Regularity of graphs in single valued neutrosophic environment

Abstract

In this paper, regular graphs are studied in the context of a single valued neutrosophic environment, where for each element the truth-membership degree, indeterminacy-membership degree and falsity-membership degree are independently assigned in [0, 1]. Firstly, the novel concepts of regular, edge regular, partially edge regular and full edge regular single valued neutrosophic graphs (SVNGs) are proposed and some of their properties are investigated. Then strongly regular SVNGs and biregular SVNGs are defined. The notion of single valued neutrosophic digraphs (SVNDGs) is introduced along with its application in multi-attribute decision making (MADM).

Keywords

Single valued neutrosophic graph edge regular single valued neutrosophic graph strongly regular single valued neutrosophic graph biregular single valued neutrosophic graph

1 Introduction

Atanassov [3] introduced the concept of intuitionistic fuzzy set (IFS), as a generalization of Zadeh’s fuzzy set [30]. Its characteristic is that a membership degree and a non-membership degree are assigned to each element in the set. Since its inception, it has been investigated by many researchers and applied in several areas, such as pattern recognition [23], decision making [13, 31], medical diagnosis [11], cluster analysis [25] and market prediction [16]. Later on, Smarandache [21] proposed neutrosophic set theory from philosophical point of view. Its prominent characteristic is that a truth-membership degree, an indeterminacy-membership degree and a falsity-membership degree, in non-standard unit interval ] 0^-, 1⁺ [, are independently assigned to each element in the set. Gradually, it has been discovered that without a specific description, neutrosophic sets are difficult to apply in the real applications. After analyzing this difficulty, Wang et al. [24] initiated the concept of a single valued neutrosophic set (SVNS) from scientific or engineering point of view, as an instance of the neutrosophic set and an extension of IFS, and provided its various properties. SVNSs represent uncertainty, incomplete, imprecise, indeterminate and inconsistent information which exist in real world. Neutrosophic set, particularly SVNS has attracted significant interest from researchers in recent years. It has been widely applied in various fields, such as information fusion in which data are combined from different sensors [6], control theory [1], image processing [12], medical diagnosis [29], decision making [28], and graph theory [8, 22], etc.

Graph representations are widely used for dealing with structural information, in different domains such as operations research, networks, systems analysis, pattern recognition, economics and image interpretation. Fuzzy graphs are designed to represent structures of relationships between objects such that the existence of a concrete object (vertex) and relationship between two objects (edge) are matters of degree. Obtaining analogs of several basic graph theoretical concepts, Rosenfeld [20] considered fuzzy relations on fuzzy sets and developed the structure of fuzzy graphs, using max and min operations. Some remarks on fuzzy graphs were given by Bhattacharya [5]. Edge regular fuzzy graphs were studied in [18] by Radha and Kumaravel. Intuitionistic fuzzy graphs with vertex sets and edge sets as IFS were first introduced by Atanassov [4] and further discussed by Akram [2]. Karunambigai et al. [15] defined the concept of edge regular intuitionistic fuzzy graphs. Recently, Borzooei et al. [7] generalized the concept of regularity of fuzzy graph to the regularity of vague graph. Mishra and Pal [17] initiated the notion of interval-valued intuitionistic fuzzy graphs. When description of the objects or their relations or both is indeterminate and inconsistent, it cannot be handled by fuzzy, intuitionistic fuzzy, vague and interval valued intuitionistic fuzzy graphs. Therefore some new theories are required. That is why, Kandasamy et al. [22] put forward the concept of neutrosophic graphs. Furthermore, Broumi et al. [8 –10] discussed several properties of SVNGs and their extensions. However, to the best of our knowledge, no work addressing the regular graphs in single valued neutrosophic setting is in literature. Therefore, the main purpose of this paper is to introduce single valued neutrosophic regular graphs and apply the concept of SVNDGs in MADM.

This paper is structured as follows: Section 2 contains a brief background about SVNSs and SVNGs. Section 3 introduces the concepts of regular, edge regular, partially edge regular, full edge regular, strongly regular and biregular SVNGs, and investigates their properties. Section 4 is devoted to the application of SVNDGs in MADM and finally we draw conclusions in Section 5.

Throughout this paper, V represents a crisp universe of generic elements, G stands for the crisp graph and $G$ is the SVNG.

2 Preliminaries

In the following, some basic concepts on SVNSs and SVNGs are reviewed to facilitate next sections.

Definition 2.1. [20, 30] Let V be a space of points (objects) with a generic element in V denoted by x. A fuzzy set X in V is defined as $X = {(x, ξ_{X} (x)) | x \in V},$ which is characterized by a membership function ξ_X : V → [0, 1] or simply ξ : V → [0, 1]. A fuzzy relation on a set V is a mapping μ : V × V → [0, 1] such that μ (x, y) ≤ ξ (x) ∧ ξ (y) for all x, y ∈ V. A fuzzy relation μ is symmetric if μ (x, y) = μ (y, x) for all x, y ∈ V. A fuzzy graph $G = (V, ξ, μ)$ is a non-empty set V together with a pair of functions ξ : V → [0, 1] and μ : V × V → [0, 1] such that μ (xy) ≤ min {ξ (x) , ξ (y)} for all x, y ∈ V. Here μ is a symmetric fuzzy relation on ξ.

Definition 2.2. [26] A fuzzy digraph $D = (V, ξ, \vec{μ})$ is a non-empty set V together with a pair of functions ξ : V → [0, 1] and $\vec{μ} : V \times V \to [0, 1]$ such that $\vec{μ} (xy) \leq min {ξ (x), ξ (y)}$ for all x, y ∈ V. Here $\vec{μ} (xy)$ denotes the membership value of the directed edge $\vec{xy}$ .

Definition 2.3. [3] An IFS X in V is an object having the form $X = {〈 x, μ_{X} (x), ν_{X} (x) 〉 | x \in V},$ where the functions μ_X : V → [0, 1] , x ∈ V, μ_X (x) ∈ [0, 1] and ν_X : V → [0, 1] , x ∈ V, ν_X (x) ∈ [0, 1] satisfy the condition 0 ≤ μ_X (x) + ν_X (x) ≤1 for all x ∈ V. μ_X (x) and ν_X (x) represent the degree of membership and degree of non-membership of the element x ∈ V to set X, respectively.

For each IFS X in V, π_X (x) =1 - μ_X (x) - ν_X (x) is called the intuitionistic fuzzy index of x ∈ X, representing the degree of hesitation of x to X.

Definition 2.4. [21] Let V be a space of points (objects), with a generic element in V denoted by x. A neutrosophic set X in V is characterized by a truth-membership function T_X, an indeterminacy-membership function I_X and a falsity-membership function F_X. T_X (x) , I_X (x) and F_X (x) are real standard or non-standard subsets of ] 0^-, 1⁺ [. That is, T_X : V →]0^-, 1⁺ [, I_X : V →]0^-, 1⁺ [ and F_X : V →]0^-, 1⁺ [.

There is no restriction on the sum of T_X (x) , I_X (x) and F_X (x), therefore 0^- ≤ sup T_X (x) + sup I_X (x) + sup F_X (x) ≤3⁺ .

Definition 2.5. [24] Let V be a space of points (objects), with a generic element in V denoted by x. A SVNS X in V is characterized by a truth-membership function T_X, an indeterminacy-membership function I_X and a falsity-membership function F_X. For each point x ∈ X, T_X (x) , I_X (x) , F_X (x) ∈ [0, 1]. Therefore, a SVNS X in V can be written as $X = {〈 x, T_{X} (x), I_{X} (x), F_{X} (x) 〉 | x \in V},$

Definition 2.6. [24] Let X, X₁ and X₂ be three SVNSs. Then their relations and operations are defined as follows:

X₁ ⊆ X₂ if and only if T_X
₁ (x) ≤ T_X
₂ (x) , I_X
₁ (x) ≥ I_X
₂ (x) and F_X
₁ (x) ≥ F_X
₂ (x) for all x ∈ V;

X₁ = X₂ if and only if X₁ ⊆ X₂ and X₂ ⊆ X₁;

$\bar{X} = {〈 x, F_{X} (x), 1 - I_{X} (x), T_{X} (x) 〉 | x \in V}$ ;

X₁ ∪ X₂ = {〈x, max {T_X
₁ (x) , T_X
₂ (x)} , min {I_X
₁ (x) , I_X
₂ (x)} , min {F_X
₁ (x) , F_X
₂ (x)} 〉 |x ∈ V};

X₁ ∩ X₂ = {〈x, min {T_X
₁ (x) , T_X
₂ (x)} , max {I_X
₁ (x) , I_X
₂ (x)} , max {F_X
₁ (x) , F_X
₂ (x)} 〉 | x ∈ V}.

