Application of neutrosophic optimal network using operations

Abstract

Neutrosophic graphs deals with more complex, uncertain problems in real-life applications which provides more flexibility and compatibility than Intuitionistic fuzzy graphs. The aim of this paper is to enrich the efficiency of the network in accordance with productivity and quality. Here we develop two Neutrosophic graphs into a fully connected Neutrosophic network using the product of graphs. Such a type of network is formed from individuals with unique aspects in every field of work among them. This study proposes extending the other graph products and forming a single valued Neutrosophic graph to find the efficient productivity in the flow of information on a single source network of a single valued Neutrosophic network. An Optimal algorithm is proposed and illustrated with an application.

1 Introduction

Graph Theory, a convenient mathematical tool has a broad spectrum of uses in various fields of Science and Technology. The graph is usually a graphical representation of practical, real-world problems. A graph is a collection of sets $(V,E)$ where $V$ is a non-empty set of vertices connected by $E$ , whose constituents are edges or links. Representing a problem as a graph provides a significant perspective and clarifies the situation. A network is typically a graph model with a set of nodes connected by edges or links. The network gives us a flexible framework for identifying and observing complex systems. The study of complex networks is a crucial concept that comprises several disciplines. Complex systems network theory provides techniques for analyzing systems of interaction structure, represented as networks [10]. These networks are generally defined by simple graphs that consist of vertex representing the objects under exploration, that are linked together by edges if there exists a relationship between them. Lofti Zadeh [19] introduced a novel concept of fuzzy set theory in 1965 to model real-world problems that are efficient, which are generally uncertain. Fuzzy is an upper version of the crisp set with varying membership value grades between [0, 1]. The membership value is a specific single value that lies between zero and one. To deal with uncertainties of complex problems, with single membership grade is important. Atanassov [5] extended the fuzzy set to an Intuitionistic fuzzy set by including a non-membership value. Smarandache [3] has presented the idea of a Neutrosophic set to capture problems that are uncertain, imprecise, and vague. A Neutrosophic set is an extension of the crisp set, fuzzy set, and Intuitionistic fuzzy set which has three different types of membership values such as truth, indeterminacy, and falsity values that are not dependent on each other and lies between [0, 1].

There exists many different of information in real-world problems that can be modeled using several types of graphs such as fuzzy graphs, Intuitionistic fuzzy graphs and Neutrosophic graphs [13, 18]. Shannon and Atanassov introduced the concept of Intuitionistic fuzzy graphs [14]. Parvathi et al. [9] proposed some operations between two Intuitionistic fuzzy graphs. Rashmanlou et al. [13] proposed graph operations such as Direct product, semi-strong product, strong product, and Lexicographic on Intuitionistic fuzzy graphs. Mahapatra et al. [15] introduced the fuzzy fractional chromatic number for calculating lexicographic product on two fuzzy graphs, also investigated m-polar fuzzy graphs and their applications [16, 17]. Neutrosophic graphs are used to model real-world problems which consist of in-consistent information. Many Scientists such as Broumi et al. [7], Yang et al. [1] and Akram [3, 4] have researched under a Neutrosophic environment. Single-valued Neutrosophic sets introduced by Haibin Wang are a subclass of Neutrosophic sets that are independent of membership values ranging from [0, 1]. Related work in the extension of the single-valued Neutrosophic set is found in [1, 6].

The main motivation of this research work is to find the most efficient optical network using different operations on single-valued Neutrosophic Graphs- (SVNG) such as Lexicographic, Symmetric difference, Residue product, and Max product based on the domination parameter presented. Further extended our study on its applications and finding the effective minimal spanning tree. In section 2 the motivation and research background is listed with preliminaries for the study. In section 3 we define the different types of operations such as Lexicographic, Symmetric difference, Residue product, Max product and examine the efficiency of the network using the score function. In section 4 the optimal network of symmetric difference is identified and its applications are given for better sales training technology.

2 Preliminaries

Definition 2.1. [2] Let $X$ be a universe of discourse. A single-valued Neutrosophic set N is defined on $X$ is given by $N=\left\{\right(x,T_N\left(x\right),I_N\left(x\right),F_N\left(x\right):x\in X\}$ where $\begin{array}{ccc}\hfill T_N\left(x\right)& \hfill :\hfill & X\to [0,1]\hfill \\ \hfill I_N\left(x\right)& \hfill :\hfill & X\to [0,1]\mathrm{and}\hfill \\ \hfill F_N\left(x\right)& \hfill :\hfill & X\to [0,1]\hfill \end{array}$ are called the degree of truth membership value, degree of indeterminancy value and degree of falsity membership value of x on N, respectively satisfying the condition $0\le T_N\left(x\right)+I_N\left(x\right)+F_N\left(x\right)\le 3,\forall x\in X.$

Definition 2.2. [2] Let $N_G=(V_G,E_G)$ be a graph, where $V_G$ be the set of vertices and $E_G$ be the set of edges. Then the single-valued Neutrosophic graph of N _G is denoted by ${\mathbb{N}}_G^{\prime }=(V_G,\sigma ,\mu )$ where σ = (T _σ, I _σ, F _σ) is a single-valued Neutrosophic set on $V_G$ and μ = (T _μ, I _μ, F _μ) is a single-valued Neutrosophic symmetric relation on $E_G\subseteq V_G\times V_G$ is defined as follows:

T _μ (x, y) ≤ T _σ (x) ∧ T _σ (y) ,    ∀ (x, y) ∈ V _G  × V _G .

I _μ (x, y) ≤ I _σ (x) ∧ I _σ (y) ,    ∀   (x, y) ∈ V _G  × V _G .

F _μ (x, y) ≥ F _σ (x) ∨ F _σ (y) ,    ∀   (x, y) ∈ V _G  × V _G .

Definition 2.3. [8] The SVNG ${\mathbb{N}}_G^{\prime }$ is called strong single-valued Neutrosophic graph if $\forall (x,y)\in E_G$ , $\begin{array}{ccc}\hfill T_{{\mu }_{{\mathbb{N}}_G^{\prime }}}(x,y)& \hfill =\hfill & T_{{\sigma }_{{\mathbb{N}}_G^{\prime }}}\left(x\right)\wedge T_{{\sigma }_{{\mathbb{N}}_G^{\prime }}}\left(y\right),\hfill \\ \hfill I_{{\mu }_{{\mathbb{N}}_G^{\prime }}}(x,y)& \hfill =\hfill & I_{{\sigma }_{{\mathbb{N}}_G^{\prime }}}\left(x\right)\wedge I_{{\sigma }_{{\mathbb{N}}_G^{\prime }}}\left(y\right)\hfill \\ \hfill F_{{\mu }_{{\mathbb{N}}_G^{\prime }}}(x,y)& \hfill =\hfill & F_{{\sigma }_{{\mathbb{N}}_G^{\prime }}}\left(x\right)\vee F_{{\sigma }_{{\mathbb{N}}_G^{\prime }}}\left(y\right)\hfill \end{array}$

Definition 2.4. [8] The SVNG ${\mathbb{N}}_G^{\prime }$ is said to be complete if $\forall x,y\in V$ , $\begin{array}{ccc}\hfill T_{{\mu }_{{\mathbb{N}}_G^{\prime }}}(x,y)& \hfill =\hfill & T_{{\sigma }_{{\mathbb{N}}_G^{\prime }}}\left(x\right)\wedge T_{{\sigma }_{{\mathbb{N}}_G^{\prime }}}\left(y\right)\hfill \\ \hfill I_{{\mu }_{{\mathbb{N}}_G^{\prime }}}(x,y)& \hfill =\hfill & I_{{\sigma }_{{\mathbb{N}}_G^{\prime }}}\left(x\right)\wedge I_{{\sigma }_{{\mathbb{N}}_G^{\prime }}}\left(y\right)\hfill \\ \hfill F_{{\mu }_{{\mathbb{N}}_G^{\prime }}}(x,y)& \hfill =\hfill & F_{{\sigma }_{{\mathbb{N}}_G^{\prime }}}\left(x\right)\vee F_{{\sigma }_{{\mathbb{N}}_G^{\prime }}}\left(y\right).\hfill \end{array}$

Definition 2.5. [8] Let u _d be a vertex in a ${\mathbb{N}}_G,G_{{\mathbb{N}}_G}=(A_N,B_N)$ . The degree of a vertex u _d is defined as the sum of the weight of the strong arcs incident at u _d and is denoted by deg (u _d ). The neighborhood of u _d is denoted by ${\mathbb{N}}_G=\{v_d\in A_N/(u_d,v_d\left)\right\}$ is a strong arc.

