{1}) + μ_{i}^{'} (z) - η_{i} (x_{1} y_{1}) μ_{i}^{'} (z) \\ \geq max (μ_{i} (x_{1}), μ_{i} (y_{1})) + μ_{i}^{'} (z) \\ - max (μ_{i} (x_{1}), μ_{i} (y_{1})) μ_{i}^{'} (z) \\ = max (μ_{i} (x_{1}) + μ_{i}^{'} (z), μ_{i} (y_{1}) + μ_{i}^{'} (z)) \\ - max (μ_{i} (x_{1}) μ_{i}^{'} (z), μ_{i} (y_{1}) μ_{i}^{'} (z)) \\ = max (μ_{i} (x_{1}) + μ_{i}^{'} (z) - μ_{i} (x_{1}) μ_{i}^{'} (z), \\ μ_{i} (y_{1}) + μ_{i}^{'} (z) - μ_{i} (y_{1}) μ_{i}^{'} (z)) \\ = max ((μ_{i} \otimes μ_{i}^{'}) (x_{1}, z), (μ_{i} \otimes μ_{i}^{'}) (y_{1}, z)), \\ (η_{f} \otimes η_{f}^{'}) ((x_{1}, z) (y_{1}, z)) \\ = η_{f} (x_{1} y_{1}) + μ_{f}^{'} (z) - η_{f} (x_{1} y_{1}) μ_{f}^{'} (z) \\ \geq max (μ_{f} (x_{1}), μ_{f} (y_{1})) + μ_{f}^{'} (z) \\ - max (μ_{f} (x_{1}), μ_{f} (y_{1})) μ_{f}^{'} (z) \\ = max (μ_{f} (x_{1}) + μ_{f}^{'} (z), μ_{f} (y_{1}) + μ_{f}^{'} (z)) \\ - max (μ_{f} (x_{1}) μ_{f}^{'} (z), μ_{f} (y_{1}) μ_{f}^{'} (z)) \\ = max (μ_{f} (x_{1}) + μ_{f}^{'} (z) - μ_{f} (x_{1}) μ_{f}^{'} (z), \\ μ_{i} (y_{1}) + μ_{f}^{'} (z) - μ_{f} (y_{1}) μ_{f}^{'} (z)) \\ = max ((μ_{f} \otimes μ_{f}^{'}) (x_{1}, z), (μ_{f} \otimes μ_{f}^{'}) (y_{1}, z)), \end{matrix}$

Proposition 3.18. Let ${\tilde{G}}_{1} = (μ, η)$ and ${\tilde{G}}_{2} = (μ^{'}, η^{'})$ be two SVN-graphs over graphs G₁ = (V₁, E₁) and G₂ = (V₂, E₂), respectively. Then, ${\tilde{G}}_{1} \hat{\otimes} {\tilde{G}}_{2}$ is SVN-subgraph of ${\tilde{G}}_{1} \hat{\times} {\tilde{G}}_{2}$ .

Proof. Let x ∈ V₁ and y ∈ V₂, we know that 0 ≤ μ_t (x) ≤1, 0 ≤ μ_i (x) ≤1, 0 ≤ μ_f (x) ≤1 and $0 \leq μ_{t}^{'} (x) \leq 1$ , $0 \leq μ_{i}^{'} (x) \leq 1$ , $0 \leq μ_{f}^{'} (x) \leq 1$ . Let us shown that $μ_{t} (x) μ_{t}^{'} (y) \leq min (μ_{t} (x), μ_{t}^{'} (y)) = \frac{μ_{t} (x) + μ_{t}^{'} (y) - | μ_{t} (x) - μ_{t}^{'} (y) |}{2}$ . ∀x ∈ V₁ and y ∈ V₂ $\begin{matrix} μ_{t} (x) \leq 1 \\ 2 μ_{t} (x) μ_{t}^{'} (y) \leq 2 μ_{t}^{'} (y) \\ - μ_{t}^{'} (y) \leq μ_{t}^{'} (y) - 2 μ_{t} (x) μ_{t}^{'} (y) \\ μ_{t} (x) - μ_{t}^{'} (y) \leq μ_{t} (x) + μ_{t}^{'} (y) - 2 μ_{t} (x) μ_{t}^{'} (y), \end{matrix}$ (3) $\begin{matrix} μ_{t}^{'} (y) \leq 1 \\ 2 μ_{t} (x) μ_{t}^{'} (y) \leq 2 μ_{t} (x) \\ - μ_{t} (x) + 2 μ_{t} (x) μ_{t}^{'} (y) \leq μ_{t} (x) \\ - μ_{t} (x) - μ_{t}^{'} (y) + 2 μ_{t} (x) μ_{t}^{'} (y) \leq μ_{t} (x) - μ_{t}^{'} (y) . \end{matrix}$ (4) from (3) and (4), we have that $| μ_{t} (x) - μ_{t} (y) | \leq μ_{t} (x) + μ_{t}^{'} (y) - 2 μ_{t} (x) μ_{t}^{'} (y)$ . Then $μ_{t} (x) μ_{t}^{'} (y) \leq \frac{μ_{t} (x) + μ_{t}^{'} (y) - | μ_{t} (x) - μ_{t}^{'} (y) |}{2} = min (μ_{t} (x), μ_{t}^{'} (y)) .$ Since $μ_{i} (x), μ_{i}^{'} (y) \in [0, 1]$ , $| μ_{i} (x) - μ_{i}^{'} (y) | \leq μ_{i} (x) + μ_{i}^{'} (y) - 2 μ_{i} (x) μ_{i}^{'} (y), | μ_{i} (x) - μ_{i}^{'} (y) | + μ_{i} (x) + μ_{i}^{'} (y) \leq μ_{i} (x) + μ_{i}^{'} (y) + μ_{i} (x) + μ_{i}^{'} (y) - 2 μ_{i} (x) μ_{i}^{'} (y), 2 μ_{i} (x) + 2 μ_{i}^{'} (y) - 2 μ_{i} (x) μ_{i}^{'} (y) \geq | μ_{i} (x) - μ_{i}^{'} (y) | + μ_{i} (x) + μ_{i}^{'} (y), μ_{i} (x) + μ_{i}^{'} (y) - μ_{i} (x) μ_{i}^{'} (y) \geq \frac{μ_{i} (x) + μ_{i}^{'} (y) + | μ_{i} (x) - μ_{i}^{'} (y) |}{2} = max (μ_{i} (x), μ_{i}^{'} (y)) .$

Similarly it can be shown that $μ_{f} (x) + μ_{f}^{'} (y) - μ_{f} (x) μ_{f}^{'} (y) \geq \frac{μ_{f} (x) + μ_{f}^{'} (y) + | μ_{f} (x) - μ_{f}^{'} (y) |}{2} = max (μ_{f} (x), μ_{f}^{'} (y))$ . Then $μ \otimes μ^{'} ⪯ μ \hat{\times} μ^{'} .$