Definition 2.7. [27] A SVNS Y in V × V is said to be a single valued neutrosophic relation in V, denoted by $Y = {〈 xy, T_{Y} (xy), I_{Y} (xy), F_{Y} (xy) 〉 | xy \in V \times V}$ where T_Y : V × V → [0, 1], I_Y : V × V → [0, 1] and F_Y : V × V → [0, 1] represent the truth-membership, indeterminacy-membership and falsity-membership function of Y, respectively.

Definition 2.8. [8] A SVNG of a (crisp) graph G = (V, E) is defined to be a pair $G = (X, Y)$ , where

the functions T_X : V → [0, 1], I_X : V → [0, 1] and F_X : V → [0, 1] denote the degree of truth-membership, indeterminacy-membership and falsity-membership of the element x ∈ V, respectively. There is no restriction on the sum of T_X (x) , I_X (x) and F_X (x), therefore 0 ≤ T_X (x) + I_X (x) + F_X (x) ≤3 for all x ∈ V,

the functions T_Y : E ⊆ V × V → [0, 1] , I_Y : E ⊆ V × V → [0, 1] and F_Y : E ⊆ V × V → [0, 1] are defined by $\begin{matrix} T_{Y} (xy) \leq min {T_{X} (x), T_{X} (y)}, \\ I_{Y} (xy) \geq max {I_{X} (x), I_{X} (y)} and \\ F_{Y} (xy) \geq max {F_{X} (x), F_{X} (y)} . \end{matrix}$ There is no restriction on the sum of T_Y (xy) , I_Y (xy) and F_Y (xy), therefore 0 ≤ T_Y (xy) + I_Y (xy) + F_Y (xy) ≤3 for all xy ∈ E .

We call X the single valued neutrosophic vertex set of $G$ and Y the single valued neutrosophic edge set of $G$ .

Definition 2.9. [8] The degree of a vertex u_i ∈ V in a SVNG $G$ is defined as $d_{G} (u_{i}) = 〈 d_{T} (u_{i}), d_{I} (u_{i}), d_{F} (u_{i}) 〉,$ where $\begin{matrix} d_{T} (u_{i}) & = & \sum_{u_{i}, u_{j} \neq u_{i} \in V} T_{Y} (u_{i} u_{j}), d_{I} (u_{i}) \\ = & \sum_{u_{i}, u_{j} \neq u_{i} \in V} I_{Y} (u_{i} u_{j}) \\ and d_{F} (u_{i}) & = & \sum_{u_{i}, u_{j} \neq u_{i} \in V} F_{Y} (u_{i} u_{j}) . \end{matrix}$

3 Regularity of graphs in single valued neutrosophic environment

In this section, based on the extension of the regularity of intuitionistic fuzzy graphs [15], we define the concepts of regularity of SVNGs, which can be used in real scientific and engineering applications.

Definition 3.1. A SVNG $G = (X, Y)$ on G, in which each vertex has the same degree is called a regular SVNG. If each vertex has degree 〈g₁, g₂, g₃〉, i.e., $d_{G} (u_{i}) = 〈 g_{1}, g_{2}, g_{3} 〉$ for all u_i ∈ V, $G$ is called 〈g₁, g₂, g₃〉-regular.

Example 3.2. Consider a graph G = (V, E), where V = {u₁, u₂, u₃, u₄} and E = {u₁u₂, u₂u₃, u₃u₄, u₄u₁}. Let $G = (X, Y)$ be a SVNG of a graph G defined by $\begin{matrix} X & = & 〈 (\frac{u_{1}}{0.3}, \frac{u_{2}}{0.6}, \frac{u_{3}}{0.4}, \frac{u_{4}}{0.5}), \\ (\frac{u_{1}}{0.2}, \frac{u_{2}}{0.1}, \frac{u_{3}}{0.3}, \frac{u_{4}}{0.1}), \\ (\frac{u_{1}}{0.1}, \frac{u_{2}}{0.3}, \frac{u_{3}}{0.2}, \frac{u_{4}}{0.3}) 〉, \\ Y & = & 〈 (\frac{u_{1} u_{2}}{0.1}, \frac{u_{2} u_{3}}{0.2}, \frac{u_{3} u_{4}}{0.1}, \frac{u_{4} u_{1}}{0.2}), \\ (\frac{u_{1} u_{2}}{0.5}, \frac{u_{2} u_{3}}{0.3}, \frac{u_{3} u_{4}}{0.5}, \frac{u_{4} u_{1}}{0.3}), \\ (\frac{u_{1} u_{2}}{0.3}, \frac{u_{2} u_{3}}{0.7}, \frac{u_{3} u_{4}}{0.3}, \frac{u_{4} u_{1}}{0.7}) 〉 . \end{matrix}$

Clearly, $d_{G} (u_{1}) = d_{G} (u_{2}) = d_{G} (u_{3}) = d_{G} (u_{4}) = 〈 0.3, 0.8, 1.0 〉 .$ So, $G$ is 〈0.3, 0.8, 1.0〉-regular SVNG. The regular SVNG is given in Fig. 1. Tabular representation of a regular SVNG is given in Table 1.

Fig.1

〈0.3, 0.8, 1.0〉-regular SVNG.

Table 1

Tabular representation of a 〈0.3, 0.8, 1.0〉-regular SVNG

	u ₁	u ₂	u ₃	u ₄
T _X	0.3	0.6	0.4	0.5
I _X	0.2	0.1	0.3	0.1
F _X	0.1	0.3	0.2	0.3
	u ₁ u ₂	u ₂ u ₃	u ₃ u ₄	u ₄ u ₁
T _Y	0.1	0.2	0.1	0.2
I _Y	0.5	0.3	0.5	0.3
F _Y	0.3	0.7	0.3	0.7

Definition 3.3. Let $G = (X, Y)$ be a SVNG. The degree of an edge u_iu_j ∈ E is defined as $d_{G} (u_{i} u_{j}) = 〈 d_{T} (u_{i} u_{j}), d_{I} (u_{i} u_{j}), d_{F} (u_{i} u_{j}) 〉,$ where $\begin{matrix} d_{T} (u_{i} u_{j}) & = & d_{T} (u_{i}) + d_{T} (u_{j}) - 2 T_{Y} (u_{i} u_{j}) \\ = & \sum_{\begin{matrix} u_{i} u_{k} \in E \\ k \neq j \end{matrix}} T_{Y} (u_{i} u_{k}) + \sum_{\begin{matrix} u_{j} u_{k} \in E \\ k \neq i \end{matrix}} T_{Y} (u_{j} u_{k}), \\ d_{I} (u_{i} u_{j}) & = & d_{I} (u_{i}) + d_{I} (u_{j}) - 2 I_{Y} (u_{i} u_{j}) \\ = & \sum_{\begin{matrix} u_{i} u_{k} \in E \\ k \neq j \end{matrix}} I_{Y} (u_{i} u_{k}) + \sum_{\begin{matrix} u_{j} u_{k} \in E \\ k \neq i \end{matrix}} I_{Y} (u_{j} u_{k}), \\ d_{F} (u_{i} u_{j}) & = & d_{F} (u_{i}) + d_{F} (u_{j}) - 2 F_{Y} (u_{i} u_{j}) \\ = & \sum_{\begin{matrix} u_{i} u_{k} \in E \\ k \neq j \end{matrix}} F_{Y} (u_{i} u_{k}) + \sum_{\begin{matrix} u_{j} u_{k} \in E \\ k \neq i \end{matrix}} F_{Y} (u_{j} u_{k}) . \end{matrix}$

Definition 3.4. Let $G$ be a SVNG.

The minimum edge degree of $G$ is $δ_{E} (G) = 〈 δ_{T} (G), δ_{I} (G), δ_{F} (G) 〉$ , where $δ_{T} (G) = min {d_{T} (u_{i} u_{j}) | u_{i} u_{j} \in E}$ , $δ_{I} (G) = max {d_{I} (u_{i} u_{j}) | u_{i} u_{j} \in E}$ and $δ_{F} (G) = max {d_{F} (u_{i} u_{j}) | u_{i} u_{j} \in E}$ .

The maximum edge degree of $G$ is $Δ_{E} (G) = 〈 Δ_{T} (G), Δ_{I} (G), Δ_{F} (G) 〉$ , where $Δ_{T} (G) = max {d_{T} (u_{i} u_{j}) | u_{i} u_{j} \in E}$ , $Δ_{I} (G) = min {d_{I} (u_{i} u_{j}) | u_{i} u_{j} \in E}$ and $Δ_{F} (G) = min {d_{F} (u_{i} u_{j}) | u_{i} u_{j} \in E}$ .

