Graph Structures Under Neutrosophic Environment

Abstract

A single-valued neutrosophic graph structure (neutrosophic graph structure, for short) is a generalization of neutrosophic graph. In this chapter, we present the notion of neutrosophic graph structures and explore some properties of neutrosophic graph structures. Moreover, we discuss the concept of $\phi$-complement of neutrosophic graph structure and present certain operations of neutrosophic graph structures elaborated with examples. Further, we discuss some applications of neutrosophic graph structures in decision-making. This chapter is due to [33, 34, 151].

A single-valued neutrosophic graph structure (neutrosophic graph structure, for short) is a generalization of neutrosophic graph. In this chapter, we present the notion of neutrosophic graph structures and explore some properties of neutrosophic graph structures. Moreover, we discuss the concept of $\phi$-complement of neutrosophic graph structure and present certain operations of neutrosophic graph structures elaborated with examples. Further, we discuss some applications of neutrosophic graph structures in decision-making. This chapter is due to [33, 34, 151].

2.1 Introduction

Sampathkumar [151] introduced the graph structure which is a generalization of undirected graph and is quite useful in studying some structures like graphs, signed graphs, labelled graphs and edge-coloured graphs.

Definition 2.1

A graph structure ${G^*} =(X, E_{1},\ldots , E_{n})$ consists of a nonempty set X together with relations $E_{1}, E_{2},\ldots , E_{n}$ on X which are mutually disjoint such that each $E_{i}$, $1 \le i \le n$, is symmetric and irreflexive.

One can represent a graph structure $G^*=(X, E_{1},\ldots , E_{n})$ in the plane just like a graph where each edge is labelled as $E_{i}$, $1 \le i \le n$.

Example 2.1

Let X = $\{r_{1}, r_{2}, r_{3}, r_{4}, r_{5}\}$ and $E_{1}$ = $\{(r_{1}, r_{2}),( r_{3}, r_{4}), (r_{1}, r_{4}) \}$, $E_{2}$ = $\{(r_{1}, r_{3}), (r_{1}, r_{5}) \}$, $E_{3}$ = $\{(r_{2}, r_{3}), (r_{4}, r_{5})\}$ be mutually disjoint, symmetric and irreflexive relations on set X. Thus $G=(X, E_{1}, E_{2}, E_{3})$ is a graph structure and is represented in plane as a graph where each edge is labelled as $E_{1}$, $E_{2}$ or $E_{3}$ (Fig. 2.1).

Fig. 2.1

Graph structure $G^*=(X, E_{1}, E_{2}, E_{3})$

Definition 2.2

Let $\phi$ be a permutation on $\{E_{1}, E_2,\ldots , E_{n}\}$. Then $\phi$-complement of a graph structure ${G^{*}}$ denoted by ${G^{*}}^{\phi c}$ is obtained by replacing $E_{i}$ by $\phi (E_{i}$), $1\le i \le n.$

${G^{*}}$ is self-complementary if it is isomorphic to ${G^{*}}^{\phi c}$, where $\phi$ is not an identity permutation. ${G^{*}}$ is totally strong self-complementary if it is identical to ${G^{*}}^{\phi c}$ for all permutations $\phi$ on $\{E_{1}, E_2,\ldots , E_{n}\}$.

Definition 2.3

If graph structure ${G^{*}}$ is connected and contains no cycle, in other words, its underlying graph is a tree, then it is called a tree. ${G^{*}}$ is an $E_{i}$-tree if subgraph structure induced by $E_{i}$-edges is a tree. Similarly, ${G^{*}}$ is an $E_{1}E_{2}\ldots E_{n}-$ tree if ${G^{*}}$ is an $E_{i}-$tree for each $i \in \{1,2,\ldots , n\}.$ ${G^{*}}$ is an $E_{i}$-forest, if subgraph structure induced by $E_{i}$-edges is a forest.

Definition 2.4

Let $G^{*}_{1} = (X, E_{1}, E_{2},\ldots , E_{n})$ and $G^{*}_{2} = (X', E'_{1}, E'_{2},\ldots , E'_{n})$ be two graph structures, Cartesian product of $G^{*}_{1}$ and $G^{*}_{2}$ is defined as: $G^{*}_{1} \times G^{*}_{2}$ = $(X \times X', E_{1} \times E'_{1}, E_{2} \times E'_{2}, \ldots , E_{n}\times E'_{n})$, where $E_{i} \times E'_{i}$ = $\{(b_{1}d, b_{2}d)\mid d \in X',$ $b_{1} b_{2} \in E_{i}\} \cup \{(bd_{1}, bd_{2})\mid b \in X, d_{1}d_{2} \in E'_{i}\}$, $i = (1,2, \ldots , n)$.

Definition 2.5

Let $G^{*}_{1} = (X, E_{1}, E_{2},\ldots , E_{n})$ and $G^{*}_{2} = (X', E'_{1}, E'_{2},\ldots , E'_{n})$ be two graph structures, cross product of $G^{*}_{1}$ and $G^{*}_{2}$ is defined as: $G^{*}_{1} * G^{*}_{2}$ = $(X * X', E_{1} * E'_{1}, E_{2} * E'_{2}, \ldots , E_{n} * E'_{n})$, where $E_{i} * E'_{i}$ = $\{(b_{1}d_{1}, b_{2}d_{2})\mid b_{1}b_{2} \in E_{i},$ $d_{1} d_{2} \in E'_{i}\}$, $i = (1,2, \ldots , n)$.

Definition 2.6

Let $G^{*}_{1} = (X, E_{1}, E_{2},\ldots , E_{n})$ and $G^{*}_{2} = (X', E'_{1}, E'_{2},\ldots , E'_{n})$ be two graph structures, lexicographic product of $G^{*}_{1}$ and $G^{*}_{2}$ is defined as: $G^{*}_{1} \bullet G^{*}_{2}$ = $(X \bullet X', E_{1} \bullet E'_{1}, E_{2} \bullet E'_{2}, \ldots , E_{n}\bullet E'_{n})$, where $E_{i} \bullet E'_{i}$ = $\{(bd_{1}, bd_{2})\mid b \in X,$ $d_{1} d_{2} \in E'_{i}\} \cup \{(b_{1}d_{1}, b_{2}d_{2})\mid b_{1}b_{2} \in E_{i}, d_{1}d_{2} \in E'_{i}\}$, $i = (1,2, \ldots , n)$.

Definition 2.7

Let $G^{*}_{1} = (X, E_{1}, E_{2},\ldots , E_{n})$ and $G^{*}_{2} = (X', E'_{1}, E'_{2},\ldots , E'_{n})$ be two graph structures, strong product of $G^{*}_{1}$ and $G^{*}_{2}$ is defined as: $G^{*}_{1} \boxtimes G^{*}_{2}$ = $(X \boxtimes X', E_{1} \boxtimes E'_{1}, E_{2} \boxtimes E'_{2}, \ldots , E_{n}\boxtimes E'_{n})$, where $E_{i} \boxtimes E'_{i}$ = $\{(b_{1}d, b_{2}d)\mid d \in X',$ $b_{1} b_{2} \in E_{i}\} \cup \{(bd_{1}, bd_{2})\mid b \in X, d_{1}d_{2} \in E'_{i}\}$ $\cup$ $\{(b_{1}d_{1}, b_{2}d_{2})\mid b_{1}b_{2} \in E_{i},$ $d_{1} d_{2} \in E'_{i}\}$, $i = (1,2, \ldots , n)$.

Definition 2.8

Let $G^{*}_{1} = (X, E_{1}, E_{2},\ldots , E_{n})$ and $G^{*}_{2} = (X', E'_{1}, E'_{2},\ldots , E'_{n})$ be two graph structures, composition of $G^{*}_{1}$ and $G^{*}_{2}$ is defined as: $G^{*}_{1} \circ G^{*}_{2}$ = $(X \circ X', E_{1} \circ E'_{1}, E_{2} \circ E'_{2}, \ldots , E_{n}\circ E'_{n})$, where $E_{i} \circ E'_{i}$ = $\{(b_{1}d, b_{2}d)\mid d \in X',$ $b_{1} b_{2} \in E_{i}\} \cup \{(bd_{1}, bd_{2})\mid b \in X, d_{1}d_{2} \in E'_{i}\}$ $\cup$ $\{(b_{1}d_{1}, b_{2}d_{2})\mid b_{1}b_{2} \in E_{i},$ $d_{1}, d_{2} \in X'$ such that $d_{1} \ne d_{2} \}$, $i = (1,2, \ldots , n)$.