Let x ∈ V₁, x₂y₂ ∈ E₂. Then, we have $\begin{matrix} (η_{t} \otimes η_{t}^{'}) ((x, x_{2}) (x, y_{2})) \\ = μ_{t} (x), η_{t}^{'} (x_{2} y_{2}) \\ \leq μ_{t} (x) min (μ_{t}^{'} (x_{2}), μ_{t}^{'} (y_{2})) \\ \leq min (μ_{t} (x), min (μ_{t}^{'} (x_{2}), μ_{t}^{'} (y_{2}))) \\ = min (min (μ_{t} (x), μ_{t}^{'} (x_{2})), min (μ_{t} (x), \\ μ_{t}^{'} (y_{2}))) \\ = min ((μ_{t} \times μ_{t}^{'}) (x, x_{2}), (μ_{t} \times μ_{t}^{'}) (x, y_{2})), \\ (η_{i} \otimes η_{i}^{'}) ((x, x_{2}) (x, y_{2})) \\ = μ_{i} (x) + η_{i}^{'} (x_{2} y_{2}) - μ_{i} (x) η_{i}^{'} (x_{2} y_{2}) \\ \geq μ_{i} (x) + max (μ_{i}^{'} (x_{2}), μ_{i}^{'} (y_{2})) \\ - μ_{i} (x) max (μ_{i}^{'} (x_{2}), μ_{i}^{'} (y_{2})) \\ \geq max (μ_{i} (x), max (μ_{i}^{'} (x_{2}), μ_{i}^{'} (y_{2}))) \\ = max (max (μ_{i} (x), μ_{i}^{'} (x_{2})), max (μ_{i} (x), \\ μ_{i}^{'} (y_{2}))) \\ = max ((μ_{i} \times μ_{i}^{'}) (x, x_{2}), (μ_{i} \times μ_{i}^{'}) (x, y_{2})), \\ (η_{f} \otimes η_{f}^{'}) ((x, x_{2}) (x, y_{2})) \\ = μ_{f} (x) + η_{f}^{'} (x_{2} y_{2}) - μ_{f} (x) η_{f}^{'} (x_{2} y_{2}) \\ \geq μ_{f} (x) + max (μ_{f}^{'} (x_{2}), μ_{f}^{'} (y_{2})) \\ - μ_{f} (x) max (μ_{f}^{'} (x_{2}), μ_{f}^{'} (y_{2})) \\ \geq max (μ_{f} (x), max (μ_{f}^{'} (x_{2}), μ_{f}^{'} (y_{2}))) \\ = max (max (μ_{f} (x), μ_{f}^{'} (x_{2})), max (μ_{f} (x), \\ μ_{f}^{'} (y_{2}))) \\ = max ((μ_{f} \times μ_{f}^{'}) (x, x_{2}), (μ_{f} \times μ_{f}^{'}) (x, y_{2})) . \end{matrix}$

Let z ∈ V₂, x₁y₁ ∈ E₁. Similarly, it can be shown that $\begin{matrix} (η_{t} \otimes η_{t}^{'}) ((x_{1}, z) (y_{1}, z)) \leq min ((μ_{t} \times μ_{t}^{'}) (x_{1}, z), \\ (μ_{t} \times μ_{t}^{'}) (y_{1}, z)), \\ (η_{t} \otimes η_{t}^{'}) ((x_{1}, z) (y_{1}, z)) \geq max ((μ_{t} \times μ_{t}^{'}) (x_{1}, z), \\ (μ_{t} \times μ_{t}^{'}) (y_{1}, z)), \\ (η_{t} \otimes η_{t}^{'}) ((x_{1}, z) (y_{1}, z)) \geq max ((μ_{t} \times μ_{t}^{'}) (x_{1}, z), \\ (μ_{t} \times μ_{t}^{'}) (y_{1}, z)) . \end{matrix}$

Thus, ${\tilde{G}}_{1} \hat{\otimes} {\tilde{G}}_{2} ⊑ {\tilde{G}}_{1} \hat{\times} {\tilde{G}}_{2}$ .

Remark 2. Let ${\tilde{G}}_{1}$ and ${\tilde{G}}_{2}$ be two SVN-graphs. Then, there is a homomorphism from ${\tilde{G}}_{1} \hat{\times} {\tilde{G}}_{2}$ to ${\tilde{G}}_{1} \hat{\otimes} {\tilde{G}}_{2}$ .

Definition 3.19. [33] Let ${\tilde{G}}_{1} = (μ, η)$ and ${\tilde{G}}_{2} = (μ^{'}, η^{'})$ be two SVN-graphs over graphs G₁ = (V₁, E₁) and G₂ = (V₂, E₂), respectively. Then, direct product of two SVN-graphs ${\tilde{G}}_{1} = (μ, η)$ and ${\tilde{G}}_{2} = (μ^{'}, η^{'})$ is denoted by ${\tilde{G}}_{1} * {\tilde{G}}_{2} = (μ * μ^{'}, η * η^{'})$ and defined as follows:

$(μ_{t} * μ_{t}^{'}) (x_{1}, x_{2}) = min (μ_{t} (x_{1}), μ_{t}^{'} (x_{2})), (μ_{i} * μ_{i}^{'}) (x_{1}, x_{2}) = max (μ_{i} (x_{1}), μ_{i}^{'} (x_{2})), (μ_{f} * μ_{f}^{'}) (x_{1}, x_{2}) = max (μ_{f} (x_{1}), μ_{f}^{'} (x_{2}))$ , (for all (x₁, x₂) ∈ V₁ × V₂),

$(η_{t} * η_{t}^{'}) ((x_{1}, x_{2}) (y_{1}, y_{2})) = min (η_{t} (x_{1} y_{1}), η_{t}^{'} (x_{2} y_{2})), (η_{i} * η_{i}^{'}) ((x_{1}, x_{2}) (y_{1}, y_{2})) = max (η_{i} (x_{1} y_{1}), η_{i}^{'} (x_{2} y_{2})), (η_{f} * η_{f}^{'}) ((x_{1}, x_{2}) (y_{1}, y_{2})) = max (η_{f} (x_{1} y_{1}), η_{f}^{'} (x_{2} y_{2}))$ , (for all x₁y₁ ∈ E₁ and x₂y₂ ∈ E₂).

Example 7. Consider two SVN-graphs ${\tilde{G}}_{1} = (μ, η)$ and ${\tilde{G}}_{2} = (μ^{'}, η^{'})$ given in Example 5. Then, direct product of SVN-graphs ${\tilde{G}}_{1}$ and ${\tilde{G}}_{2}$ is as in Fig. 7.

Fig.7

SVN-graph ${\tilde{G}}_{1} \hat{*} {\tilde{G}}_{2}$ .

Note that, in this example $(η_{t} * η_{t}^{'}) ((x_{1}, y_{1}) (x_{1}, y_{2}))$ , $(η_{t} * η_{t}^{'}) ((x_{1}, y_{1}) (x_{2}, y_{1}))$ , $(η_{t} * η_{t}^{'}) ((x_{1}, y_{1}) (x_{2}, y_{3}))$ $(η_{t} * η_{t}^{'}) ((x_{2}, y_{1}) (x_{2}, y_{2}))$ , $(η_{t} * η_{t}^{'}) ((x_{2}, y_{2}) (x_{2}, y_{1}))$ , $(η_{t} * η_{t}^{'}) ((x_{3}, y_{2}) (x_{1}, y_{2}))$ and $(η_{t} * η_{t}^{'}) ((x_{3}, y_{2}) (x_{3}, y_{1}))$ are not definable, since x₁x₁, x₂x₂, x₃x₃ ∉ E₁ and y₁y₁, y₂y₂ ∉ E₂.

Proposition 3.20. Let ${\tilde{G}}_{1} = (μ, η)$ , ${\tilde{G}}_{2} = (μ^{'}, η^{'})$ and ${\tilde{G}}_{3} = (μ^{″}, η^{″})$ be three SVN-graphs. Then, $({\tilde{G}}_{1} * {\tilde{G}}_{2}) * {\tilde{G}}_{3} = {\tilde{G}}_{1} * ({\tilde{G}}_{2} * {\tilde{G}}_{3})$ .