The minimum degree of $G_{{\mathbb{N}}_G}$ is

$\delta \left(G_{{\mathbb{N}}_G}\right)=min\left\{d_{G_{{\mathbb{N}}_G}}\left(u_d\right)/u_d\in A_N\right\}$ .

The maximum degree of $G_{{\mathbb{N}}_G}$ is

$\Delta \left(G_{{\mathbb{N}}_G}\right)=max\left\{d_{G_{{\mathbb{N}}_G}}\left(u_d\right)/u_d\in A_N\right\}.$

Definition 2.6. [12] The cardinality of a vertex $v_i\in V$ in a SVNG, ${\mathbb{N}}_G=(A,B)$ is defined by $\left|V_i\right|=T_A\left(v_i\right)+I_A\left(v_i\right)+F_A\left(v_i\right).$

The cardinality of an edge $v_i{v}_j\in E$ in a SVNG, ${\mathbb{N}}_G=(A,B)$ is defined by $\left|v_i{v}_j\right|=T_B\left(v_i{v}_j\right)+I_B\left(v_i{v}_j\right)+F_B\left(v_i{v}_j\right).$

Definition 2.7. [2] Let $C_{P1}=(A_{C_{P1}},B_{C_{P1}})$ and $C_{P2}=(A_{C_{P2}},B_{C_{P2}})$ be the two SVNG of $G_1=(V_1,E_1)$ and $G_2=(V_2,E_2)$ respectively.

The cartesian product $C_{P1}\times C_{P2}$ is defined such that

$\begin{array}{ccc}\hfill T_A(x_1,x_2)& \hfill =\hfill & min\left(T_{A_{C_{P1}}}\left(x_1\right),T_{A_{C_{P2}}}\left(x_2\right)\right);\hfill \\ \hfill I_A(x_1,x_2)& \hfill =\hfill & min\left(I_{A_{C_{P1}}}\left(x_1\right),I_{A_{C_{P2}}}\left(x_2\right)\right);\hfill \\ \hfill F_A(x_1,x_2)& \hfill =\hfill & max\left(F_{A_{C_{P1}}}\left(x_1\right),F_{A_{C_{P2}}}\left(x_2\right)\right),\hfill \\ \hfill & \hfill \hfill & \forall (x_1,x_2)\in V_1\times V_2.\hfill \\ \hfill \end{array}$

$\begin{array}{ccc}\hfill T_B\left((x,x_2)(x,y_2)\right)& \hfill =\hfill & min\left(T_{A_{C_{P1}}\left(x\right)},T_{B_{C_{P2}\left(x\right)}}\right);\hfill \\ \hfill I_B\left((x,x_2)(x,y_2)\right)& \hfill =\hfill & min\left(I_{A_{C_{P1}}\left(x\right)},I_{B_{C_{P2}}(x_2,y_2)}\right);\hfill \\ \hfill F_B\left((x,x_2)(x,y_2)\right)& \hfill =\hfill & max\left(F_{A_{C_{P1}}\left(x\right)},F_{B_{C_{P2}}\left(x_2{y}_2\right)}\right),\hfill \\ \hfill & \hfill \hfill & \forall x\in V_1,x_2{y}_2\in E_{P2}.\hfill \end{array}$

$\begin{array}{ccc}\hfill T_B\left((x_1,z)(y_1,z)\right)& \hfill =\hfill & min\left(T_{B_{C_{P1}}\left(x_1{y}_1\right)},T_{A_{C_{P2}}\left(z\right)}\right)\hfill \\ \hfill I_B\left((x_1,z)(y_1,z)\right)& \hfill =\hfill & min\left(I_{B_{C_{P1}}(x_1,y_1)},I_{A_{C_{P2}}\left(z\right)}\right)\hfill \\ \hfill F_B\left((x_1,z)(y_1,z)\right)& \hfill =\hfill & max\left(F_{B_{C_{P1}}\left(x_1{y}_1\right)},F_{A_{C_{P2}}\left(z\right)}\right),\hfill \\ \hfill & \hfill \hfill & \forall z\in V_2\mathrm{and}{x}_1{y}_1\in E_1.\hfill \end{array}$ The cartesian product $G_1\times G_2$ of two graphs $G_1$ and $G_2$ is denoted by $V\left(G_1\right)\times V\left(G_2\right)$ such that two vertices $(V_1,V_2)$ and $(V_1^{\prime },V_2^{\prime })$ are adjacent in $G_1\times G_2$ iff

$V_1=V_1^{\prime }$ and $V_2$ is adjacent to $V_2^{\prime }$ in $G_2$ .

$V_2=V_2^{\prime }$ and $V_1$ is adjacent to $V_1^{\prime }$ in $G_1$ .

Definition 2.8. [2] The Lexicographic product LP₁ · LP₂ of two graphs LP₁ = (M₁, N₁) and LP₂ = (M₂, N₂) is such that

The vertex set of LP₁ · LP₂ is the cartesian product V (LP₁) × V (LP₂).

Any two vertices (m₁, n₁) and (m₂, n₂) are adjacent in LP₁ · LP₂ iff either m₁ is adjacent to m₂ in LP₁ or m₁ = m₂ and n₁ is adjacent to n₁ in LP₂

Definition 2.9. [8] The residue product of RP₁ · RP₂ two graphs RP₁ and RP₂ is defined as

σ_RP1·RP2 (u₁, v₁) = σ _{RP 1}  (u₁) ∨ σ _{RP 2}  (v₁)

and μ_RP1·RP2 ((u₁, v₁) (u₂, v₂)) = μ _{RP 1}  (u₁u₂),

$\forall (u_1,v_1)\in V\mathrm{and}(u_1,v_1)(u_2,v_2)\in E$ .

If $u_1{u}_2\in E_1\mathit{and}{v}_1\ne v_2$ then,

μ_RP1·RP2 ((u₁, v₁) (u₂, v₂)) ≤ σ_RP1(u1,v1) ∧ σ_RP2(u2,v2).

Definition 2.10. [2] Let SD₁ = (σ₁, μ₁) and SD₂ = (σ₂, μ₂) be two SVNGs of the graphs G _{SD 1}  = (V₁, E₁) and G _{SD 2}  = (V₂, E₂) respectively. Then the symmetric difference of SD₁ and SD₂ is defined as $\begin{array}{cc}\hfill & {\mathit{SD}}_1\oplus {\mathit{SD}}_2\hfill \\ \hfill & =({\sigma }_1\oplus {\sigma }_2,{\mu }_1\oplus {\mu }_2)\mathrm{is}\mathrm{defined}\mathrm{as}\mathrm{follows}\hfill \end{array}$

∀  (x, y) ∈ V₁ × V₂. $\begin{array}{ccc}\hfill T_{{\sigma }_{{\mathit{SD}}_1}}\oplus T_{{\sigma }_{{\mathit{SD}}_2}}(x,y)& \hfill =\hfill & T_{{\sigma }_{{\mathit{SD}}_1}}\left(x\right)\wedge T_{{\sigma }_{{\mathit{SD}}_2}}\left(y\right);\hfill \\ \hfill I_{{\sigma }_{{\mathit{SD}}_1}}\oplus I_{{\sigma }_{{\mathit{SD}}_2}}(x,y)& \hfill =\hfill & I_{{\sigma }_{{\mathit{SD}}_1}}\left(x\right)\wedge I_{{\sigma }_{{\mathit{SD}}_2}}\left(y\right)\mathrm{and}\hfill \\ \hfill F_{{\sigma }_{{\mathit{SD}}_1}}\oplus F_{{\sigma }_{{\mathit{SD}}_2}}(x,y)& \hfill =\hfill & T_{{\sigma }_{{\mathit{SD}}_1}}\left(x\right)\vee F_{{\sigma }_{{\mathit{SD}}_2}}\left(y\right);\hfill \end{array}$