Definition 2.9

Let $G^{*}_{1} = (X, E_{1}, E_{2},\ldots , E_{n})$ and $G^{*}_{2} = (X', E'_{1}, E'_{2},\ldots , E'_{n})$ be two graph structures, union of $G^{*}_{1}$ and $G^{*}_{2}$ is defined as: $G^{*}_{1} \cup G^{*}_{2}$ = $(X \cup X', E_{1} \cup E'_{1}, E_{2} \cup E'_{2}, \ldots , E_{n}\cup E'_{n})$.

Definition 2.10

Let $G^{*}_{1} = (X, E_{1}, E_{2},\ldots , E_{n})$ and $G^{*}_{2} = (X', E'_{1}, E'_{2},\ldots , E'_{n})$ be two graph structures, join of $G^{*}_{1}$ and $G^{*}_{2}$ is defined as: $G^{*}_{1} + G^{*}_{2}$ = $(X + X', E_{1} + E'_{1}, E_{2} + E'_{2}, \ldots , E_{n} + E'_{n})$, where $X+X'= X \cup X'$, $E_{i} + E'_{i}$ = $E_{i} \cup E'_{i} \cup E''_{i}$ for $i = (1,2, \ldots , n)$. $E''_{i}$ contains all those edges, joining the vertices of E and $E'$.

2.2 Neutrosophic Graph Structures

Definition 2.11

Let X be a nonempty set and $E_{1}, E_{2},\ldots , E_{n}$ relations on X. $G=(A, B_{1}, B_{2},\ldots , B_{n})$ is called a single-valued neutrosophic graph structure if

$$\begin{aligned} A= \{<n,T_{i}(n),I_{i}(n), F_{i}(n)>: n \in X\} \end{aligned}$$

is a single-valued neutrosophic set on X and

$$\begin{aligned} B_{i}= \{<(m,n),T(m,n),I(m,n),F(m,n)>: (m, n) \in E_i\} \end{aligned}$$

is a single-valued neutrosophic set on $E_{i}$ such that

$$\begin{aligned} T_{i}(m,n)\le & {} \min \{T(m),T(n)\}, \,\ I_{i}(m,n)\le \min \{I(m),I(n)\},\\F_{i}(m,n)\le & {} \max \{F(m),F(n)\},\forall m, n\in X. \end{aligned}$$

Note that $T_{i}(m, n)=0=I_{i}(m,n)=F_{i}(m, n)$ for all $(m, n)\in X\times X-E_{i}$ and

$$\begin{aligned} 0\le T_{i}(m,n)+I_{i}(m,n)+F_{i}(m, n)\le 3\,\,\text {for all}\,\,(m, n)\in E_{i}, \end{aligned}$$

where X and $E_{i}$ ($i=1,2,\ldots , n$) are underlying vertex and underlying i-edge sets of G, respectively.

Throughout this chapter, we will use neutrosophic set, neutrosophic relation and neutrosophic graph structure, for short.

Definition 2.12

Let $G=(A, B_{1}, B_{2},\ldots , B_{n})$ be a neutrosophic graph structure of $G^{*}$. If ${H}=(A^{\prime }, B_{1}^{\prime }, B_{2}^{\prime },\ldots , B_{n}^{\prime })$ is a neutrosophic graph structure of $G^{*}$ such that

$$\begin{aligned} T^{\prime }(n)\le T(n), \,\ I^{\prime }(n)\le I(n), F^{\prime }(n)\ge F(n), \forall n\in X, \end{aligned}$$

$$\begin{aligned} T_{i}^{\prime }(m,n)\le T_{i}(m,n), I_{i}^{\prime }(m,n)\le I_{i}(m,n)\text { and }F_{i}^{\prime }(m,n)\ge F_{i}(m,n), \forall m, n\in E_{i}, \end{aligned}$$

where $i=1,2,\ldots , n$. Then H is called a neutrosophic subgraph structure of neutrosophic graph structure G.

Example 2.2

Let $G^{*}$ = $(X, E_{1}, E_{2})$ be a graph structure, where X = $\{q_{1}, q_{2}, q_{3},$ $q_{4}, q_{5}, q_{6}\}$, $E_{1}$ = $\{q_{1}q_{6}, q_{2}q_{3}, q_{3}q_{4},$ $q_{4}q_{5}\}$, $E_{2}$ = $\{q_{1}q_{2}, q_{5}q_{6}, q_{4}q_{6}, q_{1}q_{3}\}$. Now we define neutrosophic sets A, $B_{1}$, $B_{2}$ on X, $E_{1}$, $E_{2}$, respectively.

Let $A = \{(q_{1}, 0.3, 0.6, 0.4), (q_{2}, 0.4, 0.7, 0.5),$ $(q_{3}, 0.5, 0.7, 0.6),$ $(q_{4}, 0.6, 0.9, 0.7),$ $(q_{5}, 0.4, 0.5, 0.5),(q_{6}, 0.3, 0.4, 0.4)\}$, $B_{1}$ = $\{(q_{1}q_{6}, 0.3, 0.2, 0.3),$ $(q_{2}q_{3}, 0.3, 0.5, 0.4),$ $(q_{3}q_{4}, 0.5, 0.7, 0.6),$ $(q_{4}q_{5}, 0.3, 0.3, 0.4)\}$, $B_{2}$ = $\{(q_{1}q_{2}, 0.2, 0.6, 0.3), (q_{5}q_{6}, 0.1, 0.4, 0.2),$ $(q_{4}q_{6}, 0.1, 0.4, 0.2), (q_{1}q_{3}, 0.2, 0.4, 0.3)\}$. By direct calculations, it is easy to show that G = $(A, B_{1}, B_{2})$ is a neutrosophic graph structure of $G^{*}$ as shown in Fig. 2.2.

Fig. 2.2

Single-valued neutrosophic graph structure

Definition 2.13

A neutrosophic graph structure ${H} = (A^{\prime }, B_{1}^{\prime }, B_{2}^{\prime },\ldots , B_{n}^{\prime })$ is called an induced subgraph structure of G by a subset R of X if

$$\begin{aligned} T^{\prime }(n) = T(n), I^{\prime }(n)=I(n), F^{\prime }(n)=F(n), \forall n\in E, \end{aligned}$$

$$\begin{aligned} T_{i}^{\prime }(m,n)=T_{i}(m,n), I_{i}^{\prime }(m,n)=I_{i}(m,n) \text { and }F_{i}^{\prime }(m,n)=F_{i}(m,n), \forall m, n\in E, \end{aligned}$$

where $i=1,2,\ldots , n$.

Definition 2.14

A neutrosophic graph structure ${H}=(A^{\prime }, B_{1}^{\prime }, B_{2}^{\prime },\ldots , B_{n}^{\prime })$ is called a spanning subgraph structure of G if $A^{\prime }=A$ and

$$\begin{aligned} T_{i}^{\prime }(m,n)\le T_{i}(m,n), I_{i}^{\prime }(m,n)\le I_{i}(m,n)\text { and }F_{i}^{\prime }(m,n)\ge F_{i}(m, n), i=1,2,..., n. \end{aligned}$$

Example 2.3

Consider a graph structure $G^{*}=(X, E_{1}, E_{2})$ and let A, $B_{1}$, $B_{2}$ be neutrosophic subsets of $X, E_{1}, E_{2}$, respectively, such that

$$\begin{aligned} A= \{(n_{1}, 0.5,0.2,0.3),(n_{2}, 0.7,0.3,0.4),(n_{3}, 0.4,0.3,0.5),(n_{4}, 0.7,0.3,0.6)\}, \end{aligned}$$

$$\begin{aligned} B_{1}=\{(n_{1}n_{2}, 0.5,0.2,0.4),(n_{2}n_{4}, 0.7,0.3,0.6)\}, \end{aligned}$$

$$\begin{aligned} B_{2}=\{(n_{3}n_{4}, 0.4,0.3,0.6),(n_{1}n_{4}, 0.5,0.2,0.6)\}. \end{aligned}$$

Direct calculations show that $G=(A, B_{1}, B_{2})$ is a neutrosophic graph structure of $G^{*}$ as shown in Fig. 2.3.