Proof. Let (x₁, x₂), (y₁, y₂) ∈ V₁ × V₂ and z₁, z₂ ∈ V₃. Assume that x₁y₁ ∈ E₁, x₂y₂ ∈ E₂ and z₁z₂ ∈ E₃. Then, we have

$\begin{matrix} ((η_{t} * η_{t}^{'}) * η_{t}^{″}) (((x_{1}, x_{2}), z_{1}), ((y_{1}, y_{2}), z_{2})) \\ = min ((η_{t} * η_{t}^{'}) ((x_{1}, x_{2}), (y_{1}, y_{2})), η_{t}^{″} (z_{1} z_{2})) \\ \leq min (min (η_{t} (x_{1} y_{1}), η_{t}^{'} (x_{2} y_{2})), min (μ_{t}^{″} (z_{1}), μ_{t}^{″} (z_{2}))) \\ = min ((min (μ_{t} (x_{1}), μ (y_{1})), min (μ_{t}^{'} (x_{2}), μ_{t}^{'} (y_{2}))), \\ min (μ_{t}^{″} (z_{1}), μ_{t}^{″} (z_{2}))) \\ = min (min (μ_{t} (x_{1}), μ (y_{1})), (min (μ_{t}^{'} (x_{2}), μ_{t}^{'} (y_{2})), \\ min (μ_{t}^{″} (z_{1}), μ_{t}^{″} (z_{2})))) \\ = min (min (μ_{t} (x_{1}), μ (y_{1})), (η_{t}^{'} * η_{t}^{″}) ((x_{2}, z_{1}) (y_{2}, z_{2}))) \\ = (η_{t} * (η_{t}^{'} * η_{t}^{″})) ((x_{1}, (x_{2}, z_{1})) (y_{1}, (y_{2}, z_{2}))), \\ ((η_{i} * η_{i}^{'}) * η_{i}^{″}) (((x_{1}, x_{2}), z_{1}), ((y_{1}, y_{2}), z_{2})) \\ = max ((η_{i} * η_{i}^{'}) ((x_{1}, x_{2}), (y_{1}, y_{2})), η_{i}^{″} (z_{1} z_{2})) \\ \geq max (max (η_{i} (x_{1} y_{1}), η_{i}^{'} (x_{2} y_{2})), max (μ_{i}^{″} (z_{1}), μ_{i}^{″} (z_{2}))) \\ = max ((max (μ_{i} (x_{1}), μ (y_{1})), max (μ_{i}^{'} (x_{2}), μ_{i}^{'} (y_{2}))), \\ max (μ_{i}^{″} (z_{1}), μ_{i}^{″} (z_{2}))) \\ = max (max (μ_{i} (x_{1}), μ (y_{1})), (max (μ_{i}^{'} (x_{2}), μ_{i}^{'} (y_{2})), \\ max (μ_{i}^{″} (z_{1}), μ_{i}^{″} (z_{2})))) \\ = max (max (μ_{i} (x_{1}), μ (y_{1})), (η_{i}^{'} * η_{i}^{″}) ((x_{2}, z_{1}) (y_{2}, z_{2}))) \\ = (η_{i} * (η_{i}^{'} * η_{i}^{″})) ((x_{1}, (x_{2}, z_{1})) (y_{1}, (y_{2}, z_{2}))), \\ ((η_{f} * η_{f}^{'}) * η_{f}^{″}) (((x_{1}, x_{2}), z_{1}), ((y_{1}, y_{2}), z_{2})) \\ = max ((η_{f} * η_{f}^{'}) ((x_{1}, x_{2}), (y_{1}, y_{2})), η_{f}^{″} (z_{1} z_{2})) \\ \geq max (max (η_{f} (x_{1} y_{1}), η_{f}^{'} (x_{2} y_{2})), max (μ_{f}^{″} (y_{1}), μ_{f}^{″} (y_{2}))) \\ = max ((max (μ_{f} (x_{1}), μ (y_{1})), max (μ_{f}^{'} (x_{2}), μ_{f}^{'} (y_{2}))), \\ max (μ_{f}^{″} (y_{1}), μ_{f}^{″} (y_{1}))) \\ = max (max (μ_{f} (x_{1}), μ (y_{1})), (max (μ_{f}^{'} (x_{2}), μ_{f}^{'} (y_{2})), \\ max (μ_{f}^{″} (y_{1}), μ_{f}^{″} (y_{1})))) \\ = max (max (μ_{f} (x_{1}), μ (y_{1})), (η_{f}^{'} * η_{f}^{″}) ((x_{2}, z_{1}) (y_{2}, z_{2}))) \\ = (η_{f} * (η_{f}^{'} * η_{f}^{″})) ((x_{1}, (x_{2}, z_{1})) (y_{1}, (y_{2}, z_{2}))) . \end{matrix}$

Therefore, we concluded that $({\tilde{G}}_{1} * {\tilde{G}}_{2}) * {\tilde{G}}_{3} = {\tilde{G}}_{1} * ({\tilde{G}}_{2} * {\tilde{G}}_{3})$ .

Proposition 3.21. Let ${{\tilde{G}}_{i} = (μ^{i}, η^{i}) : i = 1, . . ., n, n \in N}$ be a family of SVN-graphs. Then, $*_{i = 1}^{n} {\tilde{G}}_{i}$ is a SVN-graph.

Proof. The proof can be made using by induction law.

Definition 3.22. Let ${\tilde{G}}_{1} = (μ, η)$ and ${\tilde{G}}_{2} = (μ^{'}, η^{'})$ be two SVN-graphs over graphs G₁ = (V₁, E₁) and G₂ = (V₂, E₂), respectively. Then, arithmetic direct product of two SVN-graphs ${\tilde{G}}_{1} = (μ, η)$ and ${\tilde{G}}_{2} = (μ^{'}, η^{'})$ is denoted by ${\tilde{G}}_{1} \hat{⊙} {\tilde{G}}_{2} = (μ ⊙ μ^{'}, η ⊙ η^{'})$ and defined as follows:

$(μ_{t} ⊙ μ_{t}^{'}) (x_{1}, x_{2}) = (μ_{t} \otimes μ_{t}^{'}) (x_{1}, x_{2}), (μ_{i} ⊙ μ_{i}^{'}) (x_{1}, x_{2}) = (μ_{i} \otimes μ_{i}^{'}) (x_{1}, x_{2}), (μ_{f} ⊙ μ_{f}^{'}) (x_{1}, x_{2}) = (μ_{f} \otimes μ_{f}^{'}) (x_{1}, x_{2})$ , (for all (x₁, x₂) ∈ V₁ × V₂),

$(η_{t} ⊙ η_{t}^{'}) ((x_{1}, x_{2}) (y_{1}, y_{2})) = (η_{t} \otimes η_{t}^{'}) (x_{1} y_{1}, x_{2} y_{2}), (η_{i} ⊙ η_{i}^{'}) ((x_{1}, x_{2}) (y_{1}, y_{2})) = (η_{i} \otimes η_{i}^{'}) (x_{1} y_{1}, x_{2} y_{2}), (η_{f} ⊙ η_{f}^{'}) ((x_{1}, x_{2}) (y_{1}, y_{2})) = (η_{f} \otimes η_{f}^{'}) (x_{1} y_{1}, x_{2} y_{2})$ , (for all x₁y₁ ∈ E₁ and x₂y₂ ∈ E₂).

Example 8. Consider two SVN-graphs ${\tilde{G}}_{1} = (μ, η)$ and ${\tilde{G}}_{2} = (μ^{'}, η^{'})$ given in Example 5. Then, direct product of SVN-graphs ${\tilde{G}}_{1}$ and ${\tilde{G}}_{2}$ is as in Fig. 8.

Fig.8

SVN-graph ${\tilde{G}}_{1} ⊙ {\tilde{G}}_{2}$ .

Proposition 3.23. Let ${\tilde{G}}_{1} = (μ, η)$ and ${\tilde{G}}_{2} = (μ^{'}, η^{'})$ be two SVN-graphs over graphs G₁ = (V₁, E₁) and G₂ = (V₂, E₂), respectively. Then, ${\tilde{G}}_{1} ⊙ {\tilde{G}}_{2}$ is a SVN-graph.

Proof. The proof can be made in similar way to proof of Proposition 3.17.

Proposition 3.24. Let ${\tilde{G}}_{1} = (μ, η)$ and ${\tilde{G}}_{2} = (μ^{'}, η^{'})$ be two SVN-graphs over graphs G₁ = (V₁, E₁) and G₂ = (V₂, E₂), respectively. Then, ${\tilde{G}}_{1} ⊙ {\tilde{G}}_{2}$ is SVN-subgraph of ${\tilde{G}}_{1} * {\tilde{G}}_{2}$ .

Proof. The proof can be made in similar way to proof of Proposition 3.18.

Remark 3. Let ${\tilde{G}}_{1}$ and ${\tilde{G}}_{2}$ be two SVN-graphs. Then, there is a homomorphism from ${\tilde{G}}_{1} * {\tilde{G}}_{2}$ to ${\tilde{G}}_{1} \hat{⊙} {\tilde{G}}_{2}$ .