∀ x ∈ V₁ and (y, z) ∈ E₂, $\begin{array}{ccc}\hfill & \hfill \hfill & (T_{{\mu }_{{\mathit{SD}}_1}}\oplus T_{{\mu }_{{\mathit{SD}}_2}})\left(\right(x,y),(x,z\left)\right)\hfill \\ \hfill & \hfill \hfill & =T_{{\sigma }_{{\mathit{SD}}_1}}\left(x\right)\wedge T_{{\mu }_{{\mathit{SD}}_2}}(y,z);\hfill \\ \hfill & \hfill \hfill & (I_{{\mu }_{{\mathit{SD}}_1}}\oplus I_{{\mu }_{{\mathit{SD}}_2}})\left(\right(x,y),(x,z\left)\right)\hfill \\ \hfill & \hfill \hfill & =I_{{\sigma }_{{\mathit{SD}}_1}}\left(x\right)\wedge I_{{\mu }_{{\mathit{SD}}_2}}(y,z);\hfill \\ \hfill & \hfill \hfill & (F_{{\mu }_{{\mathit{SD}}_1}}\oplus F_{{\mu }_{{\mathit{SD}}_2}})\left(\right(x,y),(x,z\left)\right)\hfill \\ \hfill & \hfill \hfill & =F_{{\sigma }_{{\mathit{SD}}_1}}\left(x\right)\vee F_{{\mu }_{{\mathit{SD}}_2}}(y,z);\hfill \end{array}$

∀ x ∈ V₂ and (y, z) ∈ E₁, $\begin{array}{ccc}\hfill & \hfill \hfill & (T_{{\mu }_{{\mathit{SD}}_1}}\oplus T_{{\mu }_{{\mathit{SD}}_2}})\left(\right(y,x),(z,x\left)\right)\hfill \\ \hfill & \hfill \hfill & =T_{{\mu }_{{\mathit{SD}}_1}}(y,z)\wedge T_{{\sigma }_{{\mathit{SD}}_2}}\left(x\right);\hfill \\ \hfill & \hfill \hfill & (I_{{\mu }_{{\mathit{SD}}_1}}\oplus I_{{\mu }_{{\mathit{SD}}_2}})\left(\right(y,x),(z,x\left)\right)\hfill \\ \hfill & \hfill \hfill & =I_{{\mu }_{{\mathit{SD}}_1}}(y,z)\wedge I_{{\sigma }_{{\mathit{SD}}_2}}\left(x\right);\hfill \\ \hfill & \hfill \hfill & (F_{{\mu }_{{\mathit{SD}}_1}}\oplus F_{{\mu }_{{\mathit{SD}}_2}})\left(\right(y,x),(z,x\left)\right)\hfill \\ \hfill & \hfill \hfill & =F_{{\mu }_{{\mathit{SD}}_1}}(y,z)\vee F_{{\sigma }_{{\mathit{SD}}_2}}\left(x\right);\hfill \end{array}$

∀  (x, y) ∉ E₁ and (z, w) ∈ E₂, $\begin{array}{ccc}\hfill & \hfill \hfill & (T_{{\mu }_{{\mathit{SD}}_1}}\oplus T_{{\mu }_{{\mathit{SD}}_2}})\left(\right(x,z),(y,w\left)\right)\hfill \\ \hfill & \hfill \hfill & =T_{{\sigma }_{{\mathit{SD}}_1}}\left(x\right)\wedge T_{{\sigma }_{{\mathit{SD}}_1}}\left(y\right)\wedge T_{{\mu }_{{\mathit{SD}}_2}}(z,w);\hfill \\ \hfill & \hfill \hfill & (I_{{\mu }_{{\mathit{SD}}_1}}\oplus I_{{\mu }_{{\mathit{SD}}_2}})\left(\right(x,z),(y,w\left)\right)\hfill \\ \hfill & \hfill \hfill & =I_{{\sigma }_{{\mathit{SD}}_1}}\left(x\right)\wedge I_{{\sigma }_{{\mathit{SD}}_1}}\left(y\right)\wedge I_{{\mu }_{{\mathit{SD}}_2}}(z,w);\hfill \\ \hfill & \hfill \hfill & (F_{{\mu }_{{\mathit{SD}}_1}}\oplus F_{{\mu }_{{\mathit{SD}}_2}})\left(\right(x,z),(y,w\left)\right)\hfill \\ \hfill & \hfill \hfill & =F_{{\sigma }_{{\mathit{SD}}_1}}\left(x\right)\wedge F_{{\sigma }_{{\mathit{SD}}_1}}\left(y\right)\wedge F_{{\mu }_{{\mathit{SD}}_2}}(z,w);\hfill \end{array}$

∀  (x, y) ∈ E₁ and (z, w) ∉ E₂, $\begin{array}{ccc}\hfill & \hfill \hfill & (T_{{\mu }_{{\mathit{SD}}_1}}\oplus T_{{\mu }_{{\mathit{SD}}_2}})\left(\right(x,z),(y,w\left)\right)\hfill \\ \hfill & \hfill \hfill & =T_{{\mu }_{{\mathit{SD}}_1}}(x,y)\wedge T_{{\sigma }_{{\mathit{SD}}_2}}\left(z\right)\wedge T_{{\sigma }_{{\mathit{SD}}_2}}\left(w\right);\hfill \\ \hfill & \hfill \hfill & (I_{{\mu }_{{\mathit{SD}}_1}}\oplus I_{{\mu }_{{\mathit{SD}}_2}})\left(\right(x,z),(y,w\left)\right)\hfill \\ \hfill & \hfill \hfill & =I_{{\mu }_{{\mathit{SD}}_1}}(x,y)\wedge I_{{\sigma }_{{\mathit{SD}}_2}}\left(z\right)\wedge I_{{\sigma }_{{\mathit{SD}}_2}}\left(w\right);\hfill \\ \hfill & \hfill \hfill & (F_{{\mu }_{{\mathit{SD}}_1}}\oplus F_{{\mu }_{{\mathit{SD}}_2}})\left(\right(x,z),(y,w\left)\right)\hfill \\ \hfill & \hfill \hfill & =F_{{\mu }_{{\mathit{SD}}_1}}(x,y)\vee F_{{\sigma }_{{\mathit{SD}}_2}}\left(z\right)\vee F_{{\sigma }_{{\mathit{SD}}_2}}\left(w\right).\hfill \end{array}$

Definition 2.11. [8] Let

${\mathit{MP}}_1=\left(\right({\sigma }_{{\mathit{mp}}_1},{\sigma }_{{\mathit{mp}}_2},({\mu }_{{\mathit{mp}}_1},{\mu }_{{\mathit{mp}}_2})),{\mathit{MP}}_2=(({\sigma }_{{\mathit{mp}}_1}^{\prime },{\sigma }_{{\mathit{mp}}_2}^{\prime }),({\mu }_{{\mathit{mp}}_1}^{\prime },{\mu }_{{\mathit{mp}}_2}^{\prime }))$ be two Intuitionistic fuzzy graph.