Fig. 2.3

Neutrosophic graph structure G

Example 2.4

A neutrosophic graph structure ${K}=(A^{\prime }, B_{11}, B_{12})$ shown in Fig. 2.4 is a neutrosophic subgraph structure of $G=(A, B_{1}, B_{2})$ shown in Fig. 2.3.

Fig. 2.4

Neutrosophic subgraph structure

Definition 2.15

Let $G=(A, B_{1}, B_{2},\ldots , B_{n})$ be a neutrosophic graph structure of $G^{*}$. Then $mn\in E_{i}$ is called $B_{i}$-edge or simply $B_{i}$-edge if $T_{i}(m, n)>0$ or $I_{i}(m, n)>0$ or $F_{i}(m, n)>0$ or all three conditions hold. Consequently, support of $B_{i}$ is defined as:

$$\begin{aligned} supp(B_{i})= & {} \{mn\in B_{i}:T_{i}(m, n)>0\}\cup \{mn\in B_{i}:I_{i}(m, n)>0\} \\&\left. \cup \{mn\in B_{i}:F_{i}(m, n)>0\}, i=1,2,..., n.\right. \end{aligned}$$

Definition 2.16

$B_{i}$-path in a neutrosophic graph structure $G=(A, B_{1}, B_{2},\ldots , B_{n})$ is a sequence of distinct vertices $n_{1}, n_{2},\ldots , n_{m}$ (except choice that $n_{m}=n_{1}$) in X, such that $n_{j-1}n_{j}$ is a neutrosophic $B_{i}$-edge for all $j=2,\ldots , m$.

Definition 2.17

A neutrosophic graph structure $G=(A, B_{1}, B_{2},\ldots , B_{n})$ is called $B_{i}$-strong for some $i\in \{1,2,\ldots , n\}$ if

$$\begin{aligned} T_{i}(m,n)=\min \{T(m),T(n)\}, I_{i}(m,n)=\min \{I(m), I(n)\} \end{aligned}$$

and

$$\begin{aligned} F_{i}(m,n)=\max \{F(m), F(n)\}, \forall mn\in supp(B_{i}). \end{aligned}$$

Furthermore, neutrosophic graph structure G is called strong if it is $B_{i}$-strong for all $i\in \{1,2,\ldots , n\}$.

Example 2.5

Consider a neutrosophic graph structure $G=(A, B_{1}, B_{2})$ as shown in Fig. 2.5. Then G is a strong neutrosophic graph structure since it is both $B_{1}$- and $B_{2}$-strong.

Fig. 2.5

Strong neutrosophic graph structure G = $(A, B_{1}, B_{2})$

Definition 2.18

A neutrosophic graph structure $G=(A, B_{1}, B_{2},\ldots , B_{n})$ is called complete if G is a strong neutrosophic graph structure, $supp(B_{i})\ne \phi$ for all $i=1,2,\ldots , n$ and for every pair of vertices $m, n\in X$, mn is a $B_{i}$-edge for some i.

Example 2.6

Let $G=(A, B_{1}, B_{2})$ be a neutrosophic graph structure of graph structure $G^{*}=(X, E_{1}, E_{2})$ such that $X=\{n_{1}, n_{2}, n_{3}\}$, $E_{1}=\{n_{1}n_{2}\}$ and $E_{2}=\{n_{2}n_{3}, n_{1}n_{3}\}$ as shown in Fig. 2.6. By simple calculations, it can be seen that G is a strong neutrosophic graph structure. Moreover, $supp(B_{1})\ne \phi$, $supp(B_{2})\ne \phi$, and each pair of vertices in X is either a $B_{1}$-edge or an $B_{2}$-edge. So G is a complete, i.e. $B_{1}B_{2}$-complete neutrosophic graph structure.

Fig. 2.6

Complete neutrosophic graph structure

Definition 2.19

Let $G=(A, B_{1}, B_{2},\ldots , B_{n})$ be a neutrosophic graph structure. Then truth strength, indeterminacy strength and falsity strength of a $B_{i}$-path $P_{B_{i}}=n_{1}, n_{2},\ldots , n_{m}$ are denoted by $T.P_{B_{i}}$, $I.P_{B_{i}}$ and $F.P_{B_{i}}$, respectively, and defined as

$$\begin{aligned} T.P_{B_{i}}=\bigwedge \limits _{j=2}^{m}[T_{B_{i}}^{P}(n_{j-1}n_{j})]~, I.P_{B_{i}}=\bigwedge \limits _{j=2}^{m}[I_{B_{i}}^{P}(n_{j-1}n_{j})]~, F.P_{B_{i}}=\bigvee \limits _{j=2}^{m}[F_{B_{i}}^{P}(n_{j-1}n_{j})]~. \end{aligned}$$

Example 2.7

Consider a neutrosophic graph structure $G=(A, B_{1}, B_{2})$ as shown in Fig. 2.6. We found that $P_{B_{2}}=n_{2}, n_{3}, n_{1}$ is a $B_{2}$-path. So $T.P_{B_{2}}=0.4,$ $I.P_{B_{2}}=0.4$ and $F.P_{B_{2}}=0.8.$

Definition 2.20

Let $G=(A, B_{1}, B_{2},\ldots , B_{n})$ be a neutrosophic graph structure. Then

• (i)   $B_{i}$-truth strength of connectedness between m and n is defined as: $T_{B_{i}}^{\infty }(mn)=\bigvee \limits _{j\ge 1}\{T_{B_{i}}^{j}(mn)\}$ such that $T_{B_{i}}^{j}(mn)=(T_{B_{i}}^{j-1}\circ T_{B_{i}}^{1})(mn)$ for $j\ge 2$ and

  $$\begin{aligned} T_{B_{i}}^{2}(mn)=(T_{B_{i}}^{1}\circ T_{B_{i}}^{1})(mn)=\bigvee \limits _{z}(T_{B_{i}}^{1}(mz)\wedge T_{B_{i}}^{1}(zn)). \end{aligned}$$

• (ii)   $B_{i}$-indeterminacy strength of connectedness between m and n is defined as: $I_{B_{i}}^{\infty }(mn)=\bigvee \limits _{j\ge 1}\{I_{B_{i}}^{j}(mn)\}$ such that $I_{B_{i}}^{j}(mn)=(I_{B_{i}}^{j-1}\circ I_{B_{i}}^{1})(mn)$ for $j\ge 2$ and

  $$\begin{aligned} I_{B_{i}}^{2}(mn)=(I_{B_{i}}^{1}\circ I_{B_{i}}^{1})(mn)=\bigvee \limits _{z}(I_{B_{i}}^{1}(mz)\wedge I_{B_{i}}^{1}(zn)). \end{aligned}$$

• (iii)   $B_{i}$-falsity strength of connectedness between m and n is defined as: $F_{B_{i}}^{\infty }(mn)=\bigwedge \limits _{j\ge 1}\{F_{B_{i}}^{j}(mn)\}$ such that $F_{B_{i}}^{j}(mn)=(F_{B_{i}}^{j-1}\circ F_{B_{i}}^{1})(mn)$ for $j\ge 2$ and

  $$\begin{aligned} F_{B_{i}}^{2}(mn)=(F_{B_{i}}^{1}\circ F_{B_{i}}^{1})(mn)=\bigwedge \limits _{z}(F_{B_{i}}^{1}(mz)\vee F_{B_{i}}^{1}(zn)). \end{aligned}$$