Definition 3.25. [33] Let ${\tilde{G}}_{1} = (μ, η)$ and ${\tilde{G}}_{2} = (μ^{'}, η^{'})$ be two SVN-graphs over graphs G₁ = (V₁, E₁) and G₂ = (V₂, E₂), respectively. Then, lexicographical product of two SVN-graphs ${\tilde{G}}_{1} = (μ, η)$ and ${\tilde{G}}_{2} = (μ^{'}, η^{'})$ is denoted by ${\tilde{G}}_{1} \hat{•} {\tilde{G}}_{2} = (μ • μ^{'}, η • η^{'})$ and defined as follows:

$(μ_{t} • μ_{t}^{'}) (x_{1}, x_{2}) = min (μ_{t} (x_{1}), μ_{t}^{'} (x_{2})),$ $(μ_{i} • μ_{i}^{'}) (x_{1}, x_{2}) = max (μ_{i} (x_{1}), μ_{t}^{'} (x_{2})),$ $(μ_{f} • μ_{f}^{'}) (x_{1}, x_{2}) = max (μ_{f} (x_{1}), μ_{t}^{'} (x_{2}))$ (for all (x₁, x₂) ∈ V₁ × V₂),

$(η_{t} • η_{t}^{'}) ((x, x_{2}) (x, y_{2})) = min (μ_{t} (x), η_{t}^{'} (x_{2} y_{2})), (η_{i} • η_{i}^{'}) ((x, x_{2}) (x, y_{2})) = max (μ_{i} (x), η_{i}^{'} (x_{2} y_{2})), (η_{f} • η_{f}^{'}) ((x, x_{2}) (y, y_{2})) = max (μ_{f} (x), η_{f}^{'} (x_{2} y_{2})),$ (for all x₂y₂ ∈ E₂),

$(η_{t} • η_{t}^{'}) ((x_{1}, x_{2}) (y_{1}, y_{2})) = min (η_{t} (x_{1} y_{1}), η_{t}^{'} (x_{2} y_{2})), (η_{i} • η_{i}^{'}) ((x_{1}, x_{2}) (y_{1}, y_{2})) = max (η_{i} (x_{1} y_{1}), η_{i}^{'} (x_{2} y_{2})), (η_{f} • η_{f}^{'}) ((x_{1}, x_{2}) (y_{1}, y_{2})) = max (η_{f} (x_{1} y_{1}), η_{f}^{'} (x_{2} y_{2}))$ , (for all x₁y₁ ∈ E₁ and x₂y₂ ∈ E₂).

Example 9. Consider two SVN-graphs ${\tilde{G}}_{1} = (μ, η)$ and ${\tilde{G}}_{2} = (μ^{'}, η^{'})$ given in Example 5. Then, lexicographic product of SVN-graphs ${\tilde{G}}_{1}$ and ${\tilde{G}}_{2}$ is as in Fig. 9.

Fig.9

SVN-graph ${\tilde{G}}_{1} \hat{•} {\tilde{G}}_{2}$ .

Definition 3.26. Let ${\tilde{G}}_{1} = (μ, η)$ and ${\tilde{G}}_{2} = (μ^{'}, η^{'})$ be two SVN-graphs over graphs G₁ = (V₁, E₁) and G₂ = (V₂, E₂), respectively. Then, arithmetic lexicographical product of two SVN-graphs ${\tilde{G}}_{1} = (μ, η)$ and ${\tilde{G}}_{2} = (μ^{'}, η^{'})$ is denoted by ${\tilde{G}}_{1} \hat{⊚} {\tilde{G}}_{2} = (μ ⊚ μ^{'}, η ⊚ η^{'})$ and defined as follows:

$(μ_{t} ⊚ μ_{t}^{'}) (x_{1}, x_{2}) = (μ_{t} \otimes μ_{t}^{'}) (x_{1}, x_{2}), (μ_{i} ⊚ μ_{i}^{'}) (x_{1}, x_{2}) = (μ_{i} \otimes μ_{i}^{'}) (x_{1}, x_{2}), (μ_{f} ⊚ μ_{f}^{'}) (x_{1}, x_{2}) = (μ_{f} \otimes μ_{f}^{'}) (x_{1}, x_{2})$ (for all (x₁, x₂) ∈ V₁ × V₂),

$(η_{t} ⊚ η_{t}^{'}) ((x, x_{2}) (x, y_{2})) = (μ_{t} \otimes η_{t}^{'}) (x, x_{2} y_{2}), (η_{i} ⊚ η_{i}^{'}) ((x, x_{2}) (x, y_{2})) = (μ_{i} \otimes η_{i}^{'}) (x, x_{2} y_{2}), (η_{f} ⊚ η_{f}^{'}) ((x, x_{2}) (x, y_{2})) = (μ_{f} \otimes η_{f}^{'}) (x, x_{2} y_{2})$ , (for all x₂y₂ ∈ E₂),

$(η_{t} ⊚ η_{t}^{'}) ((x_{1}, x_{2}) (y_{1}, y_{2})) = (η_{t} \otimes η_{t}^{'}) (x_{1} y_{1}, x_{2} y_{2}), (η_{i} ⊚ η_{i}^{'}) ((x_{1}, x_{2}) (y_{1}, y_{2})) = (η_{i} \otimes η_{i}^{'}) (x_{1} y_{1}, x_{2} y_{2}), (η_{f} ⊚ η_{f}^{'}) ((x_{1}, x_{2}) (y_{1}, y_{2})) = (η_{f} \otimes η_{f}^{'}) (x_{1} y_{1}, x_{2} y_{2})$ , (for all x₁y₁ ∈ E₁ and x₂y₂ ∈ E₂).

Example 10. Consider two SVN-graphs ${\tilde{G}}_{1} = (μ, η)$ and ${\tilde{G}}_{2} = (μ^{'}, η^{'})$ given in Example 5. Then, arithmetic lexicographic product of SVN-graphs ${\tilde{G}}_{1}$ and ${\tilde{G}}_{2}$ is as in Fig. 10.

Fig.10

SVN-graph ${\tilde{G}}_{1} \hat{⊚} {\tilde{G}}_{2}$ .

Proposition 3.27. Let ${\tilde{G}}_{1}$ and ${\tilde{G}}_{2}$ be two SVN-graph. Then, ${\tilde{G}}_{1} \hat{⊚} {\tilde{G}}_{2}$ is a SVN-graph.

Proof. The proof is clear from Definition 3.26.

Proposition 3.28. Let ${\tilde{G}}_{1} = (μ, η)$ and ${\tilde{G}}_{2} = (μ^{'}, η^{'})$ be two SVN-graphs over graphs G₁ = (V₁, E₁) and G₂ = (V₂, E₂), respectively. Then, ${\tilde{G}}_{1} \hat{⊚} {\tilde{G}}_{2}$ is SVN-subgraph of ${\tilde{G}}_{1} \hat{•} {\tilde{G}}_{2}$ .

Proof. The proof can be made in similar way to proof of Proposition 3.18.

Remark 4. Let ${\tilde{G}}_{1}$ and ${\tilde{G}}_{2}$ be two SVN-graphs. Then, there is a homomorphism from ${\tilde{G}}_{1} \hat{•} {\tilde{G}}_{2}$ to ${\tilde{G}}_{1} \hat{⊚} {\tilde{G}}_{2}$ .