Definition 3.5. The total edge degree of an edge u_iu_j ∈ E in a SVNG $G = (X, Y)$ is defined as $t d_{G} (u_{i} u_{j}) = 〈 t d_{T} (u_{i} u_{j}), t d_{I} (u_{i} u_{j}), t d_{F} (u_{i} u_{j}) 〉,$ where $\begin{matrix} {td}_{T} (u_{i} u_{j}) & = & d_{T} (u_{i}) + d_{T} (u_{j}) - T_{Y} (u_{i} u_{j}) \\ = & \sum_{\begin{matrix} u_{i} u_{k} \in E \\ k \neq j \end{matrix}} T_{Y} (u_{i} u_{k}) + \sum_{\begin{matrix} u_{j} u_{k} \in E \\ k \neq i \end{matrix}} T_{Y} (u_{j} u_{k}) \\ + T_{Y} (u_{i} u_{j}), \\ {td}_{I} (u_{i} u_{j}) & = & d_{I} (u_{i}) + d_{I} (u_{j}) - I_{Y} (u_{i} u_{j}) \\ = & \sum_{\begin{matrix} u_{i} u_{k} \in E \\ k \neq j \end{matrix}} I_{Y} (u_{i} u_{k}) + \sum_{\begin{matrix} u_{j} u_{k} \in E \\ k \neq i \end{matrix}} I_{Y} (u_{j} u_{k}) \\ + I_{Y} (u_{i} u_{j}), \\ {td}_{F} (u_{i} u_{j}) & = & d_{F} (u_{i}) + d_{F} (u_{j}) - F_{Y} (u_{i} u_{j}) \\ = & \sum_{\begin{matrix} u_{i} u_{k} \in E \\ k \neq j \end{matrix}} F_{Y} (u_{i} u_{k}) + \sum_{\begin{matrix} u_{j} u_{k} \in E \\ k \neq i \end{matrix}} F_{Y} (u_{j} u_{k}) \\ + F_{Y} (u_{i} u_{j}) . \end{matrix}$

Example 3.6. Consider a SVNG $G = (X, Y)$ on V = {u₁, u₂, u₃, u₄, u₅, u₆}, as shown in Fig. 2.

Fig.2

SVNG.

Then the degree of an edge u₁u₂ is $d_{G} (u_{1} u_{2}) = 〈 1.0, 1.8, 2.6 〉$ as d_T (u₁u₂) =1.0, d_I (u₁u₂) =1.8, d_F (u₁u₂) =2.6, and the total degree of an edge u₁u₂ is ${td}_{G} (u_{1} u_{2}) = 〈 1.1, 2.2, 3.3 〉$ as td_T (u₁u₂) =1.0 + 0.1 = 1.1, td_I (u₁u₂) =1.8 + 0.4 = 2.2, td_F (u₁u₂) =2.6 + 0.7 = 3.3.

Definition 3.7. A SVNG $G$ on G, in which each edge has the same degree is called an edge regular SVNG. If each edge has degree 〈f₁, f₂, f₃〉, i.e., $d_{G} (u_{i} u_{j}) = 〈 f_{1}, f_{2}, f_{3} 〉$ for all u_iu_j ∈ E, $G$ is called 〈f₁, f₂, f₃〉-edge regular.

Definition 3.8. A SVNG $G$ on G, in which each edge has the same total degree 〈t₁, t₂, t₃〉 is called a totally edge regular SVNG.

Example 3.9. Consider a SVNG $G = (X, Y)$ on V = {u₁, u₂, u₃, u₄}, as shown in Fig. 3.

Fig.3

〈1.0, 1.4, 1.2〉-edge regular SVNG.

Clearly, $d_{G} (u_{1} u_{2}) = d_{G} (u_{2} u_{3}) = d_{G} (u_{3} u_{1}) = d_{G} (u_{1} u_{4}) = d_{G} (u_{2} u_{4}) = d_{G} (u_{3} u_{4}) = 〈 1.0, 1.4, 1.2 〉 .$ So, $G$ is 〈1.0, 1.4, 1.2〉-edge regular SVNG.

Theorem 3.10. Let $G = (X, Y)$ be a SVNG on a cycle G. Then $\sum_{u_{i} \in V} d_{G} (u_{i}) = \sum_{u_{i} u_{j} \in E} d_{G} (u_{i} u_{j})$

Proof. Let $G = (X, Y)$ be a SVNG and let G be a cycle u₁u₂u₃ … u_nu₁ . Then $\begin{matrix} \sum_{i = 1}^{n} d_{G} (u_{i} u_{i + 1}) \\ = 〈 \sum_{i = 1}^{n} d_{T} (u_{i} u_{i + 1}), \sum_{i = 1}^{n} d_{I} (u_{i} u_{i + 1}), \sum_{i = 1}^{n} d_{F} (u_{i} u_{i + 1}) 〉 . \end{matrix}$

Consider $\begin{matrix} \sum_{i = 1}^{n} d_{T} (u_{i} u_{i + 1}) \\ = d_{T} (u_{1} u_{2}) + d_{T} (u_{2} u_{3}) + \dots + d_{T} (u_{n} u_{1}), \\ where u_{n + 1} = u_{1} \\ = d_{T} (u_{1}) + d_{T} (u_{2}) - 2 T_{Y} (u_{1} u_{2}) \\ + d_{T} (u_{2}) + d_{T} (u_{3}) - 2 T_{Y} (u_{2} u_{3}) \\ + \dots + d_{T} (u_{n}) + d_{T} (u_{1}) - 2 T_{Y} (u_{n} u_{1}) \\ = 2 d_{T} (u_{1}) + 2 d_{T} (u_{2}) + \dots + 2 d_{T} (u_{n}) \\ - 2 (T_{Y} (u_{1} u_{2}) + T_{Y} (u_{2} u_{3}) + \dots + T_{Y} (u_{n} u_{1})) \\ = 2 \sum_{u_{i} \in V} d_{T} (u_{i}) - 2 \sum_{i = 1}^{n} T_{Y} (u_{i} u_{i + 1}) \\ = \sum_{u_{i} \in V} d_{T} (u_{i}) + \sum_{u_{i} \in V} d_{T} (u_{i}) - 2 \sum_{i = 1}^{n} T_{Y} (u_{i} u_{i + 1}) \\ = \sum_{u_{i} \in V} d_{T} (u_{i}) + 2 \sum_{i = 1}^{n} T_{Y} (u_{i} u_{i + 1}) \\ - 2 \sum_{i = 1}^{n} T_{Y} (u_{i} u_{i + 1}) \\ = \sum_{u_{i} \in V} d_{T} (u_{i}) . \end{matrix}$

Analogously, we can show that $\sum_{i = 1}^{n} d_{I} (u_{i} u_{i + 1}) = \sum_{u_{i} \in V} d_{I} (u_{i})$ and $\sum_{i = 1}^{n} d_{F} (u_{i} u_{i + 1}) = \sum_{u_{i} \in V} d_{F} (u_{i}) .$

Hence $\sum_{u_{i} \in V} d_{G} (u_{i}) = 〈 \sum_{u_{i} \in V} d_{T} (u_{i}), \sum_{u_{i} \in V} d_{I} (u_{i}), \sum_{u_{i} \in V} d_{F} (u_{i}) 〉 = \sum_{u_{i} u_{j} \in E} d_{G} (u_{i} u_{j}) .$ □

Theorem 3.11. Let $G = (X, Y)$ be a SVNG on a regular crisp graph G. Then $\begin{matrix} \sum_{u_{i} u_{j} \in E} d_{G} (u_{i} u_{j}) \\ = 〈 \sum_{u_{i} u_{j} \in E} d_{G} (u_{i} u_{j}) T_{Y} (u_{i} u_{j}), \\ \sum_{u_{i} u_{j} \in E} d_{G} (u_{i} u_{j}) I_{Y} (u_{i} u_{j}), \\ \sum_{u_{i} u_{j} \in E} d_{G} (u_{i} u_{j}) F_{Y} (u_{i} u_{j}) 〉, \end{matrix}$ where d_G (u_iu_j) = d_G (u_i) + d_G (u_j) -2 for all u_iu_j ∈ E.