Definition 2.21

A neutrosophic graph structure $G=(A, B_{1}, B_{2},\ldots , B_{n})$ is a $B_{i}$-cycle if

$$\begin{aligned} (supp(A), supp(B_{1}), supp(B_{2}),\ldots , supp(B_{n}))\text { is a }B_{i}\text {-cycle}. \end{aligned}$$

Definition 2.22

A neutrosophic graph structure $G=(A, B_{1}, B_{2},\ldots , B_{n})$ is a $B_{i}$-cycle (for some i) if G is a $B_{i}$-cycle, no unique $B_{i}$-edge mn is in G such that

$$\begin{aligned} T_{B_{i}}(mn)=\min \{T_{B_{i}}(rs):rs\in E_{i}=supp(B_{i})\}, \end{aligned}$$

or

$$\begin{aligned} I_{B_{i}}(mn)=\min \{I_{B_{i}}(rs):rs\in E_{i}=supp(B_{i})\}, \end{aligned}$$

or

$$\begin{aligned} F_{B_{i}}(mn)=\max \{F_{B_{i}}(rs):rs\in E_{i}=supp(B_{i})\}. \end{aligned}$$

Example 2.8

Consider a neutrosophic graph structure $G=(A, B_{1}, B_{2})$ as shown in Fig. 2.5. Then G is a $B_{1}$-cycle and neutrosophic $B_{1}-cycle$, since $(supp(A), supp(B_{1}), supp(B_{2}))$ is a $B_{1}$-cycle and there is no unique $B_{1}$-edge satisfying above condition.

Definition 2.23

Let $G=(A, B_{1}, B_{2},\ldots , B_{n})$ be a neutrosophic graph structure and p be a vertex in G. Let $(A^{\prime }, B_{1}^{\prime }, B_{2}^{\prime },\ldots , B_{n}^{\prime })$ be a neutrosophic graph structure induced by $X\setminus \{p\}$ such that, for all $v\ne p, w\ne p,$

$$\begin{aligned} T_{A^{\prime }}(p){=}0{=}I_{A^{\prime }}(p){=}F_{A^{\prime }}(p), T_{B_{i}^{\prime }}(pv)=0=I_{B_{i}^{\prime }}(pv)=F_{B_{i}^{\prime }}(pv) , \forall \text {edges }pv\in G, \end{aligned}$$

$$\begin{aligned} T_{A^{\prime }}(v)=T_{A}(v),I_{A^{\prime }}(v)=I_{A}(v), F_{A^{\prime }}(v)=F_{A}(v), \end{aligned}$$

$$\begin{aligned} T_{B_{i}^{\prime }}(vw)=T_{B_{i}}(vw), I_{B_{i}^{\prime }}(vw)=I_{B_{i}}(vw)\text { and }F_{B_{i}^{\prime }}(vw)=F_{B_{i}}(vw). \end{aligned}$$

Then p is neutrosophic $B_{i}$-cut vertex for any i if

$$\begin{aligned} T_{B_{i}}^{\infty }(vw)>T_{B_{i}^{\prime }}^{\infty }(vw), I_{B_{i}}^{\infty }(vw)>I_{B_{i}^{\prime }}^{\infty }(vw)\text { and }F_{B_{i}}^{\infty }(vw)>F_{B_{i}^{\prime }}^{\infty }(vw), \end{aligned}$$

for some $v, w\in X\setminus \{p\}$. Note that p is a

• $B_{i}-T$ neutrosophic cut vertex if $T_{B_{i}}^{\infty }(vw)>T_{B_{i}^{\prime }}^{\infty }(vw)$,

• $B_{i}-I$ neutrosophic cut vertex if $I_{B_{i}}^{\infty }(vw)>I_{B_{i}^{\prime }}^{\infty }(vw)$,

• $B_{i}-F$ neutrosophic cut vertex if $F_{B_{i}}^{\infty }(vw)>F_{B_{i}^{\prime }}^{\infty }(vw)$.

Fig. 2.7

Neutrosophic graph structure G $=$ $(A, B_{1}, B_{2})$

Example 2.9

Consider a neutrosophic graph structure $G=(A, B_{1}, B_{2})$ as shown in Fig. 2.7 and let $G^{\prime }=(A^{\prime }, B_{1}^{\prime }, B_{2}^{\prime })$ be a neutrosophic subgraph structure of neutrosophic graph structure G found by deleting vertex $n_{2}$. Deleted vertex $n_{2}$ is a neutrosophic $B_{1}$-I cut vertex since

$$\begin{aligned} I_{B_{1}}^{\infty }(n_{2}n_{5})=0.4>0.3=I_{B_{1}^{\prime }}^{\infty }(n_{2}n_{5}), \, I_{B_{1}}^{\infty }(n_{3}n_{4})=0.7=I_{B_{1}^{\prime }}^{\infty }(n_{3}n_{4}), \end{aligned}$$

and

$$\begin{aligned} I_{B_{1}}^{\infty }(n_{3}n_{5})=0.4>0.3=I_{B_{1}^{\prime }}^{\infty }(n_{3}n_{5}). \end{aligned}$$

Definition 2.24

Suppose $G=(A, B_{1}, B_{2},\ldots , B_{n})$ be a neutrosophic graph structure and mn be $B_{i}$-edge. Let $(A^{\prime }, B_{1}^{\prime }, B_{2}^{\prime },\ldots , B_{n}^{\prime })$ be a neutrosophic spanning subgraph structure of G, such that $\forall$ edges $mn\ne rs$,

$$\begin{aligned} T_{B_{i}^{\prime }}(mn)=0=I_{B_{i}^{\prime }}(mn)=F_{B_{i}^{\prime }}(mn) , T_{B_{i}^{\prime }}(rs)=T_{B_{i}}(rs), \end{aligned}$$

$$\begin{aligned} I_{B_{i}^{\prime }}(rs)=I_{B_{i}}(rs)\text { and }F_{B_{i}^{\prime }}(rs)=F_{B_{i}}(rs). \end{aligned}$$

Then mn is a neutrosophic $B_{i}$-bridge if

$$ T_{B_{i}}^{\infty }(vw)>T_{B_{i}^{\prime }}^{\infty }(vw), I_{B_{i}}^{\infty }(vw)>I_{B_{i}^{\prime }}^{\infty }(vw)\, \text {and}\, F_{B_{i}}^{\infty }(vw)>F_{B_{i}^{\prime }}^{\infty }(vw), $$

for some $v, w\in X$. Note that mn is a

• $B_{i}-T$ neutrosophic bridge if $T_{B_{i}}^{\infty }(vw)>T_{B_{i}^{\prime }}^{\infty }(vw)$,

• $B_{i}-I$ neutrosophic bridge if $I_{B_{i}}^{\infty }(vw)>I_{B_{i}^{\prime }}^{\infty }(vw)$,

• $B_{i}-F$ neutrosophic bridge if $F_{B_{i}}^{\infty }(vw)>F_{B_{i}^{\prime }}^{\infty }(vw)$.

Example 2.10

Consider the neutrosophic graph structure $G=(A, B_{1}, B_{2})$ as shown in Fig. 2.7 and $G^{\prime }=(A^{\prime }, B_{1}^{\prime }, B_{2}^{\prime })$ be a neutrosophic spanning subgraph structure of neutrosophic graph structure G which is found by deleting $B_{1}$-edge $(n_{2}n_{5})$. Edge $(n_{2}n_{5})$ is a neutrosophic $B_{1}$-bridge. Since

$$\begin{aligned} T_{B_{1}}^{\infty }(n_{2}n_{5})=0.4>0.3=T_{B_{1}^{\prime }}^{\infty }(n_{2}n_{5}), \end{aligned}$$

$$\begin{aligned} I_{B_{1}}^{\infty }(n_{2}n_{5})=0.4>0.3=I_{B_{1}^{\prime }}^{\infty }(n_{2}n_{5}) \end{aligned}$$

and

$$\begin{aligned} F_{B_{1}}^{\infty }(n_{2}n_{5})=0.5>0=F_{B_{1}^{\prime }}^{\infty }(n_{2}n_{5}). \end{aligned}$$

Definition 2.25

A neutrosophic graph structure $G=(A, B_{1}, B_{2},\ldots , B_{n})$ is a $B_{i}$-tree if

$$\begin{aligned} (supp(A), supp(B_{1}), supp(B_{2}),\ldots , supp(B_{n})) \end{aligned}$$

is a $B_{i}$-tree. In other words, G is a $B_{i}$-tree if a subgraph of G induced by $supp(B_{i})$ generates a tree.