Definition 3.29. [33] Let ${\tilde{G}}_{1} = (μ, η)$ and ${\tilde{G}}_{2} = (μ^{'}, η^{'})$ be two SVN-graphs over graphs G₁ = (V₁, E₁) and G₂ = (V₂, E₂), respectively. Union of two SVN-graphs ${\tilde{G}}_{1}$ and ${\tilde{G}}_{2}$ , denoted by ${\tilde{G}}_{1} \cup {\tilde{G}}_{2} = (μ \cup μ^{'}, η \cup η^{'})$ , and is defined as follows: $\begin{matrix} (μ \cup μ^{'}) (x) \\ = {\begin{matrix} (μ_{t} (x), μ_{i} (x), μ_{f} (x)), if x \in V_{1} \ V_{2} \\ (μ_{t}^{'} (x), μ_{i}^{'} (x), μ_{f}^{'} (x)), if x \in V_{2} \ V_{1} \\ (max (μ_{t} (x), μ_{t}^{'} (x)), min (μ_{i} (x), μ_{i}^{'} (x)), \\ min (μ_{f} (x), μ_{f}^{'} (x))), if x \in V_{1} \cap V_{2}, \end{matrix} \\ (η \cup η^{'}) (xy) \\ = {\begin{matrix} (η_{t} (xy), η_{i} (xy), η_{f} (xy)), if xy \in E_{1} \ E_{2} \\ (η_{t}^{'} (xy), η_{i}^{'} (xy), η_{f}^{'} (xy)), if xy \in E_{2} \ E_{1} \\ (max (η_{t} (xy), η_{t}^{'} (xy)), min (η_{i} (xy), η_{i}^{'} (xy)), \\ min (η_{f} (xy), η_{f}^{'} (xy))), if xy \in E_{1} \cap E_{2} . \end{matrix} \end{matrix}$

Example 11. Let V₁ = {x₁, x₂, x₃} and V₂ = {x₂, x₃, x₄, x₅} be two vertex sets and μ and μ′ two SVN-set over V₁ and V₂, respectively. Also let η and η′ be two SVN-sets over E₁ = {x₁x₂, x₁x₃, x₂x₃} ⊆ V₁ × V₁ and E₂ = {x₂x₃, x₂x₅, x₃x₄, x₄x₅} ⊆ V₂ × V₂, respectively. Consider SVN-graphs ${\tilde{G}}_{1} = (μ, η)$ and ${\tilde{G}}_{2} = (μ^{'}, η^{'})$ given as in Fig. 11, respectively:

Fig.11

SVN-graphs ${\tilde{G}}_{1}$ and ${\tilde{G}}_{2}$ , respectively.

Then, ${\tilde{G}}_{1} \cup {\tilde{G}}_{2}$ is as in Fig. 12.

Fig.12

Union of SVN-graphs ${\tilde{G}}_{1}$ and ${\tilde{G}}_{2}$ .

Definition 3.30. Let ${\tilde{G}}_{1} = (μ, η)$ and ${\tilde{G}}_{2} = (μ^{'}, η^{'})$ be two SVN-graphs over graphs G₁ = (V₁, E₁) and G₂ = (V₂, E₂), respectively. Then, we denote the intersection of two SVN-graphs ${\tilde{G}}_{1}$ and ${\tilde{G}}_{2}$ of the graphs G₁ and G₂ by ${\tilde{G}}_{1} \cap {\tilde{G}}_{2} = (μ \cap μ^{'}, η \cap η^{'})$ and defined as follows:

$(μ \cap μ^{'}) (x) = (min (μ_{t} (x), μ_{t}^{'} (x)), max (μ_{i} (x), μ_{i}^{'} (x)), max (μ_{f} (x), μ_{f}^{'} (x)))$ , if x ∈ V₁ ∩ V₂,

$(η \cap η^{'}) (xy) = (min (η_{t} (xy), η_{t}^{'} (xy)), max (η_{i} (xy), η_{i}^{'} (xy)), max (η_{f} (xy), η_{f}^{'} (xy)))$ , if (xy) ∈ E₁ ∩ E₂.

Example 12. Let us consider SVN-graphs ${\tilde{G}}_{1}$ and ${\tilde{G}}_{2}$ given in Example 11. Then, intersection of SVN-graphs ${\tilde{G}}_{1}$ and ${\tilde{G}}_{2}$ is as in Fig. 13.

Fig.13

Intersection of SVN-graphs ${\tilde{G}}_{1}$ and ${\tilde{G}}_{2}$ .

Proposition 3.31. Let ${\tilde{G}}_{1} = (μ, η)$ and ${\tilde{G}}_{2} = (μ^{'}, η^{'})$ be two SVN-graphs over graphs G₁ = (V₁, E₁) and G₂ = (V₂, E₂), respectively. Then, ${\tilde{G}}_{1} \cap {\tilde{G}}_{2}$ is a SVN-graph.

Proof. The proof is obvious from Definition.

Proposition 3.32. Let ${{\tilde{G}}_{i} : i \in {1, . . ., n}}$ be a family of SVN-graphs. Then, $⋂_{i = 1}^{n} {\tilde{G}}_{i}$ is a SVN-graph.

Proof. For any x, y ∈ V, we get $\begin{matrix} ⋂_{i = 1}^{n} η_{t} (xy) = inf_{i \in {1, . . ., n}} η_{t} (xy) \\ \leq inf_{i \in {1, . . ., n}} min {{μ_{i}}_{t} (x), {μ_{i}}_{t} (y)} \\ = min {inf_{i \in {1, . . ., n}} {μ_{i}}_{t} (x), inf_{i \in {1, . . ., n}} {μ_{i}}_{t} (y)} \\ = {\cap {μ_{i}}_{t} (x), \cap {μ_{i}}_{t} (y)}, \\ ⋃_{i = 1}^{n} η_{i} (xy) = inf_{i \in {1, . . ., n}} η_{i} (xy) \\ \leq sup_{i \in {1, . . ., n}} max {{μ_{i}}_{i} (x), {μ_{i}}_{i} (y)} \\ = max {sup_{i \in {1, . . ., n}} {μ_{i}}_{i} (x), sup_{i \in {1, . . ., n}} {μ_{i}}_{i} (y)} \\ = {\cup {μ_{i}}_{i} (x), \cup {μ_{i}}_{i} (y)}, \\ ⋃_{i = 1}^{n} η_{f} (xy) = inf_{i \in {1, . . ., n}} η_{f} (xy) \\ \leq sup_{i \in {1, . . ., n}} max {{μ_{i}}_{f} (x), {μ_{i}}_{f} (y)} \\ = max {sup_{i \in {1, . . ., n}} {μ_{i}}_{f} (x), sup_{i \in {1, . . ., n}} {μ_{i}}_{f} (y)} \\ = {\cup {μ_{i}}_{f} (x), \cup {μ_{i}}_{f} (y)} . \end{matrix}$

Thus, $⋂_{i = 1}^{n} {\tilde{G}}_{i}$ is a SVN-graph. Naz et al. [33] defined joint operations in case of V₁∩ V₂ = ∅. Now, we will define two types join operations between SVN-graphs based on union operation on SVN-graphs and addition of SVN-numbers defined by Liu and Wang [28].

Definition 3.33. Let ${\tilde{G}}_{1} = (μ, η)$ and ${\tilde{G}}_{2} = (μ^{'}, η^{'})$ be two SVN-graphs over graphs G₁ = (V₁, E₁) and G₂ = (V₂, E₂), respectively. Then, we denote the join of two SVN-graphs ${\tilde{G}}_{1}$ and ${\tilde{G}}_{2}$ , denoted by ${\tilde{G}}_{1} \hat{+} {\tilde{G}}_{2} = (μ \hat{+} μ^{'}, η \hat{+} η^{'})$ , and defined as follows:

$(μ \hat{+} μ^{'}) (x) = (μ \cup μ^{'}) (x)$ , for all x ∈ V₁ ∪ V₂,

$(η \hat{+} η^{'}) (xy) = (η \cup η^{'}) (xy)$ , for all xy ∈ E₁ ∪ E₂,

$(η \hat{+} η^{'}) (xy) = (min (μ_{t} (x), μ_{t}^{'} (y)), max (μ_{i} (x), μ_{i}^{'} (y)), max (μ_{f} (x), μ_{f}^{'} (y)))$

if xy ∈ E′ \ (E₁ ∪ E₂ ∪ {xy : x = y, x ∈ V₁, y ∈ V₂} ∪ {xy : yx ∈ E₁ ∪ E₂}), where E′ = V₁ × V₂ .