Proof. Suppose that $G$ is a SVNG on a regular crisp graph G. Since $d_{G} (u_{i} u_{j}) = 〈 d_{T} (u_{i} u_{j}), d_{I} (u_{i} u_{j}), d_{F} (u_{i} u_{j}) 〉,$ where d_T (u_iu_j) , d_I (u_iu_j) and d_F (u_iu_j) is the sum of the truth-membership, indeterminacy-membership and falsity-membership values of their adjacent edges, respectively. In ∑_{u_iu_j∈E}d_T (u_iu_j), therefore every edge contributes its truth-membership values exactly degree of that edge in corresponding crisp graph times. So, ∑_{u_iu_j∈E}d_T (u_iu_j) = ∑_{u_iu_j∈E}d_G (u_iu_j) T_Y (u_iu_j) . Analogously, ∑_{u_iu_j∈E}d_I (u_iu_j) = ∑_{u_iu_j∈E}d_G (u_iu_j) I_Y (u_iu_j) and ∑_{u_iu_j∈E}d_F (u_iu_j) = ∑_{u_iu_j∈E}d_G (u_iu_j) F_Y (u_iu_j) . Hence $\sum_{u_{i} u_{j} \in E} d_{G} (u_{i} u_{j}) = 〈 \sum_{u_{i} u_{j} \in E} d_{G} (u_{i} u_{j}) T_{Y} (u_{i} u_{j}), \sum_{u_{i} u_{j} \in E} d_{G} (u_{i} u_{j}) I_{Y} (u_{i} u_{j}), \sum_{u_{i} u_{j} \in E} d_{G} (u_{i} u_{j}) F_{Y} (u_{i} u_{j}) 〉 .$ □

Proposition 3.12. Let $G = (X, Y)$ be a SVNG on a k-regular crisp graph G. Then $\begin{matrix} \sum_{u_{i} u_{j} \in E} d_{G} (u_{i} u_{j}) = 〈 (k - 1) \sum_{u_{i} \in V} d_{T} (u_{i}), \\ (k - 1) \sum_{u_{i} \in V} d_{I} (u_{i}), (k - 1) \sum_{u_{i} \in V} d_{F} (u_{i}) 〉 . \end{matrix}$

Proof. Suppose that $G$ is a SVNG on a k-regular crisp graph G. From Theorem 3.11, we have $\begin{matrix} \sum_{u_{i} u_{j} \in E} d_{G} (u_{i} u_{j}) = 〈 \sum_{u_{i} u_{j} \in E} d_{G} (u_{i} u_{j}) T_{Y} (u_{i} u_{j}), \\ \sum_{u_{i} u_{j} \in E} d_{G} (u_{i} u_{j}) I_{Y} (u_{i} u_{j}), \sum_{u_{i} u_{j} \in E} d_{G} (u_{i} u_{j}) F_{Y} (u_{i} u_{j}) 〉 \\ = 〈 \sum_{u_{i} u_{j} \in E} (d_{G} (u_{i}) + d_{G} (u_{j}) - 2) T_{Y} (u_{i} u_{j}), \\ \sum_{u_{i} u_{j} \in E} (d_{G} (u_{i}) + d_{G} (u_{j}) - 2) I_{Y} (u_{i} u_{j}), \\ \sum_{u_{i} u_{j} \in E} (d_{G} (u_{i}) + d_{G} (u_{j}) - 2) F_{Y} (u_{i} u_{j}) 〉 . \end{matrix}$

Since G is a k-regular crisp graph, d_G (u_i) = k, for all u_i ∈ V, therefore $\begin{matrix} \sum_{u_{i} u_{j} \in E} d_{G} (u_{i} u_{j}) \\ = 〈 (k + k - 2) \sum_{u_{i} u_{j} \in E} T_{Y} (u_{i} u_{j}), (k + k - 2) \\ \sum_{u_{i} u_{j} \in E} I_{Y} (u_{i} u_{j}), (k + k - 2) \sum_{u_{i} u_{j} \in E} F_{Y} (u_{i} u_{j}) 〉, \\ = 〈 2 (k - 1) \sum_{u_{i} u_{j} \in E} T_{Y} (u_{i} u_{j}), 2 (k - 1) \\ \sum_{u_{i} u_{j} \in E} I_{Y} (u_{i} u_{j}), 2 (k - 1) \sum_{u_{i} u_{j} \in E} F_{Y} (u_{i} u_{j}) 〉, \\ = 〈 (k - 1) \sum_{u_{i} \in V} d_{T} (u_{i}), (k - 1) \sum_{u_{i} \in V} d_{I} (u_{i}), \\ (k - 1) \sum_{u_{i} \in V} d_{F} (u_{i}) 〉 . □ \end{matrix}$

Proposition 3.13. Let $G = (X, Y)$ be a SVNG on a regular crisp graph G. Then $\sum_{u_{i} u_{j} \in E} {td}_{G} (u_{i} u_{j}) = 〈 \sum_{u_{i} u_{j} \in E} d_{G} (u_{i} u_{j}) T_{Y} (u_{i} u_{j}) + \sum_{u_{i} u_{j} \in E} T_{Y} (u_{i} u_{j}), \sum_{u_{i} u_{j} \in E} d_{G} (u_{i} u_{j}) I_{Y} (u_{i} u_{j}) + \sum_{u_{i} u_{j} \in E} I_{Y} (u_{i} u_{j}), \sum_{u_{i} u_{j} \in E} d_{G} (u_{i} u_{j}) F_{Y} (u_{i} u_{j}) + \sum_{u_{i} u_{j} \in E} F_{Y} (u_{i} u_{j}) 〉 .$

Theorem 3.14. Let $G = (X, Y)$ be a SVNG. Then 〈T_Y, I_Y, F_Y〉 is a constant function if and only if the following statements are equivalent:

$G$ is an edge regular SVNG;

$G$ is a totally edge regular SVNG.

Proof. Suppose that 〈T_Y, I_Y, F_Y〉 is a constant function. Then T_Y (u_iu_j) = c₁, I_Y (u_iu_j) = c₂ and F_Y (u_iu_j) = c₃ for every u_iu_j ∈ E, where c₁, c₂ and c₃ are constants.

(i) ⇒ (ii). Assume that $G$ is 〈f₁, f₂, f₃〉-edge regular SVNG. Then $d_{G} (u_{i} u_{j}) = 〈 f_{1}, f_{2}, f_{3} 〉$ for all u_iu_j ∈ E.

Consider $\begin{matrix} t d_{G} (u_{i} u_{j}) & = & 〈 d_{T} (u_{i} u_{j}) + T_{Y} (u_{i} u_{j}), d_{I} (u_{i} u_{j}) \\ + I_{Y} (u_{i} u_{j}), d_{F} (u_{i} u_{j}) + F_{Y} (u_{i} u_{j}) 〉 \\ = & 〈 f_{1} + c_{1}, f_{2} + c_{2}, f_{3} + c_{3} 〉 \\ for all u_{i} u_{j} \in E . \end{matrix}$

Therefore, SVNG $G$ is a totally edge regular.

(ii) ⇒ (i). Let $G$ be a 〈t₁, t₂, t₃〉-totally edge regular SVNG. Then ${td}_{G} (u_{i} u_{j}) = 〈 d_{T} (u_{i} u_{j}) + T_{Y} (u_{i} u_{j}), d_{I} (u_{i} u_{j}) + I_{Y} (u_{i} u_{j}), d_{F} (u_{i} u_{j}) + F_{Y} (u_{i} u_{j}) 〉 = 〈 t_{1}, t_{2}, t_{3} 〉$ for all u_iu_j ∈ E.

Now, $\begin{matrix} 〈 d_{T} (u_{i} u_{j}), d_{I} (u_{i} u_{j}), d_{F} (u_{i} u_{j}) 〉 \\ = 〈 t_{1} - T_{Y} (u_{i} u_{j}), t_{2} - I_{Y} (u_{i} u_{j}), t_{3} - F_{Y} (u_{i} u_{j}) 〉 \\ d_{G} (u_{i} u_{j}) = 〈 t_{1} - c_{1}, t_{2} - c_{2}, t_{3} - c_{3} 〉 . \end{matrix}$

Therefore, $G$ is a 〈t₁ - c₁, t₂ - c₂, t₃ - c₃〉-edge regular SVNG.

Conversely, suppose that (i) and (ii) are equivalent. We have to prove that 〈T_Y, I_Y, F_Y〉 is a constant function. Suppose that 〈T_Y, I_Y, F_Y〉 is not a constant function. Then T_Y (u_iu_j) ≠ T_Y (u_pu_q), I_Y (u_iu_j) ≠ I_Y (u_pu_q) and F_Y (u_iu_j) ≠ F_Y (u_pu_q) for at least one pair of u_iu_j, u_pu_q ∈ E. Assume that $G$ is a 〈f₁, f₂, f₃〉-edge regular SVNG. Then $d_{G} (u_{i} u_{j}) = d_{G} (u_{p} u_{q}) = 〈 f_{1}, f_{2}, f_{3} 〉 .$ Hence

$\begin{matrix} {td}_{G} (u_{i} u_{j}) \\ = 〈 d_{T} (u_{i} u_{j}) + T_{Y} (u_{i} u_{j}), d_{I} (u_{i} u_{j}) \\ + I_{Y} (u_{i} u_{j}), d_{F} (u_{i} u_{j}) + F_{Y} (u_{i} u_{j}) 〉 \\ = 〈 f_{1} + T_{Y} (u_{i} u_{j}), f_{2} + I_{Y} (u_{i} u_{j}), \\ f_{3} + F_{Y} (u_{i} u_{j}) 〉 . \\ {td}_{G} (u_{p} u_{q}) \\ = 〈 d_{T} (u_{p} u_{q}) + T_{Y} (u_{p} u_{q}), d_{I} (u_{p} u_{q}) \\ + I_{Y} (u_{p} u_{q}), d_{F} (u_{p} u_{q}) + F_{Y} (u_{p} u_{q}) 〉 \\ = 〈 f_{1} + T_{Y} (u_{p} u_{q}), f_{2} + I_{Y} (u_{p} u_{q}), \\ f_{3} + F_{Y} (u_{p} u_{q}) 〉 . \end{matrix}$

As T_Y (u_iu_j) ≠ T_Y (u_pu_q), I_Y (u_iu_j) ≠ I_Y (u_pu_q) and F_Y (u_iu_j) ≠ F_Y (u_pu_q) , so ${td}_{G} (u_{i} u_{j}) \neq_{G} (u_{p} u_{q})$ . Hence $G$ is not a totally edge regular, a contradiction. Therefore, 〈T_Y, I_Y, F_Y〉 is a constant function. Analogously, we can show that 〈T_Y, I_Y, F_Y〉 is a constant function, if $G$ is a totally edge regular SVNG. □

Theorem 3.15. If a SVNG $G$ is both edge regular and totally edge regular, then 〈T_Y, I_Y, F_Y〉 is a constant function.