The Max product of two Intuitionistic fuzzy graph MP₁, MP₂ and is denoted by

MP₁ * MP₂ $(V_1^{″}{\times }_M{V}_2^{″},E_1^{″}{\times }_M{E}_2^{″})$

where $E_1^{″}{\times }_M{E}_2^{″}=\left\{\right(p_1^{″},q_1^{″}\left)\right((p_2^{″},q_2^{″})/$ $p_1^{″}=p_2^{″}$ ; $q_1^{″}{q}_2^{″}\in E_2^{″}$ or $q_1^{″}=q_2^{″}$ ; $p_1^{″}{p}_2^{″}\in E_1^{″}\}$

${\sigma }_{{\mathit{mp}}_1}^{{\mathit{MP}}_1*{\mathit{MP}}_2}(p_1^{″},q_1^{″})={\sigma }_{{\mathit{mp}}_1}\left(p_1^{″}\right)\vee {\sigma }_{{\mathit{mp}}_1}^{\prime }\left(q_1^{″}\right)$ for all $(p_1,q_1)\in \left(V_1^{″}{\times }_M{V}_2^{″}\right)$

${\sigma }_{{\mathit{mp}}_2}^{{\mathit{MP}}_1*{\mathit{MP}}_2}(p_1^{″},q_1^{″})={\sigma }_{{\mathit{mp}}_2}\left(p_1^{″}\right)\wedge {\sigma }_{{\mathit{mp}}_2}^{\prime }\left(q_1^{″}\right)$ for all $(p_1^{″},q_1^{″})\in \left(V_1^{″}{\times }_M{V}_2^{″}\right)$ and

${\mu }_{{\mathit{mp}}_1}^{\prime }{\mathit{MP}}_1*{\mathit{MP}}_2\left(\right(p_1^{″},q_1^{″}\left)\right(p_2^{″},q_2^{″}\left)\right)=\left\{{\sigma }_{{\mathit{mp}}_1}\right(p_1^{″})\vee {\sigma }_{{\mathit{mp}}_1}(q_1^{″})$ if $p_1^{″}=p_2^{″}$ ; $q_1^{″}{q}_2^{″}\in E_2^{″},{\sigma }_{{\mathit{mp}}_1}\left(p_1^{″}\right)\vee {\mu }_{{\mathit{mp}}_1}^{\prime }\left(q_1^{″}{q}_2^{″}\right)$ if $q_1^{″}=q_2^{″};p_1^{″}{p}_2^{″}\in E_{1^{″}}$

${\mu }_{{\mathit{mp}}_2}^{\prime }{\mathit{MP}}_1*{\mathit{MP}}_2\left(\right(p_1^{″},q_1^{″}\left)\right(p_2^{″},q_2^{″}\left)\right)=\left\{{\sigma }_{{\mathit{mp}}_2}\right(p_1^{″})\wedge {\sigma }_{{\mathit{mp}}_2}(q_1^{″})$ if $p_1^{″}=p_2^{″};q_1^{″}{q}_2^{″}\in E_2^{″}$

${\sigma }_{{\mathit{mp}}_2}\left(p_1^{″}\right)\wedge {\mu }_{{\mathit{mp}}_2}^{\prime }\left(q_1^{″}{q}_2^{″}\right)$ if $q_1^{″}=q_2^{″};p_1^{″}{p}_2^{″}\in E_1^{″}$ }

Definition 2.12. [12] In a SVNG, N _G  = (A, B), the domination number is defined by the minimum cardinality among all the minimal dominating set of N _G and it is denoted by γ _SVN  (N _G ).

Definition 2.13. [20] Let $A=(T,I,F)$ be a SVNG, then the score function S is defined as follows

$S\left(A\right)=\frac{2+T-I-F}{3}$

Let $G_1=({\mathcal{V}}_1,{\mathcal{E}}_1)$ and $G_2=({\mathcal{V}}_2,{\mathcal{E}}_2)$ be two SVNG’s, while applying any operation ’*’ on the SVNG’s $G_1$ and $G_2$ such that the required SVNG $G_1*G_2$ contains the vertex set $({\mathcal{V}}_1*{\mathcal{V}}_2)$ , cartesian product of ${\mathcal{V}}_1$ and ${\mathcal{V}}_2$ . Selecting any such operation, which satifies the above condition will be applied on any two SVNG’s and it can be constructed into a network. In this paper, we present such different operations on SVNG like lexicographic, symmetric difference, maximal product, and residue product are presented with appropriate examples. We have modeled a real-life problem for selecting the best optimal network using SVNG and those operators are used to find the most efficient one.

The operations on single-valued Neutrosophic Graphs (SVNG) such as Lexicographic, Symmetric difference, Residue product and Max product are studied from [2, 8].

3 Domination on operations of single-valued Neutrosophic graphs

3.1 Lexicographic product of two single-valued Neutrosophic graphs

3.1.1 Definition

Let LP₁ = (M₁, N₁) and LP₂ = (M₂, N₂) be two single-valued Neutrosophic networks of the graphs G _{LP 1}  = (V₁, E₁) and G _{LP 2}  = (V₂, E₂) respectively. The Lexicographic product graph is denoted as ${L\mathbb{P}}_1·{L\mathbb{P}}_2$ is the pair (M, N) of single-valued Neutrosophic graph such that

$\begin{array}{ccc}\hfill T_M(x_1,x_2)& \hfill =\hfill & min\left(T_{M_1}\left(x_1\right),T_{M_2}\left(x_2\right)\right)\hfill \\ \hfill I_M(x_1,x_2)& \hfill =\hfill & min\left(I_{M_1}\left(x_1\right),I_{M_2}\left(x_2\right)\right)\hfill \\ \hfill F_M\left(x_1,x_2\right)& \hfill =\hfill & max\left(F_{M_1}\left(x_1\right),F_{M_2}\left(x_2\right)\right)\hfill \\ \hfill & \hfill \hfill & \forall (x_1,x_2)\in M_1\times M_2.\hfill \end{array}$

$\begin{array}{ccc}\hfill & \hfill \hfill & T_N\left((x,x_2)(x,y_2)\right)\hfill \\ \hfill & \hfill \hfill & =min\left(T_{M_1}\left(x\right),T_{M_2}\left(x_2{y}_2\right)\right)\hfill \\ \hfill & \hfill \hfill & I_N\left((x,x_2)(x,y_2)\right)\hfill \\ \hfill & \hfill \hfill & =min\left(I_{M_1}\left(x\right),I_{N_2\left(x_2{y}_2\right)}\right)\hfill \\ \hfill & \hfill \hfill & F_N\left((x,x_2)(x,y_2)\right)\hfill \\ \hfill & \hfill \hfill & =max\left(F_{M_1}\left(x\right),F_{N_2}\left(x_2{y}_2\right)\right)\hfill \\ \hfill & \hfill \hfill & \forall x\in M_1,x_2{y}_2\in N_2.\hfill \end{array}$

$\begin{array}{ccc}\hfill & \hfill \hfill & T_N\left((x_1,x_2)(y_1,y_2)\right)\hfill \\ \hfill & \hfill \hfill & =min\left(T_{N_1}\left(x_1{y}_1\right),T_{N_2}\left(x_2{y}_2\right)\right)\hfill \\ \hfill & \hfill \hfill & I_N\left((x_1,x_2)(y_1,y_2)\right)\hfill \\ \hfill & \hfill \hfill & =min\left(I_{N_1}\left(x_1{y}_1\right),I_{N_2}\left(x_2{y}_2\right)\right)\hfill \\ \hfill & \hfill \hfill & F_N\left((x_1,x_2)(y_1,y_2)\right)\hfill \\ \hfill & \hfill \hfill & =max\left(F_{N_1}\left(x_1{y}_1\right),F_{N_2}\left(x_2{y}_2\right)\right),\hfill \\ \hfill & \hfill \hfill & \forall x_1{y}_1\in N_1\mathrm{and}{x}_2{y}_2\in N_2.\hfill \end{array}$

3.1.2 Example

Let SVNG $G_1$ and SVNG $G_2$ be two Lexicographic SVNN shown in Fig 1 and Fig 2 of the graphs G _{LP 1}  = (V₁, E₁) and G _{LP 2}  = (V₂, E₂) respectively. The Lexicographic product of single-valued Neutrosophic network $L{\mathbb{P}}_1·L{\mathbb{P}}_2$ is shown in Fig. 3.

Fig. 1

SVNG $G_1$ .

Fig. 2

SVNG $G_2$ .

Fig. 3

$L{\mathbb{P}}_1·L{\mathbb{P}}_2$ .

To analyze the optimal network from the constructed network, we define an efficient score function to find the minimum domination number of the weighted SVNG network. The score function defined by us is more efficient than the existing score function defined in 2.13 since, Indeterminacy value (I) does not depend on both Truth (T) and Falsity (F) value because I is not a complement of T and F and the values of T, I, F are independent of each other. Even though the value of indeterminacy is uncertain, we assume it by taking 0.5 as both chances of truth and falsity which makes our work the significant advantage of defining efficient networks.