Example 13. Let us consider SVN-graphs ${\tilde{G}}_{1}$ and ${\tilde{G}}_{2}$ given as in Fig. 14. Here, V₁ = {x₁, x₂, x₃}, V₂ = {x₄, x₅, x₆, x₇}, E₁ = {x₁x₂, x₂x₃, x₁x₃}, E₂ = {x₄x₅, x₄x₇, x₅x₆, x₆x₇} and E′ = {x₁x₄, x₁x₅, x₁x₆, x₁x₇, x₂x₄, x₂x₅, x₂x₆, x₂x₇, x₃x₄, x₃x₅, x₃x₆, x₃x₇}. Then, join of two SVN-graphs ${\tilde{G}}_{1}$ and ${\tilde{G}}_{2}$ is as in Fig. 15.

Fig.14

SVN-graphs ${\tilde{G}}_{1}$ and ${\tilde{G}}_{2}$ , respectively.

Fig.15

${\tilde{G}}_{1} \hat{+} {\tilde{G}}_{2}$ .

Let us consider SVN-graphs ${\tilde{G}}_{1}$ and ${\tilde{G}}_{2}$ given in Example 11. Note that V₁∩ V₂ ¬ = ∅. Then ${\tilde{G}}_{1} \hat{+} {\tilde{G}}_{2}$ is in Fig. 16.

Fig.16

${\tilde{G}}_{1} \hat{+} {\tilde{G}}_{2}$ .

Proposition 3.34. Let ${\tilde{G}}_{1} = (μ, η)$ and ${\tilde{G}}_{2} = (μ^{'}, η^{'})$ be two SVN-graphs over graphs G₁ = (V₁, E₁) and G₂ = (V₂, E₂), respectively. Then, ${\tilde{G}}_{1} \hat{+} {\tilde{G}}_{2}$ is a SVN-graph.

Proof. Let xy ∈ E₁ ∪ E₂. By Definition 3.29, we know that if xy ∈ E₁ \ E₂, $(η \hat{+} η^{'}) (xy) = (η \cup η^{'}) (xy) = η (xy)$ and if xy ∈ E₂ \ E₁, then $(η \hat{+} η^{'}) (xy) = (η \cup η^{'}) (xy) = η^{'} (xy)$ . If V₁∩ V₂ = ∅, E₁∩ E₂ = ∅. Therefore the proof is clear.

Let V₁∩ V₂ ¬ = ∅ and xy ∈ E′. Then, $(η \hat{+} η^{'}) (xy) = (min (μ_{t} (x), μ_{t}^{'} (y)), max (μ_{i} (x), μ_{i}^{'} (y)), max (μ_{f} (x), μ_{f}^{'} (y)))$ $\begin{matrix} (η_{t} \hat{+} η_{t}^{'}) (xy) \\ = min (μ_{t} (x), μ_{t}^{'} (y)) \\ \leq min (max (μ_{t} (x), μ_{t} (y)), max (μ_{t}^{'} (x), μ_{t}^{'} (y))) \\ = min (max (μ_{t} (x), μ_{t}^{'} (x)), max (μ_{t} (y), μ_{t}^{'} (y))) \\ = min ((μ_{t} \cup μ_{t}^{'}) (x), (μ_{t} \cup μ_{t}^{'}) (y)), \\ (η_{i} \hat{+} η_{i}^{'}) (xy) \\ = max (μ_{i} (x), μ_{i}^{'} (y)) \\ \geq max (min (μ_{i} (x), μ_{i} (y)), min (μ_{i}^{'} (x), μ_{i}^{'} (y))) \\ = max (min (μ_{i} (x), μ_{i}^{'} (x)), min (μ_{i} (y), μ_{i}^{'} (y))) \\ = max ((μ_{i} \cap μ_{i}^{'}) (x), (μ_{i} \cap μ_{i}^{'}) (y)), \\ (η_{f} \hat{+} η_{f}^{'}) (xy) \\ = max (μ_{f} (x), μ_{f}^{'} (y)) \\ \geq max (min (μ_{f} (x), μ_{f} (y)), min (μ_{f}^{'} (x), μ_{f}^{'} (y))) \\ = max (min (μ_{f} (x), μ_{f}^{'} (x)), min (μ_{f} (y), μ_{f}^{'} (y))) \\ = max ((μ_{f} \cap μ_{f}^{'}) (x), (μ_{f} \cap μ_{f}^{'}) (y)) . \end{matrix}$

This completes the proof.

Definition 3.35. Let ${\tilde{G}}_{1} = (μ, η)$ and ${\tilde{G}}_{2} = (μ^{'}, η^{'})$ be two SVN-graphs over graphs G₁ = (V₁, E₁) and G₂ = (V₂, E₂), respectively. Then, direct join of ${\tilde{G}}_{1}$ and ${\tilde{G}}_{2}$ , denoted by ${\tilde{G}}_{1} \hat{\oplus} {\tilde{G}}_{2} = (η \hat{\oplus} η^{'}, μ \oplus μ^{'})$ , is defined by as follows:

$(μ \hat{\oplus} μ^{'}) (x) = {\begin{matrix} μ (x), if x \in V_{1} \ V_{2} \\ μ^{'} (x), if x \in V_{2} \ V_{1} \\ (μ (x) \oplus μ^{'} (x)), if x \in V_{1} \cap V_{2}, \end{matrix}$

$(η \hat{\oplus} η^{'}) (xy) = {\begin{matrix} η (xy), if xy \in E_{1} \ E_{2} \\ η^{'} (xy), if xy \in E_{2} \ E_{1} \\ (η (xy) \oplus η^{'} (xy)), if xy \in E_{1} \cap E_{2} . \end{matrix}$

$(η \hat{\oplus} η^{'}) (xy) = ((η_{t} \otimes η_{t}^{'}) (xy), (η_{i} \oplus η_{i}^{'}) (xy), (η_{f} \oplus η_{f}^{'}) (xy))$ for xy ∈ E′ \ (E₁ ∪ E₂ ∪ {xy : x = y, x ∈ V₁, y ∈ V₂} ∪ {xy : yx ∈ E₁ ∪ E₂}), where E′ = V₁ × V₂ .

Here $(η_{t} \otimes η_{t}^{'}) (xy) = μ_{t} (x) μ_{t}^{'} (y)$ , $(η_{i} \oplus η_{i}^{'}) (xy) = μ_{i} (x) + μ_{i}^{'} (y) - μ_{i} (x) μ_{i}^{'} (y)$ and $(η_{f} \oplus η_{f}^{'}) (xy) = μ_{f} (x) + μ_{f}^{'} (y) - μ_{f} (x) μ_{f}^{'} (y)$ .

Example 14. Consider SVN-graphs ${\tilde{G}}_{1}$ and ${\tilde{G}}_{2}$ given in Example 11. Then, the direct join of SVN-graphs ${\tilde{G}}_{1}$ and ${\tilde{G}}_{2}$ is as in Fig. 17.

Fig.17

${\tilde{G}}_{1} \hat{\oplus} {\tilde{G}}_{2}$ .

Proposition 3.36. Let ${\tilde{G}}_{1} = (μ, η)$ and ${\tilde{G}}_{2} = (μ^{'}, η^{'})$ be two SVN-graphs over graphs G₁ = (V₁, E₁) and G₂ = (V₂, E₂), respectively. Then, ${\tilde{G}}_{1} \hat{\oplus} {\tilde{G}}_{2}$ is a SVN-graph.