Proof. Obvious, therefore omitted. □

Theorem 3.16. Let $G = (X, Y)$ be a SVNG on a k-regular crisp graph G. Then 〈T_Y, I_Y, F_Y〉 is a constant function if and only if $G$ is both regular and totally edge regular SVNG.

Proof. Let $G$ be a SVNG on a k-regular graph G. Assume that 〈T_Y, I_Y, F_Y〉 is a constant function, that is, T_Y (u_iu_j) = c₁, I_Y (u_iu_j) = c₂ and F_Y (u_iu_j) = c₃ for all u_iu_j ∈ E, where c₁, c₂ and c₃ are constants.

By definition of vertex degree, we have

$\begin{array}{l} d_{G} (u_{i}) = 〈 d_{T} (u_{i}), d_{I} (u_{i}), d_{F} (u_{i}) 〉 \\ = 〈 \sum_{u_{i} u_{j} \in E} T_{Y} (u_{i} u_{j}), \sum_{u_{i} u_{j} \in E} I_{Y} (u_{i} u_{j}), \sum_{u_{i} u_{j} \in E} F_{Y} (u_{i} u_{j}) 〉 \\ = 〈 \sum_{u_{i} u_{j} \in E} c_{1}, \sum_{u_{i} u_{j} \in E} c_{2}, \sum_{u_{i} u_{j} \in E} c_{3} 〉 \\ = 〈 k c_{1}, k c_{2}, k c_{3} 〉 f o r a l l u_{i} \in V . \end{array}$

Therefore, $G$ is regular SVNG.

Now, $t d_{G} (u_{i} u_{j}) = 〈 t d_{T} (u_{i} u_{j}), t d_{I} (u_{i} u_{j}), t d_{F} (u_{i} u_{j}) 〉$ , where ${td}_{T} (u_{i} u_{j}) = \sum_{\begin{matrix} u_{i} u_{k} \in E \\ k \neq j \end{matrix}} T_{Y} (u_{i} u_{k}) + \sum_{\begin{matrix} u_{j} u_{k} \in E \\ k \neq i \end{matrix}} T_{Y} (u_{j} u_{k}) + T_{Y} (u_{i} u_{j}) = \sum_{\begin{matrix} u_{i} u_{k} \in E \\ k \neq j \end{matrix}} c_{1} + \sum_{\begin{matrix} u_{j} u_{k} \in E \\ k \neq i \end{matrix}} c_{1} + c_{1} = c_{1} (k - 1) + c_{1} (k - 1) + c_{1} = c_{1} (2 k - 1)$ for all u_iu_j ∈ E . Similarly, it is easy to show that td_I (u_iu_j) = c₂ (2k - 1) and td_F (u_iu_j) = c₃ (2k - 1) for all u_iu_j ∈ E . Hence $G$ is a totally edge regular SVNG.

Conversely, suppose that $G$ is both regular and totally edge regular SVNG. We have to prove that 〈T_Y, I_Y, F_Y〉 is a constant function. Since SVNG $G$ is regular, $d_{G} (u_{i}) = 〈 g_{1}, g_{2}, g_{3} 〉$ for all u_i ∈ V. Also, $G$ is a totally edge regular, ${td}_{G} (u_{i} u_{j}) = 〈 t_{1}, t_{2}, t_{3} 〉$ for all u_iu_j ∈ E . By definition of total edge degree, we have ${td}_{G} (u_{i} u_{j}) = 〈 {td}_{T} (u_{i} u_{j}), {td}_{I} (u_{i} u_{j}), {td}_{F} (u_{i} u_{j}) 〉,$ where $\begin{matrix} {td}_{T} (u_{i} u_{j}) & = & d_{T} (u_{i}) + d_{T} (u_{j}) - T_{Y} (u_{i} u_{j}) \\ t_{1} & = & g_{1} + g_{1} - T_{Y} (u_{i} u_{j}) \\ T_{Y} (u_{i} u_{j}) & = & 2 g_{1} - t_{1} for all u_{i} u_{j} \in E . \end{matrix}$

Similarly, we can show that I_Y (u_iu_j) =2g₂ - t₂ and F_Y (u_iu_j) =2g₃ - t₃ for all u_iu_j ∈ E . Hence 〈T_Y, I_Y, F_Y〉 is a constant function. □

Theorem 3.17. Let $G = (X, Y)$ be a SVNG on a crisp graph G. If 〈T_Y, I_Y, F_Y〉 is a constant function, then $G$ is an edge regular SVNG if and only if G is an edge regular.

Proof. Assume that 〈T_Y, I_Y, F_Y〉 is a constant function, i.e., T_Y (u_iu_j) = c₁, I_Y (u_iu_j) = c₂ and F_Y (u_iu_j) = c₃ for all u_iu_j ∈ E, where c₁, c₂ and c₃ are constants. Suppose that $G$ is an edge regular SVNG. We have to show that G is an edge regular. Suppose on contrary that G is not an edge regular. i.e., d_G (u_iu_j) ≠ d_G (u_lu_m) for at least on pair of u_iu_j, u_lu_m ∈ E. According to the definition of edge degree of a SVNG, $d_{G} (u_{i} u_{j}) = 〈 d_{T} (u_{i} u_{j}), d_{I} (u_{i} u_{j}), d_{F} (u_{i} u_{j}) 〉,$ where $\begin{matrix} d_{T} (u_{i} u_{j}) & = & \sum_{\begin{matrix} u_{i} u_{k} \in E \\ k \neq j \end{matrix}} T_{Y} (u_{i} u_{k}) + \sum_{\begin{matrix} u_{j} u_{k} \in E \\ k \neq i \end{matrix}} T_{Y} (u_{j} u_{k}) \\ = & \sum_{\begin{matrix} u_{i} u_{k} \in E \\ k \neq j \end{matrix}} c_{1} + \sum_{\begin{matrix} u_{j} u_{k} \in E \\ k \neq i \end{matrix}} c_{1} \\ = & c_{1} (d_{G} (u_{i}) - 1) + c_{1} (d_{G} (u_{j}) - 1) \\ = & c_{1} (d_{G} (u_{i}) + d_{G} (u_{j}) - 2) \\ = & c_{1} d_{G} (u_{i} u_{j}) for all u_{i} u_{j} \in E . \end{matrix}$

Analogously, we can show that d_I (u_iu_j) = c₂d_G (u_iu_j) and d_F (u_iu_j) = c₃d_G (u_iu_j) for all u_iu_j ∈ E . Therefore $d_{G} (u_{i} u_{j}) = 〈 c_{1} d_{G} (u_{i} u_{j}), c_{2} d_{G} (u_{i} u_{j}), c_{3} d_{G} (u_{i} u_{j}) 〉,$ $d_{G} (u_{l} u_{m}) = 〈 c_{1} d_{G} (u_{l} u_{m}), c_{2} d_{G} (u_{l} u_{m}), c_{3} d_{G} (u_{l} u_{m}) 〉 .$ Since d_G (u_iu_j) _neqdG (u_lu_m), so $d_{G} (u_{i} u_{j}) \neq d_{G} (u_{l} u_{m})$ . Thus $G$ is not an edge regular, a contradiction. Hence G is an edge regular.