Definition 2.26

A neutrosophic graph structure $G=(A, B_{1}, B_{2},\ldots , B_{n})$ is $B_{i}$-tree if G has a neutrosophic spanning subgraph structure $H=(A^{\prime }, B_{1}^{\prime }, B_{2}^{\prime },\ldots , B_{n}^{\prime })$ such that for all $B_{i}$-edges mn not in H, H is a $B_{i}^{\prime }$-tree,

$$\begin{aligned} T_{B_{i}}(mn)<T_{B_{i}^{\prime }}^{\infty }(mn), I_{B_{i}}(mn)<I_{B_{i}^{\prime }}^{\infty }(mn)\text { and } F_{B_{i}}(mn)>F_{B_{i}^{\prime }}^{\infty }(mn). \end{aligned}$$

In particular, G is a:

• neutrosophic $B_{i}$-T tree if $T_{B_{i}}(mn)<T_{B_{i}^{\prime }}^{\infty }(mn)$,

• neutrosophic $B_{i}$-I tree if $I_{B_{i}}(mn)<I_{B_{i}^{\prime }}^{\infty }(mn)$,

• neutrosophic $B_{i}$-F tree if $F_{B_{i}}(mn)>F_{B_{i}^{\prime }}^{\infty }(mn)$.

Example 2.11

Consider the neutrosophic graph structure $G=(A, B_{1}, B_{2})$ as shown in Fig. 2.8, which is a $B_{2}$-tree. It is not a $B_{1}$-tree but a neutrosophic $B_{1}$-tree since it has a neutrosophic spanning subgraph $(A^{\prime }, B_{1}^{\prime }, B_{2}^{\prime })$ as a $B_{1}^{\prime }$-tree, which is obtained by deleting $B_{1}$-edge $n_{2}n_{5}$ from G.

Moreover,

$$\begin{aligned} T_{B_{1}}(n_{2}n_{5})=0.2<0.3=T_{B_{1}^{\prime }}^{\infty }(n_{2}n_{5}), \, I_{B_{1}}(n_{2}n_{5})=0.1<0.3=I_{{B_{1}}^{\prime }}^{\infty }(n_{2}n_{5}) \end{aligned}$$

and

$$\begin{aligned} F_{B_{1}}(n_{2}n_{5})=0.6>0.5=F_{{B_{1}}^{\prime }}^{\infty }(n_{2}n_{5}). \end{aligned}$$

Fig. 2.8

Neutrosophic $B_{1}$-tree

Definition 2.27

A neutrosophic graph structure $G_{1}=(A_{1}, B_{11}, B_{12},\ldots , B_{1n})$ of the graph structure $G^{*}_{1}=(X_{1}, E_{11}, E_{12},\ldots , E_{1n})$ is isomorphic to neutrosophic graph structure $G=(A_{2}, B_{21}, B_{22},\ldots , B_{2n})$ of the graph structure $G^{*}_{2}=(X_{2}, E_{21}, B_{22},\ldots , E_{2n})$ if we have $(f,\phi )$ where $f:X_{1}\rightarrow X_{2}$ is a bijection and $\phi$ is a permutation on set $\{1,2,\ldots , n\}$ and following relations are satisfied

$$\begin{aligned} T_{A_{1}}(m)=T_{A_{2}}(f(m)), \, I_{A_{1}}(m)=I_{A_{2}}(f(m)), F_{A_{1}}(m)=F_{A_{2}}(f(m)), \end{aligned}$$

for all $m\in X_{1}$ and

$$\begin{aligned} T_{B_{1i}}(mn)=T_{B_{2\phi (i)}}(f(m)f(n)), \, I_{B_{1i}}(mn)=I_{B_{2 \phi (i)}}(f(m)f(n), \end{aligned}$$

$$\begin{aligned} F_{B_{1i}}(mn)=F_{B_{2\phi (i)}}(f(m)f(n)), \end{aligned}$$

for all $mn\in E_{1i}$, $i=1,2,\ldots , n.$

Example 2.12

Let $G_{1}=(A, B_{1}, B_{2})$ and $G_{2}=(A^{\prime }, B_{1}^{\prime }, B_{2}^{\prime })$ be two neutrosophic graph structures as shown in Fig. 2.9. $G_{1}$ is isomorphic $G_{2}$ under $(f,\phi )$ where $f:X\rightarrow X^{\prime }$ is a bijection and $\phi$ is a permutation on set $\{1,2\}$ defined as $\phi (1)=2$, $\phi (2)=1$ and following relations are satisfied

$$\begin{aligned} T_{A}(n_{i})=T_{A^{\prime }}(f(n_{i})),I_{A}(n_{i})=I_{A^{\prime }}(f(n_{i})) ,\, F_{A}(n_{i})=F_{A^{\prime }}(f(n_{i})), \end{aligned}$$

for all $n_{i}\in X$, and

$$\begin{aligned} T_{B_{i}}(n_{i}n_{j})=T_{B_{\phi (i)}^{\prime }}(f(n_{i})f(n_{j})), I_{B_{i}}(n_{i}n_{j})=I_{B_{\phi (i)}^{\prime }}(f(n_{i})f(n_{j})), \end{aligned}$$

$$\begin{aligned} F_{B_{i}}(n_{i}n_{j})=F_{B_{\phi (i)}^{\prime }}(f(n_{i})f(n_{j})), \end{aligned}$$

$\forall n_{i}n_{j}\in E_{i}$ and $i=1,2.$

Fig. 2.9

Isomorphic neutrosophic graph structures

Definition 2.28

A neutrosophic graph structure $G_{1}=(A_{1}, B_{11}, B_{12},\ldots , B_{1n})$ of the graph structure $G^{*}_{1}=(X_{1}, E_{11}, E_{12},\ldots , E_{1n})$ is identical to neutrosophic graph structure $G_{2}=(A_{2}, B_{21}, B_{22},\ldots , B_{2n})$ of graph structure $G^{*}_{2}=(X_{2}, E_{21}, B_{22},\ldots , E_{2n})$ if $f:X_{1}\rightarrow X_{2}$ is a bijection and following relations are satisfied:

$$\begin{aligned} T_{A_{1}}(m)=T_{A_{2}}(f(m)), \, I_{A_{1}}(m)=I_{A_{2}}(f(m)), \, F_{A_{1}}(m)=F_{A_{2}}(f(m)), \end{aligned}$$

for all $m\in X_{1}$ and

$$\begin{aligned} T_{B_{1i}}(mn)=T_{B_{2i}}(f(m)f(n)), \, I_{B_{1i}}(mn)=I_{B_{2i}}(f(m)f(n)), \end{aligned}$$

$$\begin{aligned} F_{B_{1i}}(mn)=F_{B_{2i}}(f(m)f(n)), \end{aligned}$$

for all $mn\in E_{1i}$ and $i=1,2,\ldots , n.$

Example 2.13

Let $G_{1}=(A, B_{1}, B_{2})$ and ${G} _{2}=(A^{\prime }, B_{1}^{\prime }, B_{2}^{\prime })$ be two neutrosophic graph structures of graph structures $G^{*}_{1}=(X, E_{1}, E_{2})$ and $G^{*}_{2}=(X^{\prime }, E_{1}^{\prime }, E_{2}^{\prime })$, respectively, as shown in Figs. 2.10 and 2.11. Neutrosophic graph structure $G_{1}$ is identical to $G_{2}$ under $f:X\rightarrow X^{\prime }$ defined as