Proof. Let xy ∈ E₁ ∪ E₂. From Definition 3.33, we know that if xy ∈ E₁ \ E₂, $(η \hat{\oplus} η^{'}) (xy) = (η \oplus η^{'}) (xy) = η (xy)$ and if if xy ∈ E₂ \ E₁, $(η \hat{\oplus} η^{'}) (xy) = (η \oplus η^{'}) (xy) = η^{'} (xy)$ . Let xy ∈ E₁ ∩ E₂, then $\begin{matrix} (η_{t} \hat{\oplus} η_{t}^{'}) (xy) \\ = η_{t} (xy) + η_{t}^{'} (xy) - η_{t} (xy) η_{t}^{'} (xy)) \\ \leq min (μ_{t} (x), μ_{t} (y)) + min (μ_{t}^{'} (x), μ_{t}^{'} (y)) \\ - min (μ_{t} (x), μ_{t} (y)) min (μ_{t}^{'} (x), μ_{t}^{'} (y)) \\ \leq min ((μ_{t} (x) + μ_{t}^{'} (x) - μ_{t} (x) μ_{t}^{'} (x)), \\ (μ_{t} (y) + μ_{t}^{'} (y) - μ_{t} (y) μ_{t}^{'} (y))) \\ = min ((μ_{t} \oplus μ_{t}^{'}) (x), (μ_{t} \oplus μ_{t}^{'}) (y)), \\ (η_{i} \hat{\oplus} η_{i}^{'}) (xy) \\ = η_{i} (xy) η_{i}^{'} (xy) \\ \geq max (μ_{i} (x), μ_{i} (y)) max (μ_{i}^{'} (x), μ_{i}^{'} (y)) \\ \geq max ((μ_{i} (x) μ_{i}^{'} (x)), (μ_{i} (y) μ_{i}^{'} (y))) \\ = max ((μ_{i} \oplus μ_{i}^{'}) (x), (μ_{i} \oplus μ_{i}^{'}) (y)), \\ (η_{f} \hat{\oplus} η_{f}^{'}) (xy) \\ = η_{f} (xy) η_{f}^{'} (xy) \\ \geq max (μ_{f} (x), μ_{f} (y)) max (μ_{f}^{'} (x), μ_{f}^{'} (y)) \\ \geq max ((μ_{f} (x) μ_{f}^{'} (x)), (μ_{f} (y) μ_{f}^{'} (y))) \\ = max ((μ_{f} \oplus μ_{f}^{'}) (x), (μ_{f} \oplus μ_{f}^{'}) (y)) . \end{matrix}$

Let xy ∈ E′. Then, $\begin{matrix} (η \hat{\oplus} η^{'}) (xy) \\ = ((η_{t} \otimes η_{t}^{'}) (xy), (η_{i} \oplus η_{i}^{'}) (xy), \\ (η_{f} \oplus η_{f}^{'}) (xy))), \\ (η_{t} \otimes η_{t}^{'}) (xy) \\ = μ_{t} (x) μ_{t}^{'} (y) \leq min (μ_{t} (x), μ_{t}^{'} (y)), \\ (η_{i} \oplus η_{i}^{'}) (xy) \\ = μ_{i} (x) + μ_{i}^{'} (y) - μ_{i} (x) μ_{i}^{'} (y) \\ \geq max (μ_{i} (x), μ_{i}^{'} (y)), \\ (η_{f} \oplus η_{f}^{'}) (xy) \\ = μ_{f} (x) + μ_{f}^{'} (y) - μ_{f} (x) μ_{f}^{'} (y) \\ \geq max (μ_{f} (x), μ_{f}^{'} (y)) . \end{matrix}$

Then, $(η \hat{\oplus} η^{'}) (xy) ⪯ ((μ \otimes μ^{'}) (x) ⋏ (μ \otimes μ^{'}) (y))$ and this completes the proof.

4 Decision making method

In this section, we firstly present some definitions which we will use in suggested method. Then we develop a decision making method and give its algorithm in order to choose element which has dominate degree in group, and to show process of proposed decision making method we give an illustrative example.

Definition 4.1. [35] Let $μ_{j} = 〈 μ_{j}^{t}, μ_{j}^{i}, μ_{j}^{f} 〉$ (j = 1, 2, . . . , n) be a collection of single-valued neutrosophic numbers, SNNWA : SNNⁿ → SNN

${SNNWA}_{w} (μ_{1}, μ_{2}, . . ., μ_{n}) = \sum_{j = 1}^{n} w_{j} μ_{j},$ (5) the SNNWA operator is called the simplified neutrosophic number weighted averaging operator of dimension n, where w = (w₁, w₂, . . . , w_n) is the weight vector of μ_j (j = 1, 2, . . . , n), with w_j ≥ 0 (j = 1, 2, . . . , n) and $\sum_{j = 1}^{n} w_{j} = 1$ (see [35]). Here if it is taken all of w_j (j = 1, 2, . . . , n) as equal, then Equation (5) can be written as follows:

$SNNWA (μ_{1}, μ_{2}, . . ., μ_{n}) = {oplus}_{j = 1}^{n} μ_{j} .$ (6)

Equation (6) can be written as more explicit, respectively, as follows: ${oplus}_{j = 1}^{n} μ_{j} = 〈 1 - \prod_{j = 1}^{n} (1 - μ_{j}^{t}), \prod_{j = 1}^{n} μ_{j}^{i}, \prod_{j = 1}^{n} μ_{j}^{f} 〉$

Definition 4.2. [35] Let μ = 〈μ_t, μ_i, μ_f〉 be a SVN-number. Then, the score function of SVN-number μ, denoted by s (μ), is defined as follows: $s (μ) = \frac{μ_{t} + 1 - μ_{i} + 1 - μ_{f}}{3}$

Based on Definitions 4.1 and 4.2, we define following concepts:

Definition 4.3. Let ${\tilde{G}}_{1} = (μ, η)$ be a directed SVN-graphs over graphs G = (V, E) and v_r, r = 1, 2, . . . , n be adjacent SVN-vertices of v_p ∈ V. Out-degree of v_p, denoted by $O (v_{p})$ , is defined as follows:

$\begin{matrix} O (v_{p}) & = & 〈 1 - \prod_{r = 1}^{n} (1 - η_{t} (v_{p} v_{r})), \\ \prod_{r = 1}^{n} η_{i} (v_{p} v_{r}), \prod_{r = 1}^{n} η_{f} (v_{p} v_{r}) 〉 \end{matrix}$ (7)

Similarly, in-degree of v_p, denoted by $I (v_{p})$ is defined as follows:

$\begin{matrix} I (v_{p}) & = & 〈 1 - \prod_{r = 1}^{n} (1 - η_{t} (v_{r} v_{p})), \\ \prod_{r = 1}^{n} η_{i} (v_{r} v_{p}), \prod_{r = 1}^{n} η_{f} (v_{r} v_{p}) 〉 \end{matrix}$ (8)

Score of out-degree and in-degree of a vertex v_p is calculated by using Definitions 4.1 and 4.2, respectively as follows: $\begin{matrix} s (O (v_{p})) = 1 \\ - \frac{\prod_{r = 1}^{n} (1 - η_{t} (v_{p} v_{r})) + \prod_{r = 1}^{n} η_{i} (v_{p} v_{r}) + \prod_{r = 1}^{n} η_{f} (v_{p} v_{r})}{3} \end{matrix}$ (9) $\begin{matrix} s (I (v_{p})) = 1 \\ - \frac{\prod_{r = 1}^{n} (1 - η_{t} (v_{r} v_{p})) + \prod_{r = 1}^{n} η_{i} (v_{r} v_{p}) + \prod_{r = 1}^{n} η_{f} (v_{r} v_{p})}{3} \end{matrix}$ (10)

Let $s (O (v_{p}))$ and $s (I (v_{p}))$ be scores of out-degree and in-degree of a vertex v_p, respectively. Then $s (O (v_{p})) - s (I (v_{p}))$ is called domination degree of vertex v_p and denoted by $DD (v_{p})$ .

Algorithm

Step 1: Input the set of vertices I = {I₁, I₂, . . . , I_n} and a SVN-set A which is defined on set I.

Step 2: Input the set of edges J = {J₁, J₂, . . . , J_n} .

Step 3: Compute the truth-membership degree, indeterminacy-membership degree and falsity-membership degree of edge using η_t (xy) ≤ min(μ_t (x), μ_t (y)), η_i (xy) ≥ max(μ_i (x), μ_i (y)) and η_f (xy) ≥ max(μ_f (x), μ_f (y)).

Step 4: Compute out-degree and in-degree for each of vertices by using Equations (7) and (8).

Step 5: Compute score functions of out-degree and in-degree for each of vertices by using Equations (9) and (10).

Step 6: Find domination degree for each of vertices.

Step 7: Rank vertices according to real domination degrees of them.