Conversely, suppose that G is an edge regular graph. To prove that $G$ is an edge regular SVNG. Consider $G$ is not an edge regular SVNG. i.e., $d_{G} (u_{i} u_{j}) \neq d_{G} (u_{p} u_{q})$ for at least one pair of u_iu_j, u_pu_q ∈ E, 〈d_T (u_iu_j) , d_I (u_iu_j) , d_F (u_iu_j) 〉 ≠ 〈d_T (u_pu_q) , d_I (u_pu_q) , d_F (u_pu_q) 〉 . Now d_T (u_iu_j) ≠ d_T (u_pu_q) implies $\sum_{\begin{matrix} u_{i} u_{k} \in E \\ k \neq j \end{matrix}} T_{Y} (u_{i} u_{k}) + \sum_{\begin{matrix} u_{j} u_{k} \in E \\ k \neq i \end{matrix}} T_{Y} (u_{j} u_{k}) \neq \sum_{\begin{matrix} u_{p} u_{s} \in E \\ s \neq q \end{matrix}} T_{Y} (u_{p} u_{s}) + \sum_{\begin{matrix} u_{s} u_{q} \in E \\ s \neq p \end{matrix}} T_{Y} (u_{s} u_{q})$ ,since T_Y is a constant function, so d_G (u_iu_j) ≠ d_G (u_pu_q), a contradiction. Hence $G$ is an edge regular SVNG. □

Theorem 3.18. Let $G = (X, Y)$ be a regular SVNG. Then $G$ is an edge regular SVNG if and only if 〈T_Y, I_Y, F_Y〉 is a constant function.

Proof. Let $G$ be a 〈g₁, g₂, g₃〉-regular SVNG i.e., $d_{G} (u_{i}) = 〈 g_{1}, g_{2}, g_{3} 〉$ for all u_i ∈ V. Suppose that 〈T_Y, I_Y, F_Y〉 is a constant function. Then T_Y (u_iu_j) = c₁, I_Y (u_iu_j) = c₂ and F_Y (u_iu_j) = c₃ for all u_iu_j ∈ E, where c₁, c₂ and c₃ are constants. According to the definition of edge degree of a SVNG, $d_{G} (u_{i} u_{j}) = 〈 d_{T} (u_{i} u_{j}), d_{I} (u_{i} u_{j}), d_{F} (u_{i} u_{j}) 〉,$ where $\begin{matrix} d_{T} (u_{i} u_{j}) \\ = d_{T} (u_{i}) + d_{T} (u_{j}) - 2 T_{Y} (u_{i} u_{j}) \\ = g_{1} + g_{1} - 2 c_{1} \\ = 2 (g_{1} - c_{1}) for all u_{i} u_{j} \in E . \end{matrix}$

Similarly, we can show that, d_I (u_iu_j) =2 (g₂ - c₂) and d_F (u_iu_j) =2 (g₃ - c₃) for all u_iu_j ∈ E. Hence $G$ is an edge regular SVNG.

Conversely, suppose that $G$ is an edge regular SVNG, i.e., $d_{G} (u_{i} u_{j}) = 〈 f_{1}, f_{2}, f_{3} 〉$ for all u_iu_j ∈ E . we have to show that 〈T_Y, I_Y, F_Y〉 is a constant function. Since $d_{G} (u_{i} u_{j}) = 〈 d_{T} (u_{i} u_{j}), d_{I} (u_{i} u_{j}), d_{F} (u_{i} u_{j}) 〉$ , where $\begin{matrix} d_{T} (u_{i} u_{j}) & = & d_{T} (u_{i}) + d_{T} (u_{j}) - 2 T_{Y} (u_{i} u_{j}) \\ f_{1} & = & g_{1} + g_{1} - 2 T_{Y} (u_{i} u_{j}) \\ T_{Y} (u_{i} u_{j}) & = & \frac{(2 g_{1} - f_{1})}{2} for all u_{i} u_{j} \in E . \end{matrix}$

Similarly, it is easy to show that $I_{Y} (u_{i} u_{j}) = \frac{(2 g_{2} - f_{2})}{2}$ and $F_{Y} (u_{i} u_{j}) = \frac{(2 g_{3} - f_{3})}{2}$ for all u_iu_j ∈ E. Hence 〈T_Y, I_Y, F_Y〉 is a constant function. □

Definition 3.19. A SVNG $G$ is said to be a

partially regular, if the underlying graph G is regular.

full regular, if $G$ is both regular and partially regular.

partially edge regular, if the underlying graph G is an edge regular.

full edge regular, if $G$ is both edge regular and partially edge regular.

Theorem 3.20. Let $G = (X, Y)$ be a SVNG on G such that (T_Y, I_Y, F_Y) is a constant function. If $G$ is full regular SVNG, then $G$ is full edge regular SVNG.

Proof. Suppose that 〈T_Y, I_Y, F_Y〉 is a constant function. Then T_Y (u_iu_j) = c₁, I_Y (u_iu_j) = c₂ and F_Y (u_iu_j) = c₃ for all u_iu_j ∈ E, where c₁, c₂ and c₃ are constants. Let $G$ be a full regular SVNG, then d_G (u_i) = k and $d_{G} (u_{i}) = 〈 g_{1}, g_{2}, g_{3} 〉$ for all u_i ∈ V, where k, g₁, g₂ and g₃ are constants. Therefore d_G (u_iu_j) = d_G (u_i) + d_G (u_j) -2 = 2k - 2 = constant. Hence G is an edge regular graph.

Now, $d_{G} (u_{i} u_{j}) = 〈 d_{T} (u_{i} u_{j}), d_{I} (u_{i} u_{j}), d_{F} (u_{i} u_{j}) 〉$ for all u_iu_j ∈ E, where $\begin{matrix} d_{T} (u_{i} u_{j}) & = & d_{T} (u_{i}) + d_{T} (u_{j}) - 2 T_{Y} (u_{i} u_{j}) \\ = & g_{1} + g_{1} - 2 c_{1} \\ = & 2 g_{1} - 2 c_{1} = constant . \end{matrix}$

Similarly, it is easy to show that d_I (u_iu_j) =2g₂ - 2c₂ = constant and d_F (u_iu_j) =2g₃ - 2c₃ =constant, for all u_iu_j ∈ E . Hence $G$ is an edge regular SVNG. Thus $G$ is full edge regular SVNG. □

Theorem 3.21. Let $G = (X, Y)$ be a t-totally edge regular and -partially edge regular SVNG. Then $S (G) = \frac{r t}{1 + t^{'}}$ , where r = |E|.

Proof. The size of SVNG $G$ is

$\begin{matrix} S (G) \\ = 〈 \sum_{u_{i} u_{j} \in E} T_{Y} (u_{i} u_{j}), \sum_{u_{i} u_{j} \in E} I_{Y} (u_{i} u_{j}), \sum_{u_{i} u_{j} \in E} F_{Y} (u_{i} u_{j}) 〉 . \end{matrix}$

Since $G$ is t-totally edge regular and -partially edge regular SVNG, i.e., ${td}_{G} (u_{i} u_{j}) = t$ and d_G (u_iu_j)=^t′, respectively. Therefore,

$\begin{matrix} \sum_{u_{i} u_{j} \in E} {td}_{G} (u_{i} u_{j}) \\ = 〈 \sum_{u_{i} u_{j} \in E} d_{G} (u_{i} u_{j}) T_{Y} (u_{i} u_{j}) + \sum_{u_{i} u_{j} \in E} T_{Y} (u_{i} u_{j}), \\ \sum_{u_{i} u_{j} \in E} d_{G} (u_{i} u_{j}) I_{Y} (u_{i} u_{j}) + \sum_{u_{i} u_{j} \in E} I_{Y} (u_{i} u_{j}), \\ \sum_{u_{i} u_{j} \in E} d_{G} (u_{i} u_{j}) F_{Y} (u_{i} u_{j}) + \sum_{u_{i} u_{j} \in E} F_{Y} (u_{i} u_{j}) 〉 \\ = 〈 \sum_{u_{i} u_{j} \in E} d_{G} (u_{i} u_{j}) T_{Y} (u_{i} u_{j}), \sum_{u_{i} u_{j} \in E} d_{G} (u_{i} u_{j}) I_{Y} (u_{i} u_{j}), \\ \sum_{u_{i} u_{j} \in E} d_{G} (u_{i} u_{j}) F_{Y} (u_{i} u_{j}) 〉 + S (G) \\ rt = t^{'} S (G) + S (G) \\ S (G) = \frac{rt}{1 + t^{'}} . □ \end{matrix}$

Definition 3.22. A SVNG $G = (X, Y)$ on n vertices which is regular of degree g = 〈g₁, g₂, g₃〉 is said to be a strongly regular SVNG, if it satisfies the following properties:

The sum of truth-membership values, indeterminacy-membership values and falsity-membership values of the common neighbourhood vertices of any two adjacent vertices of $G$ have exactly λ = 〈λ₁, λ₂, λ₃〉 weight;

The sum of truth-membership values, indeterminacy-membership values and falsity-membership values of the common neighbourhood vertices of any two non-adjacent vertices of $G$ have exactly χ = 〈χ₁, χ₂, χ₃〉 weight.