$$\begin{aligned} f(n_{1})=m_{2}, \, f(n_{2})=m_{1},\, f(n_{3})=m_{4}, \, f(n_{4})=m_{3}, \, f(n_{5})=m_{5}, \, f(n_{6})=m_{8}, \end{aligned}$$

$$\begin{aligned} f(n_{7})=m_{7}, \, f(n_{8})=m_{6}, \, T_{A}(n_{i})=T_{A^{\prime }}(f(n_{i})), \end{aligned}$$

$$\begin{aligned} I_{A}(n_{i})=I_{A^{\prime }}(f(n_{i})), \, F_{A}(n_{i})=F_{A^{\prime }}(f(n_{i})), \end{aligned}$$

for all $n_{i}\in X$ and

$$\begin{aligned} T_{B_{i}}(n_{i}n_{j})=T_{B_{i}^{\prime }}(f(n_{i})f(n_{j})),\, I_{B_{i}}(n_{i}n_{j})=I_{B_{i}^{\prime }}(f(n_{i})f(n_{j})),\, F_{B_{i}}(n_{i}n_{j})=F_{B_{i}^{\prime }}(f(n_{i})f(n_{j})), \end{aligned}$$

for all $n_{i}n_{j}\in E_{i}$ and $i=1,2.$

Fig. 2.10

Neutrosophic graph structure $G_{1}$

Fig. 2.11

Neutrosophic graph structure $G_{2}$

Definition 2.29

Let $G=(A, B_{1}, B_{2},\ldots , B_{n})$ be a neutrosophic graph structure and $\phi$ be a permutation on $\{B_{1}, B_{2},\ldots , B_{n}\}$ and on $\{1,2,\ldots , n\}$ defined by $\phi (B_{i})=B_{j}$ if and only if $\phi (i)=j$ for all i. If $mn\in B_{i}$ for any i and

$$\begin{aligned} T_{B_{i}^{\phi }}(mn)=T_{A}(m)\wedge T_{A}(n)-\bigvee \limits _{j\ne i}T_{\phi (B_{j})}(mn),\, I_{B_{i}^{\phi }}(mn)=I_{A}(m)\wedge I_{A}(n)-\bigvee \limits _{j\ne i}I_{\phi (B_{j})}(mn), \end{aligned}$$

$$\begin{aligned} F_{B_{i}^{\phi }}(mn)=F_{A}(m)\vee F_{A}(n)-\bigwedge \limits _{j\ne i}T_{\phi (B_{j})}(mn),\, i=1,2,\ldots , n, \end{aligned}$$

then $mn\in B_{k}^{\phi }$, where k is selected such that

$$\begin{aligned} T_{B_{k}^{\phi }}(mn)\ge T_{B_{i}^{\phi }}(mn),\, I_{B_{k}^{\phi }}(mn)\ge I_{B_{i}^{\phi }}(mn)\text { and }F_{B_{k}^{\phi }}(mn)\ge F_{B_{i}^{\phi }}(mn)\text { for all }i, \end{aligned}$$

then neutrosophic graph structure $(A, B_{1}^{\phi }, B_{2}^{\phi },\ldots , B_{n}^{\phi })$ is called $\phi$-complement of G and denoted by ${G} ^{\phi c}$.

Example 2.14

Let $G=(A, B_{1}, B_{2}, B_{3})$ be a neutrosophic graph structure shown in Fig. 2.12 and $\phi$ be a permutation on $\{1,2,3\}$ defined as:

$\phi (1)=2,$ $\phi (2)=3,$ $\phi (3)=1$. By direct calculations, we found that

$n_{1}n_{3}\in B_{3}^{\phi }$, $n_{2}n_{3}\in B_{1}^{\phi }$, $n_{1}n_{2}\in B_{2}^{\phi }$. So, $G^{\phi c}=(A, B_{1}^{\phi }, B_{2}^{\phi }, B_{3}^{\phi })$ is $\phi$-complement of neutrosophic graph structure G as shown in Fig. 2.12.

Fig. 2.12

Neutrosophic graph structures G, $G^{\phi c}$

Proposition 2.1

$\phi$-complement of a neutrosophic graph structure ${G}=(A, B_{1}, B_{2},\ldots , B_{n})$ is always a strong neutrosophic graph structure. Moreover, if $\phi (i)=k$, where $i, k\in \{1,2,\ldots , n\}$, then all $B_{k}$-edges in neutrosophic graph structure $(A, B_{1}, B_{2},\ldots , B_{n})$ become $B_{i}^{\phi }$-edges in

$(A, B_{1}^{\phi }, B_{2}^{\phi },\ldots , B_{n}^{\phi })$.

Proof

According to the definition of $\phi$-complement,

$$\begin{aligned} T_{B_{i}^{\phi }}(mn)=T_{A}(m)\wedge T_{A}(n)-\bigvee \limits _{j\ne i}T_{\phi (B_{j})}(mn), \end{aligned}$$

$$\begin{aligned} I_{B_{i}^{\phi }}(mn)=I_{A}(m)\wedge I_{A}(n)-\bigvee \limits _{j\ne i}I_{\phi (B_{j})}(mn), \end{aligned}$$

$$\begin{aligned} F_{B_{i}^{\phi }}(mn)=F_{A}(m)\vee F_{A}(n)-\bigwedge \limits _{j\ne i}F_{\phi (B_{j})}(mn), \end{aligned}$$

for $i\in \{1,2,\ldots , n\}$. For expression of truthness in $\phi$-complement:

Since

$$\begin{aligned} T_{A}(m)\wedge T_{A}(n)\ge 0,\bigvee \limits _{j\ne i}T_{\phi (B_{j})}(mn)\ge 0\text { and }T_{B_{i}}(mn)\le T_{A}(m)\wedge T_{A}(n) , \, \forall B_{i}, \end{aligned}$$

we see that

$$\begin{aligned} \bigvee \limits _{j\ne i}T_{\phi (B_{j})}(mn)\le T_{A}(m)\wedge T_{A}(n), \end{aligned}$$

which implies that

$$\begin{aligned} T_{A}(m)\wedge T_{A}(n)-\bigvee \limits _{j\ne i}T_{\phi (B_{j})}(mn)\ge 0. \end{aligned}$$

Therefore, $T_{B_{i}^{\phi }}(mn)\ge 0$ $\forall i$. Moreover, $T_{B_{i}^{\phi }}(mn)$ achieves its maximum value when $\bigvee \limits _{j \ne i}T_{\phi (B_{j})}(mn)$ is zero. It is obvious that when $\phi (B_{i})=B_{k}$ and mn is a $B_{k}$-edge then $\bigvee \limits _{j\ne i}T_{\phi (B_{j})}(mn)$ gets zero value. So

$$\begin{aligned} T_{B_{i}^{\phi }}(mn)=T_{A}(m)\wedge T_{A}(n),\,\, for\,(mn)\in B_{k},\,\, \phi (B_{i})=B_{k}. \end{aligned}$$

Similarly, we have

$$\begin{aligned} I_{B_{i}^{\phi }}(mn)=I_{A}(m)\wedge I_{A}(n),\,\, for\,(mn)\in B_{k},\,\, \phi (B_{i})=B_{k}. \end{aligned}$$

In the similar way for expression of falsity in $\phi$-complement:

Since

$$\begin{aligned} F_{A}(m)\vee F_{A}(n)\ge 0,\, \bigwedge \limits _{j\ne i}F_{\phi (B_{j})}(mn)\ge 0\text { and }F_{B_{i}}(mn)\le F_{A}(m)\vee F_{A}(n)\forall B_{i}{,} \end{aligned}$$

we see that

$$\begin{aligned} \bigwedge \limits _{j\ne i}F_{\phi (B_{j})}(mn)\le F_{A}(m)\vee F_{A}(n), \end{aligned}$$

which implies that

$$\begin{aligned} F_{A}(m)\vee F_{A}(n)-\bigwedge \limits _{j\ne i}F_{\phi (B_{j})}(mn)\ge 0. \end{aligned}$$

Therefore, $F_{B_{i}^{\phi }}(mn)$ is nonnegative for all i. Moreover, $F_{B_{i}^{\phi }}(mn)$ attains its maximum value when $\bigwedge \limits _{j \ne i}F_{\phi (B_{j})}(mn)$ becomes zero. It is clear that when $\phi (B_{i})=B_{k}$ and mn is a $B_{k}$-edge then $\bigwedge \limits _{j\ne i}F_{\phi (B_{j})}(mn)$ gets zero value. So

$$\begin{aligned} F_{B_{i}^{\phi }}(mn)=F_{A}(m)\vee F_{A}(n)\text { for }(mn)\in B_{k},\,\phi (B_{i})=B_{k}. \end{aligned}$$

This completes the proof.