Step 8: Choose vertex which has maximum domination degrees.

4.1 Illustrative example

Let us consider the application in [8]. There are seven person which are influence each other on a social group on whatsapp. Let P = {P₁, P₂, P₃, P₄, P₅, P₆, P₇} be set of seven person in a social group on whatsapp.

Step 1: A = {(P₁, 〈0.7, 0.4, 0.5〉), P₂ (〈0.5, 0.5, 0.7〉), (P₃, 〈0.6, 0.5, 0.7〉), (P₄, 〈0.3, 0.8, 0.4〉), (P₅, 〈0.8, 0.4, 0.3〉), (P₆, 〈0.9, 0.3, 0.3〉), (P₇, 〈0.4, 0.2, 0.6〉)} is the SVN-set on the set I, where truth value of person represents his good influence on others, falsity value represents his bad influence on others, and indeterminacy value represents uncertainty in his influence.

Step 2: J = {(P₆, P₃), (P₆, P₄), (P₆, P₇), (P₆, P₂), (P₁, P₃), (P₂, P₃), (P₇, P₁), (P₅, P₄), (P₅, P₂), (P₂, P₇)} is the set of edges.

Step 3: Let B be SVN-set on the set J as shown in Table 4.

Table 4
SVN-set B of edges

Edge t i f

(P₁, P₃) 0.6 0.5 0.7

(P₁, P₆) 0.6 0.5 0.5

(P₁, P₇) 0.3 0.6 0.8

(P₂, P₁) 0.4 0.6 0.7

(P₂, P₃) 0.4 0.6 0.8

(P₃, P₇) 0.3 0.6 0.8

(P₃, P₄) 0.2 0.8 0.8

(P₄, P₇) 0.3 0.9 0.7

(P₅, P₄) 0.2 0.8 0.5

(P₅, P₂) 0.5 0.9 0.8

(P₆, P₃) 0.5 0.6 0.8

(P₆, P₄) 0.2 0.8 0.4

(P₆, P₇) 0.4 0.4 0.7

(P₆, P₂) 0.4 0.5 0.7

(P₇, P₅) 0.4 0.5 0.7

Edge	t	i	f
(P₁, P₃)	0.6	0.5	0.7
(P₁, P₆)	0.6	0.5	0.5
(P₁, P₇)	0.3	0.6	0.8
(P₂, P₁)	0.4	0.6	0.7
(P₂, P₃)	0.4	0.6	0.8
(P₃, P₇)	0.3	0.6	0.8
(P₃, P₄)	0.2	0.8	0.8
(P₄, P₇)	0.3	0.9	0.7

The truth, indeterminacy and falsity values of each edge are calculated using:

η_t (xy) ≤ min(μ_t (x), μ_t (y)), η_i (xy) ≥ max(μ_i (x), μ_i (y)) and η_f (xy) ≥ max(μ_f (x), μ_f (y)).

The SVN-diagraph $\tilde{G} = (μ, η)$ is shown in Fig. 18.

Fig.18

Single-valued neutrosophic diagraph.

Step 4: By using Equations (7) and (8), out-degrees and in-degrees of each person are obtained as follows: $\begin{matrix} O (P_{1})) = 〈 0.888, 0.150, 0.280 〉, \\ I (P_{1}) = 〈 0.400, 0.600, 0.700 〉 \\ O (P_{2})) = 〈 0.640, 0.360, 0.560 〉, \\ I (P_{2}) = 〈 0.700, 0.450, 0.560 〉 \\ O (P_{3})) = 〈 0.940, 0.480, 0.560 〉, \\ I (P_{3}) = 〈 0.880, 0.180, 0.448 〉 \\ O (P_{4})) = 〈 0.300, 0.900, 0.700 〉, \\ I (P_{4}) = 〈 0.488, 0.512, 0.140 〉 \\ O (P_{5})) = 〈 0.600, 0.720, 0.400 〉, \\ I (P_{5}) = 〈 0.400, 0.500, 0.700 〉 \\ O (P_{6})) = 〈 0.856, 0.096, 0.157 〉, \\ I (P_{6}) = 〈 0.600, 0.500, 0.500 〉 \\ O (P_{7})) = 〈 0.400, 0.500, 0.700 〉, \\ I (P_{7}) = 〈 0, 794, 0.130, 0.314 〉 . \end{matrix}$

Step 5: By using Equations (9) and (10), we obtain scores of out-degrees and in-degrees of each person are obtained as follows: $\begin{matrix} s (O (P_{1})) = 0.819, s (I (P_{1})) = 0.367 \\ s (O (P_{2})) = 0.573, s (I (P_{2})) = 0.563 \\ s (O (P_{3})) = 0.633, s (I (P_{3})) = 0.751 \\ s (O (P_{4})) = 0.233, s (I (P_{4})) = 0.612 \\ s (O (P_{5})) = 0.493, s (I (P_{5})) = 0.400 \\ s (O (P_{6})) = 0.868, s (I (P_{6})) = 0.533 \\ s (O (P_{7})) = 0.400, s (I (P_{7})) = 0.784 . \end{matrix}$

Step 6: Real domination degrees of each person are obtained as follows: $\begin{matrix} DD (P_{1}) = 0.819 - 0.367 = 0.452 \\ DD (P_{2}) = 0.573 - 0.563 = 0.010 \\ DD (P_{3}) = 0.633 - 0.751 = - 0.118 \\ DD (P_{4}) = 0.233 - 0.612 = - 0.379 \\ DD (P_{5}) = 0.493 - 0.400 = 0.041 \\ DD (P_{6}) = 0.868 - 0.533 = 0.335 \\ DD (P_{7}) = 0.400 - 0.784 = - 0.384 . \end{matrix}$

Step 7: Real domination degrees of peoples in this group can be rank in form P₁ ≥ P₆ ≥ P₅ ≥ P₂ ≥ P₃ ≥ P₄ ≥ P₅ .

Step 8: P₁ is a person who has real dominate effect in the group.

5 Conclusion

The paper is devoted to present some novel concepts and operations related to single-valued neutrosophic graphs, and application in decision making. Also some properties of the defined novel concepts and operations are obtained. In the future, we will give applications of single-valued neutrosophic graphs in decision making and clustering analysis based on defined new concept and operations. Furthermore, many concepts existing in graph theory may be defined and investigated by researchers on single-valued neutrosophic environment, rough set [37] soft set [30], hybrid soft set [29], soft rough set [54, 55], soft rough fuzzy [56].

Conflict of interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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(P₅, P₄)	0.2	0.8	0.5
(P₅, P₂)	0.5	0.9	0.8
(P₆, P₃)	0.5	0.6	0.8
(P₆, P₄)	0.2	0.8	0.4
(P₆, P₇)	0.4	0.4	0.7
(P₆, P₂)	0.4	0.5	0.7
(P₇, P₅)	0.4	0.5	0.7

Properties of single-valued neutrosophic graphs

Abstract

Keywords

1 Introduction

2 Preliminaries

2.1 Single-valued neutrosophic sets

2.2 Single-valued neutrosophic relations

3 Single-valued neutrosophic graphs

Table 1 SVN-graph G ˜ x 1 x 2 x 3 x 4 μ t 0.7 0.5 0.6 0.8 μ i 0.5 0.4 0.2 0.7 μ f 0.6 0.2 0.5 0.4 x 1 x 2 x 2 x 3 x 3 x 1 x 2 x 4 η t 0.3 0.4 0.6 0.2 η i 0.6 0.5 0.7 0.8 η f 0.9 1.0 0.9 0.6

Algorithm

Conflict of interests

References

Table 1
SVN-graph $\tilde{G}$

x ₁ x ₂ x ₃ x ₄

μ _t 0.7 0.5 0.6 0.8

μ _i 0.5 0.4 0.2 0.7

μ _f 0.6 0.2 0.5 0.4

x ₁ x ₂ x ₂ x ₃ x ₃ x ₁ x ₂ x ₄

η _t 0.3 0.4 0.6 0.2

η _i 0.6 0.5 0.7 0.8

η _f 0.9 1.0 0.9 0.6