A strongly regular SVNG $G$ is denoted by $G = (n, g, λ, χ) .$

Example 3.23. Consider a SVNG $G = (X, Y)$ on V = {u₁, u₂, u₃, u₄, u₅, u₆}, as given in Fig. 4.

Fig.4

Strongly regular SVNG.

Here n = 6, g = 〈g₁, g₂, g₃〉 = 〈0.6, 2.4, 2.6〉, λ = 〈λ₁, λ₂, λ₃〉 = 〈0.9, 0.7, 0.3〉, χ = 〈χ₁, χ₂, χ₃〉 = 〈1.8, 1.4, 0.6〉. Therefore $G$ is a strongly regular SVNG.

Theorem 3.24. Let $G = (X, Y)$ be a complete SVNG with 〈T_X, I_X, F_X〉 and 〈T_Y, I_Y, F_Y〉 as constant functions. Then $G$ is a strongly regular SVNG.

Proof. Suppose that $G = (X, Y)$ is a complete SVNG on n vertices. Since 〈T_X, I_X, F_X〉 and 〈T_Y, I_Y, F_Y〉 are constant functions, so $T_{X} (u_{i}) = c_{1}^{'}$ , $I_{X} (u_{i}) = c_{2}^{'}$ and $F_{X} (u_{i}) = c_{3}^{'}$ for all u_i ∈ V, T_Y (u_iu_j) = c₁, I_Y (u_iu_j) = c₂ and F_Y (u_iu_j) = c₃ for all u_iu_j ∈ E, where $c_{1}^{'}, c_{2}^{'}$ , $c_{3}^{'}$ c₁, c₂ and c₃ are constants. To prove that $G$ is a strongly regular SVNG, we have to show that $G$ is g = 〈g₁, g₂, g₃〉-regular SVNG and the adjacent vertices have the same common neighborhood λ = 〈λ₁, λ₂, λ₃〉 and non-adjacent vertices have the same common neighborhood χ = 〈χ₁, χ₂, χ₃〉. Now, since $G$ is complete SVNG $\begin{matrix} d_{G} (u_{i}) \\ = 〈 d_{T} (u_{i}), d_{I} (u_{i}), d_{F} (u_{i}) 〉 \\ = 〈 \sum_{u_{i}, u_{j} \neq u_{i} \in V} T_{Y} (u_{i} u_{j}), \sum_{u_{i}, u_{j} \neq u_{i} \in V} I_{Y} (u_{i} u_{j}), \\ \sum_{u_{i}, u_{j} \neq u_{i} \in V} F_{Y} (u_{i} u_{j}) 〉 \\ = 〈 (n - 1) c_{1}, (n - 1) c_{2}, (n - 1) c_{3} 〉 . \end{matrix}$

Hence $G$ is a 〈 (n - 1) c₁, (n - 1) c₂, (n - 1) c₃〉-regular SVNG. Also, the sum of truth-membership values, indeterminacy-membership values and falsity-membership values of common neighborhood vertices of any two adjacent vertices $λ = 〈 (n - 2) c_{1}^{'}, (n - 2) c_{2}^{'}, (n - 2) c_{3}^{'} 〉$ are the same and the sum of truth-membership values, indeterminacy-membership values and falsity-membership values of common neighborhood vertices of any two non-adjacent vertices χ = 〈0, 0, 0〉 are the same. □

Definition 3.25. A SVNG $G = (X, Y)$ is said to be bipartite if the vertex set V can be partitioned into two nonempty sets V₁ and V₂ such that T_Y (u_iu_j) =0, I_Y (u_iu_j) =0 and F_Y (u_iu_j) =0if u_i, u_j ∈ V₁ or u_i, u_j ∈ V₂ . Further if T_Y (u_iu_j) = min {T_X (u_i) , T_X (u_j)} , I_Y (u_iu_j) = max {I_X (u_i) , I_X (u_j)} and F_Y (u_iu_j) = max {F_X (u_i) , F_X (u_j)} for all u_i ∈ V₁ and u_j ∈ V₂, then $G$ is called a complete bipartite SVNG, denoted by K_X₁,X₂, where X₁ and X₂ are, respectively, the restrictions of X to V₁ and V₂.

Definition 3.26. A bipartite SVNG $G$ is said to be a biregular SVNG if every vertex in V₁ has same degree φ = 〈φ₁, φ₂, φ₃〉 and every vertex in V₂ has same degree ψ = 〈ψ₁, ψ₂, ψ₃〉, where φ and ψ are constants.

Example 3.27. Consider a SVNG $G = (X, Y)$ on V = {u₁, u₂, u₃, u₄, u₅, u₆}, as given in Fig. 5.

Fig.5

Biregular SVNG.

Clearly, SVNG $G$ is bipartite and degree of vertices $d_{G} (u_{1}) = d_{G} (u_{2}) = 〈 0.9, 1.5, 1.8 〉$ and $d_{G} (u_{3}) = d_{G} (u_{4}) = d_{G} (u_{5}) = 〈 0.6, 1.0, 1.2 〉$ . SVNG $G$ is therefore biregular.

4 Single valued neutrosophic digraphs and its application in MADM

In this section, we introduce the concept of SVNDGs along with its application inMADM.

Definition 4.1. A SVNDG of a (crisp) digraph $D = (V, \vec{E})$ is a pair $D = (X, \vec{Y})$ , where

the functions $T_{\vec{Y}} : E \subseteq V \times V \to [0, 1], I_{\vec{Y}} : E \subseteq V \times V \to [0, 1]$ and $F_{\vec{Y}} : E \subseteq V \times V \to [0, 1]$ are defined by $\begin{matrix} T_{\vec{Y}} (xy) \leq min {T_{X} (x), T_{X} (y)}, \\ I_{\vec{Y}} (xy) \geq max {I_{X} (x), I_{X} (y)} and \\ F_{\vec{Y}} (xy) \geq max {F_{X} (x), F_{X} (y)} . \end{matrix}$ There is no restriction on the sum of $T_{\vec{Y}} (xy), I_{\vec{Y}} (xy)$ and $F_{\vec{Y}} (xy)$ , therefore $0 \leq T_{\vec{Y}} (xy) + I_{\vec{Y}} (xy) + F_{\vec{Y}} (xy) \leq 3$ for all xy ∈ E .

Where X is a single valued neutrosophic vertex set of $D$ and $\vec{Y}$ is a single valued neutrosophic directed edge set of $D$ . Also $T_{\vec{Y}} (xy)$ , $I_{\vec{Y}} (xy)$ and $F_{\vec{Y}} (xy)$ represent, respectively, the truth-membership, indeterminacy-membership and falsity-membership values of the directededge $\vec{xy}$ .

Example 4.2. Consider a digraph $D = (V, \vec{E})$ , where V = {u₁, u₂, u₃, u₄, u₅, u₆} and $\vec{E} = {u_{1} u_{2}, u_{2} u_{3}, u_{3} u_{4}, u_{4} u_{5}, u_{5} u_{2}, u_{6} u_{4}}$ . Let $D = (X, \vec{Y})$ be a SVNDG of (crisp) digraph D, as given in Fig. 6.

Fig.6

SVNDG.

The corresponding adjacent matrix R is asfollows: $R = [\begin{matrix} 〈 0, 0, 0 〉 & 〈 0.1, 0.5, 0.4 〉 & 〈 0, 0, 0 〉 & 〈 0, 0, 0 〉 & 〈 0, 0, 0 〉 & 〈 0, 0, 0 〉 \\ 〈 0, 0, 0 〉 & 〈 0, 0, 0 〉 & 〈 0.2, 0.6, 0.4 〉 & 〈 0, 0, 0 〉 & 〈 0, 0, 0 〉 & 〈 0, 0, 0 〉 \\ 〈 0, 0, 0 〉 & 〈 0, 0, 0 〉 & 〈 0, 0, 0 〉 & 〈 0.3, 0.4, 0.5 〉 & 〈 0, 0, 0 〉 & 〈 0, 0, 0 〉 \\ 〈 0, 0, 0 〉 & 〈 0, 0, 0 〉 & 〈 0, 0, 0 〉 & 〈 0, 0, 0 〉 & 〈 0.4, 0.3, 0.6 〉 & 〈 0, 0, 0 〉 \\ 〈 0, 0, 0 〉 & 〈 0.1, 0.6, 0.2 〉 & 〈 0, 0, 0 〉 & 〈 0, 0, 0 〉 & 〈 0, 0, 0 〉 & 〈 0, 0, 0 〉 \\ 〈 0, 0, 0 〉 & 〈 0, 0, 0 〉 & 〈 0, 0, 0 〉 & 〈 0.1, 0.4, 0.7 〉 & 〈 0, 0, 0 〉 & 〈 0, 0, 0 〉 \end{matrix}]$

Since in decision making problems, there is a number of uncertainties and in some situations, there exist some relations among attribute in a MADM problems. So, it is an interesting area of applications in neutrosophic graph theory. A MADM problem is solved under the general framework of SVNDGs.