Hence, we define the Edge score function (ESF) and Vertex score function (VSF) of a single-valued Neutrosophic graph to find the minimum weight of the spanning tree as follows:

$\mathit{ESF}=\frac{2+T_{\mu }(x,y)-\left(0.5\right)I_{\mu }(x,y)-F_{\mu }(x,y)}{3}$ $\mathit{VSF}=\frac{2+T_{\sigma }\left(x\right)-\left(0.5\right)I_{\sigma }\left(x\right)-F_{\sigma }\left(x\right)}{3}$

The weighted $L{\mathbb{P}}_1·L{\mathbb{P}}_2$ represented in Fig 4, and it’s minimal dominating sets are as follows; $\begin{array}{ccc}\hfill S_1& \hfill =\hfill & \{ay,cz,dy\}\hfill \\ \hfill S_2& \hfill =\hfill & \{ay,bz,dy\}\hfill \\ \hfill S_3& \hfill =\hfill & \{ay,bx,dy\}\hfill \\ \hfill S_4& \hfill =\hfill & \{ay,cz,dy\}\hfill \\ \hfill S_5& \hfill =\hfill & \{by,cy,dy,ay\}.\hfill \end{array}$ The vertex cardinality of the members of above dominating sets are ay = 0.4666, cz = 0.5, dy = 0.5333, bz = 0.52, bx = 0.5166, by = 0.55, cy = 0.5 .

Fig. 4

$L{\mathbb{P}}_1·L{\mathbb{P}}_2$ with minimum score function.

The domination number of the dominating sets S₁, S₂, S₃, S₄, S₅ are $\begin{array}{ccc}\hfill & \hfill \hfill & S_1(L{\mathbb{P}}_1·L{\mathbb{P}}_2)\hfill \\ \hfill & \hfill \hfill & =0.4666+0.5+0.5333=1.4999\hfill \\ \hfill & \hfill \hfill & S_2(L{\mathbb{P}}_1·L{\mathbb{P}}_2)\hfill \\ \hfill & \hfill \hfill & =0.4666+0.5166+0.5333=1.5165\hfill \\ \hfill & \hfill \hfill & S_3(L{\mathbb{P}}_1·L{\mathbb{P}}_2)\hfill \\ \hfill & \hfill \hfill & =0.4666+0.55+0.5333=1.5495\hfill \\ \hfill & \hfill \hfill & S_4(L{\mathbb{P}}_1·L{\mathbb{P}}_2)\hfill \\ \hfill & \hfill \hfill & =0.4666+0.5+0.5333=1.4999\hfill \\ \hfill & \hfill \hfill & S_5(L{\mathbb{P}}_1·L{\mathbb{P}}_2)\hfill \\ \hfill & \hfill \hfill & =0.55+0.5+0.5333+0.4666=2.049\hfill \end{array}$ The domination number of $L{\mathbb{P}}_1·L{\mathbb{P}}_2$ is 1.4999 which is obtained from the dominating sets S₁ and S₄.

The minimal spanning tree of the weighted network $L{\mathbb{P}}_1·L{\mathbb{P}}_2$ is found using the Kruskal’s algorithm is shown in Fig 5 and hence, its minimum weight of spanning tree is 5.1996.

Fig. 5

Minimum Spanning Tree of $L{\mathbb{P}}_1·L{\mathbb{P}}_2$ .

3.2 Symmetric difference of two single-valued Neutrosophic graphs

Let SD₁ = (σ₁, μ₁) and SD₂ = (σ₂, μ₂) be two SVNGs of the graphs G _{SD 1}  = (V₁, E₁) & G _{SD 2}  = (V₂, E₂) respectively. Then the symmetric difference of SD₁ & SD₂ is defined and denoted as $\begin{array}{cc}\hfill & S{D}_1\oplus S{D}_2\hfill \\ \hfill & =({\sigma }_1\oplus {\sigma }_2,{\mu }_1\oplus {\mu }_2)\mathrm{is}\mathrm{defined}\mathrm{as}\mathrm{follows}\hfill \end{array}$

∀  (x, y) ∈ V₁ × V₂. $\begin{array}{ccc}\hfill & \hfill \hfill & T_{{\sigma }_{{\mathit{SD}}_1}}\oplus T_{{\sigma }_{{\mathit{SD}}_2}}(x,y)\hfill \\ \hfill & \hfill \hfill & =T_{{\sigma }_{{\mathit{SD}}_1}}\left(x\right)\wedge T_{{\sigma }_{{\mathit{SD}}_2}}\left(y\right),\hfill \\ \hfill & \hfill \hfill & I_{{\sigma }_{{\mathit{SD}}_1}}\oplus I_{{\sigma }_{{\mathit{SD}}_2}}(x,y)\hfill \\ \hfill & \hfill \hfill & =I_{{\sigma }_{{\mathit{SD}}_1}}\left(x\right)\wedge I_{{\sigma }_{{\mathit{SD}}_2}}\left(y\right)\mathrm{and}\hfill \\ \hfill & \hfill \hfill & F_{{\sigma }_{{\mathit{SD}}_1}}\oplus F_{{\sigma }_{{\mathit{SD}}_2}}(x,y)\hfill \\ \hfill & \hfill \hfill & =T_{{\sigma }_{{\mathit{SD}}_1}}\left(x\right)\vee F_{{\sigma }_{{\mathit{SD}}_2}}\left(y\right);\hfill \end{array}$

∀ x ∈ V₁ and (y, z) ∈ E₂,

(T _{μ SD 1}  ⊕ T _{μ SD 2} ) ((x, y) , (x, z)) = T _{σ SD 1}  (x) ∧ T _{μ SD 2}  (y, z);

(I _{μ SD 1}  ⊕ I _{μ SD 2} ) ((x, y) , (x, z)) = I _{σ SD 1}  (x) ∧ I _{μ SD 2}  (y, z);

(F _{μ SD 1}  ⊕ F _{μ SD 2} ) ((x, y) , (x, z)) = F _{σ SD 1}  (x) ∨ F _{μ SD 2}  (y, z);

∀ x ∈ V₂ and (y, z) ∈ E₁, $\begin{array}{ccc}\hfill & \hfill \hfill & (T_{{\mu }_{{\mathit{SD}}_1}}\oplus T_{{\mu }_{{\mathit{SD}}_2}})\left(\right(y,x),(z,x\left)\right)\hfill \\ \hfill & \hfill \hfill & =T_{{\mu }_{{\mathit{SD}}_1}}(y,z)\wedge T_{{\sigma }_{{\mathit{SD}}_2}}\left(x\right);\hfill \\ \hfill & \hfill \hfill & (I_{{\mu }_{{\mathit{SD}}_1}}\oplus I_{{\mu }_{{\mathit{SD}}_2}})\left(\right(y,x),(z,x\left)\right)\hfill \\ \hfill & \hfill \hfill & =I_{{\mu }_{{\mathit{SD}}_1}}(y,z)\wedge I_{{\sigma }_{{\mathit{SD}}_2}}\left(x\right);\hfill \\ \hfill & \hfill \hfill & (F_{{\mu }_{{\mathit{SD}}_1}}\oplus F_{{\mu }_{{\mathit{SD}}_2}})\left(\right(y,x),(z,x\left)\right)\hfill \\ \hfill & \hfill \hfill & =F_{{\mu }_{{\mathit{SD}}_1}}(y,z)\vee F_{{\sigma }_{{\mathit{SD}}_2}}\left(x\right);\hfill \end{array}$