Definition 2.30

Let $G=(A, B_{1}, B_{2},\ldots , B_{n})$ be a neutrosophic graph structure and $\phi$ be a permutation on $\{1,2,\ldots , n\}$. Then

• (i)   If G is isomorphic to $G^{\phi c}$, then G is said to be self-complementary.

• (ii)   If G is identical to $G^{\phi c}$, then G is said to be strong self-complementary.

Definition 2.31

Suppose $G=(A, B_{1}, B_{2},\ldots , B_{n})$ be a neutrosophic graph structure. Then

• (i)   If G is isomorphic to $G^{\phi c}$, for all permutations $\phi$ on $\{1,2,\ldots , n\}$, then G is totally self-complementary.

• (ii)   If G is identical to $G^{\phi c}$, for all permutations $\phi$ on $\{1,2,\ldots , n\}$, then G is totally strong self-complementary.

Remark 2.1

All strong neutrosophic graph structures are self-complementary or totally self-complementary neutrosophic graph structures.

Example 2.15

A neutrosophic graph structure $G=(A, B_{1}, B_{2}, B_{3})$ in Fig. 2.13 is a totally strong self-complementary neutrosophic graph structure.

Fig. 2.13

Totally strong self-complementary neutrosophic graph structure

Theorem 2.1

A neutrosophic graph structure is totally self-complementary if and only if it is strong neutrosophic graph structure.

Proof

Consider a strong neutrosophic graph structure G and a permutation $\phi$ on $\{1,2,\ldots , n\}$. By Proposition 2.1, $\phi$-complement of a neutrosophic graph structure ${G}=(A, B_{1}, B_{2},\ldots , B_{n})$ is always a strong neutrosophic graph structure. Moreover, if $\phi (i)=k$, where i, k $\in \{1,2,\ldots , n\}$, then all $B_{k}$-edges in neutrosophic graph structure $(A, B_{1}, B_{2},\ldots , B_{n})$ become $B_{i}^{\phi }$-edges in $(A, B_{1}^{\phi }, B_{2}^{\phi },\ldots , B_{n}^{\phi })$. This leads

$$\begin{aligned} T_{B_{k}}(mn)=T_{A}(m)\wedge T_{A}(n)=T_{B_{i}^{\phi }}(mn),\, I_{B_{k}}(mn)=I_{A}(m)\wedge I_{A}(n)=I_{B_{i}^{\phi }}(mn) \end{aligned}$$

and

$$\begin{aligned} F_{B_{k}}(mn)=F_{A}(m)\vee F_{A}(n)=F_{B_{i}^{\phi }}(mn). \end{aligned}$$

Hence, under the mapping (identity mapping) $f:X\rightarrow X$, G and $G^{\phi }$ are isomorphic such that

$$\begin{aligned} T_{A}(m)=T_{A}(f(m)),\, I_{A}(m)=I_{A}(f(m)),\, F_{A}(m)=F_{A}(f(m)), \end{aligned}$$

$$\begin{aligned} T_{B_{k}}(mn)=T_{B_{i}^{\phi }}(f(m)f(n))=T_{B_{i}^{\phi }}(mn),\, I_{B_{k}}(mn)=I_{B_{i}^{\phi }}(f(m)f(n))=I_{B_{i}^{\phi }}(mn), \end{aligned}$$

$$\begin{aligned} F_{B_{k}}(mn)=F_{B_{i}^{\phi }}(f(m)f(n))=F_{B_{i}^{\phi }}(mn), \end{aligned}$$

for all $mn\in E_{k}$, $\phi ^{-1}(k)=i$ and $k=1,2,\ldots , n$. This is satisfied for every permutation $\phi$ on $\{1,2,\ldots , n\}$. Hence, G is totally self-complementary neutrosophic graph structure. Conversely, let for every permutation $\phi$ on $\{1,2,\ldots , n\}$, G and ${G} ^{\phi }$ are isomorphic. Then according to the definition of isomorphism of neutrosophic graph structures and $\phi$-complement of neutrosophic graph structure,

$$\begin{aligned} T_{B_{k}}(mn)=T_{B_{i}^{\phi }}(f(m)f(n))=T_{A}(f(m))\wedge T_{A}(f(n))=T_{A}(m)\wedge T_{A}(n), \end{aligned}$$

$$\begin{aligned} I_{B_{k}}(mn)=I_{B_{i}^{\phi }}(f(m)f(n))=I_{A}(f(m))\wedge I_{A}(f(n))=T_{A}(m)\wedge I_{A}(n), \end{aligned}$$

$$\begin{aligned} F_{B_{k}}(mn)=F_{B_{i}^{\phi }}(f(m)f(n))=F_{A}(f(m))\vee T_{A}(f(n))=F_{A}(m)\wedge T_{A}(n), \end{aligned}$$

for all $mn\in E_{k}$ and $k=1,2,\ldots , n$. Hence, G is strong neutrosophic graph structure.

Remark 2.2

Every self-complementary neutrosophic graph structure is totally self-complementary.

Theorem 2.2

If $G^{*}=(X, E_{1}, E_{2},\ldots , E_{n})$ is a totally strong self-complementary graph structure and $A=(T_{A},I_{A}, F_{A})$ is a neutrosophic subset of X where $T_{A},I_{A}, F_{A}$ are constant valued functions, then a strong neutrosophic graph structure of $G^{*}$ with neutrosophic vertex set A is always a totally strong self-complementary neutrosophic graph structure.

Proof

Consider three constants p, q, r $\in [0,1]$, such that $T_{A}(m)=p,I_{A}(m)=q, F_{A}(m)=r$ $\forall m\in X$. Since $G^{*}$ is totally self-complementary strong graph structure, so there is a bijection $f:X\rightarrow X$ for any permutation $\phi ^{-1}$ on $\{1,2,\ldots , n\}$, such that for any $E_{k}$-edge (mn), (f(m)f(n)) [an $E_{i}$-edge in $G^{*}$ ] is an $E_{k}$-edge in $G^{*\phi ^{-1}c}$. Hence, for every $B_{k}$-edge (mn), (f(m)f(n)) [a $B_{i}$-edge in G ] is a $B_{k}^{\phi }$-edge in ${G}^{\phi ^{-1}c}$. Moreover, G is strong neutrosophic graph structure. Thus,

$$\begin{aligned} T_{A}(m)=p=T_{A}(f(m)),\ I_{A}(m)=q=I_{A}(f(m)),\ F_{A}(m)=r=F_{A}(f(m)),\ \forall m\in X, \end{aligned}$$

$$\begin{aligned} T_{B_{k}}(mn)=T_{A}(m)\wedge T_{A}(n)=T_{A}(f(m))\wedge T_{A}(f(n))=T_{B_{i}^{\phi }}(f(m)f(n)), \end{aligned}$$

$$\begin{aligned} I_{B_{k}}(mn)=I_{A}(m)\wedge I_{A}(n)=I_{A}(f(m))\wedge I_{A}(f(n))=I_{B_{i}^{\phi }}(f(m)f(n)), \end{aligned}$$

$$\begin{aligned} F_{B_{k}}(mn)=F_{A}(m)\vee I_{A}(n)=F_{A}(f(m))\vee F_{A}(f(n))=F_{B_{i}^{\phi }}(f(m)f(n)), \end{aligned}$$

for all $mn\in E_{i}$ and $i=1,2,\ldots , n$. This shows that G is self-complementary strong neutrosophic graph structure. Every permutation $\phi$ and $\phi ^{-1}$ on $\{1,2,\ldots , n\}$ satisfy above expressions; thus G is totally strong self-complementary neutrosophic graph structure.