A military unit is planning to purchase new artillery weapons and there are six feasible artillery weapons (alternatives) x_i (i = 1, 2, 3, 4, 5, 6) to be selected. When making a decision, the attributes considered are as follows: (1) a₁- assault fire capability indices; (2) a₂- reaction capability indices; (3) a₃- mobility indices; and (4) a₄- survival ability indices. Among these four attributes, a₁, a₂, a₄ are of benefit type (beneficial); and a₃ is of cost type (non-beneficial) given in Table 2.

Table 2

Single valued neutrosophic decision matrix $R^{'} = {(r_{i j}^{'})}_{6 \times 4}$

Weapons	a ₁	a ₂	a ₃	a ₄
x ₁	〈0.5, 0.3, 0.6〉	〈0.6, 0.3, 0.2〉	〈0.4, 0.5, 0.1〉	〈0.1, 0.7, 0.5〉
x ₂	〈0.6, 0.1, 0.2〉	〈0.2, 0.1, 0.4〉	〈0.2, 0.3, 0.4〉	〈0.3, 0.4, 0.1〉
x ₃	〈0.1, 0.5, 0.3〉	〈0.3, 0.2, 0.5〉	〈0.7, 0.2, 0.1〉	〈0.5, 0.1, 0.2〉
x ₄	〈0.3, 0.4, 0.2〉	〈0.4, 0.5, 0.1〉	〈0.3, 0.1, 0.4〉	〈0.5, 0.3, 0.4〉
x ₅	〈0.1, 0.2, 0.4〉	〈0.2, 0.7, 0.3〉	〈0.1, 0.3, 0.5〉	〈0.2, 0.1, 0.5〉
x ₆	〈0.5, 0.1, 0.7〉	〈0.5, 0.1, 0.4〉	〈0.3, 0.2, 0.6〉	〈0.4, 0.2, 0.6〉

Normalized values of an attribute assigned to the alternatives (see Table 3) are calculated by using the following: $r_{ij} = 〈 T_{ij}, I_{ij}, F_{ij} 〉 = {\begin{matrix} r_{ij}^{'} & for beneficial attribute, \\ {\bar{r}}_{ij}^{'} & for non - beneficial attribute . \end{matrix}$ i = 1, 2, …, n ; j = 1, 2, …, m. Where ${\bar{r}}_{ij}^{'}$ is the complement of $r_{ij}^{'}$ , such that ${\bar{r}}_{ij}^{'} = 〈 F_{ij}, 1 - I_{ij}, T_{ij} 〉$ .

Table 3

Single valued neutrosophic decision matrix R = (r_ij) _6×4 of normalized data

Weapons	a ₁	a ₂	a ₃	a ₄
x ₁	〈0.5, 0.3, 0.6〉	〈0.6, 0.3, 0.2〉	〈0.1, 0.5, 0.4〉	〈0.1, 0.7, 0.5〉
x ₂	〈0.6, 0.1, 0.2〉	〈0.2, 0.1, 0.4〉	〈0.4, 0.7, 0.2〉	〈0.3, 0.4, 0.1〉
x ₃	〈0.1, 0.5, 0.3〉	〈0.3, 0.2, 0.5〉	〈0.1, 0.8, 0.7〉	〈0.5, 0.1, 0.2〉
x ₄	〈0.3, 0.4, 0.2〉	〈0.4, 0.5, 0.1〉	〈0.4, 0.9, 0.3〉	〈0.5, 0.3, 0.4〉
x ₅	〈0.1, 0.2, 0.4〉	〈0.2, 0.7, 0.3〉	〈0.5, 0.7, 0.1〉	〈0.2, 0.1, 0.5〉
x ₆	〈0.5, 0.1, 0.7〉	〈0.5, 0.1, 0.4〉	〈0.6, 0.8, 0.3〉	〈0.4, 0.2, 0.6〉

Relative importance of attributes is also assigned (see Table 2 in [19]). Let the decision maker select the following assignments: $\begin{matrix} \begin{matrix} a_{1} & a_{2} & a_{3} & a_{4} \end{matrix} \\ R = & \begin{matrix} a_{1} \\ a_{2} \\ a_{3} \\ a_{4} \end{matrix} [\begin{matrix} - - - & 〈 0.045, 0.410, 0.865 〉 & 〈 0.665, 0.045, 0.335 〉 & 〈 0.045, 0.590, 0.745 〉 \\ 〈 0.865, 0.590, 0.045 〉 & - - - & 〈 0.135, 0.665, 0.335 〉 & 〈 0.590, 0.410, 0.255 〉 \\ 〈 0.335, 0.955, 0.665 〉 & 〈 0.335, 0.335, 0.135 〉 & - - - & 〈 0.410, 0.255, 0.135 〉 \\ 〈 0.745, 0.410, 0.045 〉 & 〈 0.255, 0.590, 0.590 〉 & 〈 0.135, 0.745, 0.410 〉 & - - - \end{matrix}] . \end{matrix}$

The weapon selection attributes SVNDG given in Fig. 7, represents the presence as well as relative importance of four attributes a₁, a₂, a₃ and a₄ which are the vertices of the digraph. The weapon selection index is calculated using the values of A_i and a_ij for each alternative weapon, where A_i is the value of i-th attribute represented by the weapon x_i and a_ij is the relative importance of the i-th attribute over j-th atribute.

Fig.7

Weapon selection attributes SVNDG.

For first weapon x₁, substituting values of A₁, A₂, A₃ and A₄ in above matrix R, we get $\begin{matrix} \begin{matrix} a_{1} & a_{2} & a_{3} & a_{4} \end{matrix} \\ R_{1} = & \begin{matrix} a_{1} \\ a_{2} \\ a_{3} \\ a_{4} \end{matrix} [\begin{matrix} 〈 0.5, 0.3, 0.6 〉 & 〈 0.045, 0.410, 0.865 〉 & 〈 0.665, 0.045, 0.335 〉 & 〈 0.045, 0.590, 0.745 〉 \\ 〈 0.865, 0.590, 0.045 〉 & 〈 0.6, 0.3, 0.2 〉 & 〈 0.135, 0.665, 0.335 〉 & 〈 0.590, 0.410, 0.255 〉 \\ 〈 0.335, 0.955, 0.665 〉 & 〈 0.335, 0.335, 0.135 〉 & 〈 0.1, 0.5, 0.4 〉 & 〈 0.410, 0.255, 0.135 〉 \\ 〈 0.745, 0.410, 0.045 〉 & 〈 0.255, 0.590, 0.590 〉 & 〈 0.135, 0.745, 0.410 〉 & 〈 0.1, 0.7, 0.5 〉 \end{matrix}] . \end{matrix}$

Now we calculate the permanent function value of above matrix using computer program, that is, per(R₁) = 〈0.4117, 1.3482, 0.4884〉. The permanent function is nothing but the determinant of a matrix but considering all the determinant terms as positive terms [14]. So, the weapon selection index values of different weapons are: $\begin{matrix} x_{1} = 〈 0.4117, 1.3482, 0.4884 〉, \\ x_{2} = 〈 0.4224, 1.0522, 0.3415 〉, \\ x_{3} = 〈 0.4098, 1.1991, 0.4782 〉, \\ x_{4} = 〈 0.5173, 1.5801, 0.3468 〉, \\ x_{5} = 〈 0.3272, 1.3426, 0.4429 〉, \\ x_{6} = 〈 0.6113, 0.9950, 0.6179 〉 . \end{matrix}$

Calculate the score function s (x_i) = T_i + 1 - I_i + 1 - F_i [32] of the weapons x_i (i = 1, 2, 3, 4, 5, 6), respectively: s (x₁) = 0.5751, s (x₂) = 1.0287, s (x₃) = 0.7325, s (x₄) = 0.5904, s (x₅) = 0.5417, s (x₆) =0.9984 . Thus, we can rank the weapons:

$x_{2} ≻ x_{6} ≻ x_{3} ≻ x_{4} ≻ x_{1} ≻ x_{5} .$

Therefore, the best choice is the second weapon (x₂).

5 Conclusion

SVNS as an instance of a neutrosophic set provide an additional possibility to represent imprecise, uncertainty, inconsistent and incomplete information which exist in real situations. Single valued neutrosophic models are more flexible and practical than fuzzy and intuitionistic fuzzy models. Hence, we have introduced the concepts of regular, edge regular, partially edge regular, full edge regular, strongly regular and biregular graphs under single valued neutrosophic environment and investigated their properties. We have also introduced the notion of SVNDGs and presented its application in MADM. In further work, it is necessary and meaningful to extend the regularity of SVNGs to interval neutrosophic regular graphs and their applications, such as decision making, pattern recognition, and medical diagnosis.

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