∀  (x, y) ∉ E₁ and (z, w) ∈ E₂, $\begin{array}{ccc}\hfill & \hfill \hfill & (T_{{\mu }_{{\mathit{SD}}_1}}\oplus T_{{\mu }_{{\mathit{SD}}_2}})\left(\right(x,z),(y,w\left)\right)\hfill \\ \hfill & \hfill \hfill & =min\left\{T_{{\sigma }_{{\mathit{SD}}_1}}\right(x),T_{{\sigma }_{{\mathit{SD}}_1}}(y),T_{{\mu }_{{\mathit{SD}}_2}}(z,w\left)\right\};\hfill \\ \hfill & \hfill \hfill & (I_{{\mu }_{{\mathit{SD}}_1}}\oplus I_{{\mu }_{{\mathit{SD}}_2}})\left(\right(x,z),(y,w\left)\right)\hfill \\ \hfill & \hfill \hfill & =min\left\{I_{{\sigma }_{{\mathit{SD}}_1}}\right(x),I_{{\sigma }_{{\mathit{SD}}_1}}(y),I_{{\mu }_{{\mathit{SD}}_2}}(z,w\left)\right\};\hfill \\ \hfill & \hfill \hfill & (F_{{\mu }_{{\mathit{SD}}_1}}\oplus F_{{\mu }_{{\mathit{SD}}_2}})\left(\right(x,z),(y,w\left)\right)\hfill \\ \hfill & \hfill \hfill & =max\left\{F_{{\sigma }_{{\mathit{SD}}_1}}\right(x),F_{{\sigma }_{{\mathit{SD}}_1}}(y),F_{{\mu }_{{\mathit{SD}}_2}}(z,w\left)\right\};\hfill \end{array}$

∀  (x, y) ∈ E₁ and (z, w) ∉ E₂, $\begin{array}{ccc}\hfill a)& \hfill \hfill & (T_{{\mu }_{{\mathit{SD}}_1}}\oplus T_{{\mu }_{{\mathit{SD}}_2}})\left(\right(x,z),(y,w\left)\right)\hfill \\ \hfill & \hfill \hfill & =min\left\{T_{\mathit{SD}{\mu }_1}\right(x,y),T_{{\sigma }_{{\mathit{SD}}_2}}(z),T_{{\sigma }_{{\mathit{SD}}_2}}(w\left)\right\};\hfill \\ \hfill & \hfill \hfill & (I_{\mathit{SD}{\mu }_1}\oplus I_{{\mu }_{{\mathit{SD}}_2}})\left(\right(x,z),(y,w\left)\right)\hfill \\ \hfill & \hfill \hfill & =min\left\{I_{\mathit{SD}{\mu }_1}\right(x,y),I_{{\sigma }_{{\mathit{SD}}_2}}(z),I_{{\sigma }_{{\mathit{SD}}_2}}(w\left)\right\};\hfill \\ \hfill & \hfill \hfill & (F_{\mathit{SD}{\mu }_1}\oplus F_{{\mu }_{{\mathit{SD}}_2}})\left(\right(x,z),(y,w\left)\right)\hfill \\ \hfill & \hfill \hfill & =max\left\{F_{\mathit{SD}{\mu }_1}\right(x,y),F_{{\sigma }_{{\mathit{SD}}_2}}(z),F_{{\sigma }_{{\mathit{SD}}_2}}(w\left)\right\}.\hfill \end{array}$

3.2.1 Example

Let SVNG $G_1$ and SVNG $G_2$ be two Symmetric difference SVNN shown in Fig 1 and Fig 2 of the graphs G _{SD 1}  = (V₁, E₁) and G _{SD 2}  = (V₂, E₂) respectively. The Symmetric difference of single-valued Neutrosophic network $S{D}_1\oplus S{D}_2$ is shown in Fig. 6.

Fig. 6

$S{D}_1\oplus S{D}_2$ .

The weighted $S{D}_1\oplus S{D}_2$ is represented in Fig 7, and its corresponding dominating sets are as follows;

Fig. 7

$S{D}_1\oplus S{D}_2$ with minimum score function.

$\begin{array}{ccc}\hfill S_1& \hfill =\hfill & \{ay,cz,dy\}\hfill \\ \hfill S_2& \hfill =\hfill & \{bx,by,dy\}\hfill \\ \hfill S_3& \hfill =\hfill & \{cx,cy,ay\}\hfill \\ \hfill S_4& \hfill =\hfill & \{ay,cz,dx\}.\hfill \end{array}$

The domination number of the dominating sets S₁, S₂, S₃, S₄ are $\begin{array}{ccc}\hfill & \hfill \hfill & S_1(S{D}_1\oplus S{D}_2)\hfill \\ \hfill & \hfill \hfill & =0.4666+0.5+0.5333=1.4999\hfill \\ \hfill & \hfill \hfill & S_2(S{D}_1\oplus S{D}_2)\hfill \\ \hfill & \hfill \hfill & =0.55+0.55+0.5333=1.6333\hfill \\ \hfill & \hfill \hfill & S_3(S{D}_1\oplus S{D}_2)\hfill \\ \hfill & \hfill \hfill & =0.4666+0.55+0.5=1.5166\hfill \\ \hfill & \hfill \hfill & S_4(S{D}_1\oplus S{D}_2)\hfill \\ \hfill & \hfill \hfill & =0.4666+0.5+0.5833=1.5166.\hfill \end{array}$ The domination number of $S{D}_1\oplus S{D}_2$ is 1.4999 which is obtained from the dominating set S₁.

The minimal spanning tree of the weighted network $S{D}_1\oplus S{D}_2$ is shown in Fig 8 and hence, its weight of minimum spanning tree is 4.6498.

Fig. 8

Minimum Spanning Tree of $S{D}_1\oplus S{D}_2$ .

3.3 Residue product of two single-valued Neutrosophic graphs

Let RP₁ = (σ₁, μ₁) and RP₂ = (σ₂, μ₂) be two single-valued Neutrosophic networks of the graphs G _{RP 1}  = (V₁, E₁) and G _{RP 2}  = (V₂, E₂) respectively. Then the Residue product $ℝ{\mathbb{P}}_1·ℝ{\mathbb{P}}_2=\left({\sigma }_1·{\sigma }_2,{\mu }_1·{\mu }_2\right)$ is defined as

∀  (x, y) ∈ V₁ × V₂, $\begin{array}{ccc}\hfill T_{{\sigma }_1}·T_{{\sigma }_2}(x,y)& \hfill =\hfill & T_{{\sigma }_1}\left(x\right)\wedge T_{{\sigma }_2}\left(y\right),\hfill \\ \hfill I_{{\sigma }_1}·I_{{\sigma }_2}(x,y)& \hfill =\hfill & I_{{\sigma }_1}\left(x\right)\wedge I_{{\sigma }_2}\left(y\right)\mathrm{and}\hfill \\ \hfill F_{{\sigma }_1}·F_{{\sigma }_2}(x,y)& \hfill =\hfill & F_{{\sigma }_1}\left(x\right)\vee F_{{\sigma }_2}\left(y\right),\hfill \end{array}$

∀  (x, y) ∈ E₁ and z ≠ w ∈ V₂,

$\begin{array}{ccc}\hfill (T_{{\mu }_1}·T_{{\mu }_2})\left(\right(x,z),(y,w\left)\right)& \hfill =\hfill & T_{{\mu }_1}(x,y);\hfill \\ \hfill (I_{{\mu }_1}·I_{{\mu }_2})\left(\right(x,z),(y,w\left)\right)& \hfill =\hfill & I_{{\mu }_1}(x,y)\mathrm{and}\hfill \\ \hfill (F_{{\mu }_1}·F_{{\mu }_2})\left(\right(x,z),(y,w\left)\right)& \hfill =\hfill & F_{{\mu }_1}(x,y).\hfill \end{array}$

3.3.1 Example

Let SVNG $G_1$ and SVNG $G_2$ be two Residue product SVNN shown in Fig 1 and Fig 2 of the graphs G _{RP 1}  = (V₁, E₁) and G _{RP 2}  = (V₂, E₂) respectively. The Residue product of single-valued Neutrosophic network $ℝ{\mathbb{P}}_1·ℝ{\mathbb{P}}_2$ is shown in Fig 9.

Fig. 9

$ℝ{\mathbb{P}}_1·ℝ{\mathbb{P}}_2$ .

The weighted $ℝ{\mathbb{P}}_1·ℝ{\mathbb{P}}_2$ represented in Fig 10, and it’s corresponding dominating sets are as follows;

Fig. 10

$ℝ{\mathbb{P}}_1·ℝ{\mathbb{P}}_2$ with minimum score function.

$\begin{array}{ccc}\hfill S_1& \hfill =\hfill & \{bx,cx,bz,cz\}\hfill \\ \hfill S_2& \hfill =\hfill & \{by,cy,dy,bz\}\hfill \\ \hfill S_3& \hfill =\hfill & \{ay,by,cy,dy\}\hfill \\ \hfill S_4& \hfill =\hfill & \{cx,az,bz,cz\}.\hfill \end{array}$

The domination number of the dominating sets S₁, S₂, S₃, S₄ are $\begin{array}{ccc}\hfill S_1(ℝ{\mathbb{P}}_1·ℝ{\mathbb{P}}_2)& \hfill =\hfill & 0.55+0.55+0.5166+0.5\hfill \\ \hfill & \hfill =\hfill & 2.1166\hfill \\ \hfill S_2(ℝ{\mathbb{P}}_1·ℝ{\mathbb{P}}_2)& \hfill =\hfill & 0.55+0.5+0.5333+0.5166\hfill \\ \hfill & \hfill =\hfill & 2.0999\hfill \\ \hfill S_3(ℝ{\mathbb{P}}_1·ℝ{\mathbb{P}}_2)& \hfill =\hfill & 0.4833+0.55+0.5+0.5333\hfill \\ \hfill & \hfill =\hfill & 2.0666\hfill \\ \hfill S_4(ℝ{\mathbb{P}}_1·ℝ{\mathbb{P}}_2)& \hfill =\hfill & 0.55+0.4666+0.5166+0.5\hfill \\ \hfill & \hfill =\hfill & 2.0332.\hfill \end{array}$

The domination number of $ℝ{\mathbb{P}}_1·ℝ{\mathbb{P}}_2$ is 2.0332 which is obtained from the dominating set S₄.

The minimal spanning tree of the weighted network $ℝ{\mathbb{P}}_1·ℝ{\mathbb{P}}_2$ is shown in Fig 11 and hence, its minimum weight of the spanning tree is 5.4333.

Fig. 11

Minimal Spanning Tree of $ℝ{\mathbb{P}}_1·ℝ{\mathbb{P}}_2$ .

3.4 Max product of two single-valued Neutrosophic graphs

Let MP₁ = (σ _{mp 1} , μ _{mp 1} ) and MP₂ = (σ _{mp 2} , μ _{mp 2} ) be two single-valued Neutrosophic networks of the graphs G _{MP 1}  = (V _{mp 1} , E _{mp 1} ) and G _{MP 2}  = (V _{mp 2} , E _{mp 2} ) respectively. Then the maximal product of the graphs MP₁ and MP₂ is denoted by $M{\mathbb{P}}_1*M{\mathbb{P}}_2=\left({\sigma }_{{\mathit{mp}}_1}*{\sigma }_{{\mathit{mp}}_2},{\mu }_{{\mathit{mp}}_1}*{\mu }_{{\mathit{mp}}_2}\right)$ and defined as:

∀  (x, y) ∈ V _{mp 1}  × V _{mp 2} , $\begin{array}{ccc}\hfill (T_{{\sigma }_{{\mathit{mp}}_1}}*T_{{\sigma }_{{\mathit{mp}}_2}})(x,y)& \hfill =\hfill & T_{{\sigma }_{{\mathit{mp}}_1}}\left(x\right)\vee T_{{\sigma }_{{\mathit{mp}}_2}}\left(y\right),\hfill \\ \hfill (I_{{\sigma }_{{\mathit{mp}}_1}}*I_{{\sigma }_{{\mathit{mp}}_2}})(x,y)& \hfill =\hfill & I_{{\sigma }_{{\mathit{mp}}_1}}\left(x\right)\vee I_{{\sigma }_{{\mathit{mp}}_2}}\left(y\right),\hfill \\ \hfill & \hfill \hfill & \mathrm{and}\hfill \\ \hfill (F_{{\sigma }_{{\mathit{mp}}_1}}*F_{{\sigma }_{{\mathit{mp}}_2}})(x,y)& \hfill =\hfill & T_{{\sigma }_{{\mathit{mp}}_1}}\left(x\right)\wedge F_{{\sigma }_{{\mathit{mp}}_2}}\left(y\right);\hfill \end{array}$

∀ x ∈ V _{mp 1} and (y, z) ∈ E _{mp 2} ,

(T _{μ mp 1}  * T _{μ mp 2} ) ((x, y) , (x, z)) = T _{σ mp 1}  (x) ∨ T _{μ mp 2}  (y, z);

(I _{μ mp 1}  * I _{μ mp 2} ) ((x, y) , (x, z)) = I _{σ mp 1}  (x) ∨ I _{μ mp 2}  (y, z);

(F _{μ mp 1}  * F _{μ mp 2} ) ((x, y) , (x, z)) = F _{σ mp 1}  (x) ∧ F _{μ mp 2}  (y, z);

∀ x ∈ V _{mp 2} and (y, z) ∈ E _{mp 1} ,

(T _{μ mp 1}  * T _{μ mp 2} ) ((y, x) , (z, x)) = T _{μ mp 1}  (y, z) ∨ T _{σ mp 2}  (x);

(I _{μ mp 1}  * I _{μ mp 2} ) ((y, x) , (z, x)) = I _{μ mp 1}  (y, z) ∨ I _{σ mp 2}  (x);

(F _{μ mp 1}  * F _{μ mp 2} ) ((y, x) , (z, x)) = F _{μ mp 1}  (z, y) ∧ F _{σ mp 2}  (x);

3.4.1 Example

Let SVNG $G_1$ and SVNG $G_2$ be two Max product SVNN shown in Fig 1 and Fig 2 of the graphs G _{MP 1}  = (V₁, E₁) and G _{MP 2}  = (V₂, E₂) respectively. The Max product of single-valued Neutrosophic network $M{\mathbb{P}}_1*M{\mathbb{P}}_2$ is shown in Fig 12.

Fig. 12

$M{\mathbb{P}}_1*M{\mathbb{P}}_2$ .

The maximal product of $M{\mathbb{P}}_1*M{\mathbb{P}}_2$ is given as follows.

The weighted $M{\mathbb{P}}_1*M{\mathbb{P}}_2$ is represented in Fig 13, and it’s corresponding dominating sets are as follows;

Fig. 13

$M{\mathbb{P}}_1*M{\mathbb{P}}_2$ with minimal score function.

$\begin{array}{ccc}\hfill S_1& \hfill =\hfill & \{bx,dy,az,cz\}\hfill \\ \hfill S_2& \hfill =\hfill & \{ax,cx,ay,dz\}\hfill \\ \hfill S_3& \hfill =\hfill & \{cx,ay,dy,cz\}\hfill \\ \hfill S_4& \hfill =\hfill & \{ay,dy,bx,bz\}\hfill \end{array}$

The domination number of the dominating sets S₁, S₂, S₃, S₄ are

$\begin{array}{ccc}\hfill S_1(M{\mathbb{P}}_1*M{\mathbb{P}}_2)& \hfill =\hfill & 0.6166+0.5833+0.5166\hfill \\ \hfill & \hfill \hfill & +0.55=2.2665\hfill \\ \hfill S_2(M{\mathbb{P}}_1*M{\mathbb{P}}_2)& \hfill =\hfill & 0.5666+0.5666+0.5333\hfill \\ \hfill & \hfill \hfill & +0.5833=2.2498\hfill \\ \hfill S_3(M{\mathbb{P}}_1*M{\mathbb{P}}_2)& \hfill =\hfill & 0.5666+0.5333+0.5833\hfill \\ \hfill & \hfill \hfill & +0.55=2.2332\hfill \\ \hfill S_4(M{\mathbb{P}}_1*M{\mathbb{P}}_2)& \hfill =\hfill & 0.5333+0.5833+0.6166\hfill \\ \hfill & \hfill \hfill & +0.5833=2.3165.\hfill \end{array}$

The domination number of $M{\mathbb{P}}_1*M{\mathbb{P}}_2$ is 2.2332 which is obtained from the dominating set S₃.

The minimal spanning tree of the weighted network $M{\mathbb{P}}_1*M{\mathbb{P}}_2$ is shown in Fig 14 and hence, its minimum weight of spanning tree is 5.8996.

Fig. 14

Minimal Spanning Tree of $M{\mathbb{P}}_1*M{\mathbb{P}}_2$ .

4 Application

4.1 An application of symmetric difference network

Technology salespeople fulfil responsibilities throughout their workday to help consumers find the technology that can benefit them the most. Technology sales are the result of connecting customers with technology that can provide a solution to a specific problem.

Technology sales professionals face a unique set of challenges, such as needing a deep understanding of the complex products they sell and possessing the people’s skills needed to build trust as well as sales abilities to close deals with prospects.

A sales training program is designed to help sales professionals achieve sales success for themselves or for their organizations. Most sales training programs help to develop the sales skills and techniques needed to approach leads, create new sales opportunities, close deals, and build rapport with clients and customers.

Sales team members have the right combination of technical knowledge and practical sales know how to simultaneously do well. For this reason, sales training designed specifically for technology companies is important. Especially whether selling a new technology or in a highly competitive market, these training can help the technology sales team develop the sales skills needed to serve more, reach decision makers and take deals off the line to maximize revenue.

Let us consider a group of experts who will train the group of trainees to develop their sales skills. Assume that Network $S{D}_1$ as trainees has a concern for persons (nodes) whom they have a flow arising from their knowledge or skills. Nodes a (.2, . 4, . 6) , b (.4, . 3, . 5) , c (.3, . 4, . 5) , d (.4, . 5, . 4) are represented as Trainee 1, Trainee 2, Trainee 3 and Trainee 4 respectively.

Let us assume that Network $S{D}_2$ consists of expects x(.3,.3,.4), y(.4,.4,.6), z(.2,.5,.4), whose role is to train the trainees with their skills so that each trainee can attain a new skill when trained by the experts.

The role of each expert in training is different from one another. So, when a skill is trained by an expert to a trainee a new skill is developed by them and also their existing skill will make the sales training more effective in technology. A trainee therefore is trained by experts and does attain other skills expect their own core competency so that the trainee can have a cleaver focus on what they can do the trust to attain and wider the scope to capture high-value opportunities in sales technology.

The experts of Network $S{D}_2$ plays a different role in sales training. Each expert (or) trainer is well-developed with special sales training. For example, expert ’x(.3,.3,.4)’ is good at training inside and field sales for the trainees. Expert ’y(.4,.4,.6)’ is used in service sales training skills and expert ’z(.2,.5,.4)’ is prone to sales management skills. These experts combine their sales training to develop the trainees for the letter sales development to achieve the business objectives through effective management.

For example, ax (.2, . 3, . 6) be the sales executive of the Network developed by the expert ’x(.3,.3,.4)’ with training in inside sales for a trainee who is good at effective communication when ’ax’ is trained they are built into a better sales executive with their existing skill ay (.2, . 4 . , .6) as insurance sales officer with a skill of better communication and training of service sales expert and az (.2, . 4, . 6) the account manager who is trained by the sales management expert by (.2, . 3, . 65) is attained by the trainee ’b(.4,.3,.5)’ with good networking skills who is trained by the expert with service sales [Sales Development Representative] and so on the expertise in each field are developed by the experts to the trainees in sales technology.

The roles of each node are different from one another, when these nodes are connected into a Symmetric difference, the above Network $S{D}_1\oplus S{D}_2$ is obtained from the two small networks $S{D}_1$ and $S{D}_2$ respectively.

Symmetric difference Network of the sales technology allows effective management of business and pursue network. Organizational structure in the first place.

Flexibility is one of the main reasons for trained employees to engage in a network organization by outsourcing work. This allows them to complete the tasks in a minimal duration of time without facing major problems.

The neutrosophic network nodes are linked to one another for the flow of information in less time to other nodes. The truth-membership degree of each node indicates the better-skilled person trained in the organization. The indeterminacy-membership degree of each node demonstrates how much the person’s skill is uncertain. The falsity-membership degree of each node tells the fewer skills gained by the person. The flow of information from one node to another the node in the network takes place in effective time management. The truth-membership degree, the indeterminacy-membership degree and the falsity-memb-ership degree of each link is given by effective time management of the node in collaboration. From the above single-valued Neutrosophic network models, we find the Optimal network whose minimal spanning tree make the network more flexible with the minimum possible weights with effective score function are found and thus the optimal network with minimum optimal value increase in profits of the organizations.

The limitations of the study is, an effective optimum network is obtained from the each constructed network with a minimum weight of spanning tree using score function. The score function defined in our study gives an Optimal value from which the effective optimal network is chosen from the various operations applied on single valued Neutrosophic graphs. This study can also be extended to different operations applied on graphs.

4.2 Optimal network algorithm

Step-1: Constructed a set of finite networks say N = N₁, N₂, ⋯ , N _r using the distinct operations on network with vertex set V = V₁ × V₂.

Step-2: Find the value of score function of each nodes and links of the constructed networks N₁, N₂, ⋯ , N _r .

Step-3: Find the minimal dominating set and dominating number of each constructed networks N₁, N₂, ⋯ , N _r .

Step-4: Let the domination number of the constructed network N₁, N₂, ⋯ , N _r be D _{N 1} , D _{N 2} , ⋯ , D _{N r} respectively.

Step-5: Discover the minimal spanning trees of the constructed networks and Let it be T _{ST 1} , T _{ST 2} , ⋯ , T _{ST R} of the networks N₁, N₂, ⋯ , N _r respectively and find the minimum weights of T _{ST 1} , T _{ST 2} , ⋯ , T _{ST R} using score function.

Step-6: Let the minimum weight of T _{ST 1} , T _{ST 2} , ⋯ , T _{ST R} be W _{ST 1} , W _{ST 2} , ⋯ , W _{ST R} .

Step-7: Compute the optimal value for each constructed network N₁, N₂, ⋯ , N _r , where the optimal value is defined as the minimum value of the sum of the domination number and the minimum weight of the spanning tree. $\begin{array}{ccc}\hfill O_{N_i}& \hfill =\hfill & D_{N_i}+W_{{\mathit{ST}}_i}\hfill \\ \hfill \mathit{OptimalValue},O_{V_N}& \hfill =\hfill & \underset{i}{min}\left\{O_{N_i}\right\},i=1,2,\cdots ,r.\hfill \end{array}$

Step-8: Among these values which network gives the optimal value is said to be the optimal network.

From section-3 we arrived at the following;

The domination number of $L{\mathbb{P}}_1·L{\mathbb{P}}_2$ is 1.4999.

The domination number of $S{D}_1\oplus S{D}_2$ is 1.4999.

The domination number of $ℝ{\mathbb{P}}_1·ℝ{\mathbb{P}}_2$ is 2.0332.

The domination number of $M{\mathbb{P}}_1*M{\mathbb{P}}_2$ is 2.2332.

Minimum weight of spanning tree $L{\mathbb{P}}_1·L{\mathbb{P}}_2$ = 5.1996.

Minimum weight of spanning tree $S{D}_1\oplus S{D}_2$ = 4.6498.

Minimum weight of spannin,g tree $ℝ{\mathbb{P}}_1·ℝ{\mathbb{P}}_2$ = 5.4333.

Minimum weight of spanning tree $M{\mathbb{P}}_1*M{\mathbb{P}}_2$ = 5.8996.

Using the Optimal network algorithm, symmetric difference $S{D}_1\oplus S{D}_2$ network has the minimum domination number and its weight of the spanning tree is minimum, which gives the best optimal network of all the other networks constructed here.

5 Conclusion

The single-valued Neutrosophic models give more precision, flexibility, and compatibility to the system as compared to the classical, fuzzy, intuitionistic fuzzy and Neutrosophic models. In this paper, the authors arrive at some operations such as Lexicographic, Symmetric difference, Residue product and Max product on single-Valued Neutrosophic graphs. Also, investigated some of their properties to find their efficiency and discussed the real-world application of the Symmetric difference network with a minimum spanning tree algorithm which is generated to achieve the minimum efficient productivity to complete the tasks in a social network. In the future, the study will be extended to other operations along with strategies to achive the efficiency of the constructed network.