Remark 2.3

Converse of Theorem 2.2 may not be true, for example a neutrosophic graph structure shown in Fig. 2.13 is a totally strong self-complementary, it is strong and its underlying graph structure is a totally strong self-complementary but $T_{A}$, $I_{A}$, $F_{A}$ are not constant functions.

2.3 Operations on Neutrosophic Graph Structures

In this section, we present the operations on neutrosophic graph structures.

Definition 2.32

Let $G_{1}$ = $(A_{1}, B_{11}, B_{12},\ldots , B_{1n})$ and $G_{2}$ = $(A_{2}, B_{21}, B_{22},\ldots , B_{2n})$ be neutrosophic graph structures of the graph structures $G^{*}_{1}$ = $(X_{1}, E_{11}, E_{12}, \ldots , E_{1n})$ and $G^{*}_{2}$ = $(X_{2}, E_{21}, E_{22},\ldots , E_{2n})$, respectively. The Cartesian product of $G_{1}$ and $G_{2}$, denoted by

$$ G_{1} \times G_{2} = (A_{1}\times A_{2}, B_{11}\times B_{21}, B_{12}\times B_{22},\ldots , B_{1n}\times B_{2n}), $$

is defined by the following:

• (i)   $\left\{ \begin{array}{ll} T_{(A_{1}\times A_{2})}(qr) = (T_{A_{1}}\times T_{A_{2}})(qr) = T_{A_{1}}(q) \wedge T_{A_{2}}(r)\\ I_{(A_{1}\times A_{2})}(qr) = (I_{A_{1}}\times I_{A_{2}})(qr) = I_{A_{1}}(q) \wedge I_{A_{2}}(r)\\ F_{(A_{1}\times A_{2})}(qr) = (F_{A_{1}}\times F_{A_{2}})(qr) = F_{A_{1}}(q) \vee F_{A_{2}}(r)\\ \end{array}\right.$ for all $qr \in E_{1} \times E_{2}$,

• (ii)   $\left\{ \begin{array}{ll} T_{(B_{1i}\times B_{2i})}(qr_{1})(qr_{2}) = (T_{B_{1i}}\times T_{B_{2i}})(qr_{1})(qr_{2}) = T_{A_{1}}(q) \wedge T_{B_{2i}}(r_{1}r_{2})\\ I_{(B_{1i}\times B_{2i})}(qr_{1})(qr_{2}) = (I_{B_{1i}}\times I_{B_{2i}})(qr_{1})(qr_{2}) = I_{A_{1}}(q) \wedge I_{B_{2i}}(r_{1}r_{2})\\ F_{(B_{1i}\times B_{2i})}(qr_{1})(qr_{2}) = (F_{B_{1i}}\times F_{B_{2i}})(qr_{1})(qr_{2}) = F_{A_{1}}(q) \vee F_{B_{2i}}(r_{1}r_{2})\\ \end{array}\right.$ for all $q \in X_{1}$, $r_{1}r_{2} \in E_{2i},$

• (iii)   $\left\{ \begin{array}{ll} T_{(B_{1i}\times B_{2i})}(q_{1}r)(q_{2}r) = (T_{B_{1i}}\times T_{B_{2i}})(q_{1}r)(q_{2}r) = T_{A_{2}}(r) \wedge T_{B_{1i}}(q_{1}q_{2})\\ I_{(B_{1i}\times B_{2i})}(q_{1}r)(q_{2}r) = (I_{B_{1i}}\times I_{B_{2i}})(q_{1}r)(q_{2}r) = I_{A_{2}}(r) \wedge I_{B_{1i}}(q_{1}q_{2})\\ F_{(B_{1i}\times B_{2i})}(q_{1}r)(q_{2}r) = (F_{B_{1i}}\times F_{B_{2i}})(q_{1}r)(q_{2}r) = F_{A_{2}}(r) \vee F_{B_{1i}}(q_{1}q_{2})\\ \end{array}\right.$ for all $r \in X_{2}$, $q_{1}q_{2} \in E_{1i}$.

Example 2.16

Consider $G_{1}$ = $(A_{1}, B_{11}, B_{12})$ and $G_{2}$ = $(A_{2}, B_{21}, B_{22})$ are neutrosophic graph structures of graph structures ${G}^{*}_{1}$ = $(X_{1}, E_{11}, E_{12})$ and ${G}^{*}_{2}$ = $(X_{2}, E_{21}, E_{22})$, respectively, as shown in Fig. 2.14, where $E_{11} = \{q_{1}q_{2}\}$, $E_{12} = \{q_{3}q_{4}\}$, $E_{21} = \{r_{1}r_{2}\}$, $E_{22} = \{r_{2}r_{3}\}$.

Fig. 2.14

Neutrosophic graph structures

Cartesian product of $G_{1}$ and $G_{2}$ defined as $G_{1} \times G_{2}$ = $\{A_{1}\times A_{2}, B_{11}\times B_{21}, B_{12}\times B_{22}\}$ is shown in the Fig. 2.15.

Fig. 2.15

Cartesian product of two neutrosophic graph structures

Theorem 2.3

The Cartesian product $G_{1} \times G_{2}$ = $(A_{1}\times A_{2}, B_{11}\times B_{21}, B_{12}\times B_{22},\ldots , B_{1n}\times B_{2n})$ of two neutrosophic graph structures $G_{1}$ and $G_{2}$ of the graph structures $G^{*}_{1}$ and $G^{*}_{2}$ is a neutrosophic graph structure of $G^{*}_{1} \times G^{*}_{2}$.

Proof

According to the definition of Cartesian product, there are two cases:

 

Case 1.:

When $q \in X_{1}$, $r_{1}r_{2} \in E_{2i}$

$$\begin{aligned} T_{(B_{1i}\times B_{2i})}((qr_{1})(qr_{2}))&= T_{A_{1}}(q) \wedge T_{B_{2i}}(r_{1}r_{2})\\&\le T_{A_{1}}(q) \wedge [T_{A_{2}}(r_{1}) \wedge T_{A_{2}}(r_{2})]~\\&= [T_{A_{1}}(q) \wedge T_{A_{2}}(r_{1})]~ \wedge [T_{A_{1}}(q) \wedge T_{A_{2}}(r_{2})]~\\&= T_{(A_{1}\times A_{2})}(qr_{1})\wedge T_{(A_{1}\times A_{2})}(qr_{2}), \end{aligned}$$

$$\begin{aligned} I_{(B_{1i}\times B_{2i})}((qr_{1})(qr_{2}))&= I_{A_{1}}(q) \wedge I_{B_{2i}}(r_{1}r_{2})\\&\le I_{A_{1}}(q) \wedge [I_{A_{2}}(r_{1}) \wedge I_{A_{2}}(r_{2})]~\\&= [I_{A_{1}}(q) \wedge I_{A_{2}}(r_{1})]~ \wedge [I_{A_{1}}(q) \wedge I_{A_{2}}(r_{2})]~\\&= I_{(A_{1}\times A_{2})}(qr_{1})\wedge I_{(A_{1}\times A_{2})}(qr_{2}), \end{aligned}$$

$$\begin{aligned} F_{(B_{1i}\times B_{2i})}((qr_{1})(qr_{2}))&= F_{A_{1}}(q) \vee F_{B_{2i}}(r_{1}r_{2})\\&\le F_{A_{1}}(q) \vee [F_{A_{2}}(r_{1}) \vee F_{A_{2}}(r_{2})]~\\&= [F_{A_{1}}(q) \vee F_{A_{2}}(r_{1})]~ \vee [F_{A_{1}}(q) \vee F_{A_{2}}(r_{2})]~\\&= F_{(A_{1}\times A_{2})}(qr_{1})\vee F_{(A_{1}\times A_{2})}(qr_{2}), \end{aligned}$$

for $qr_{1}, qr_{2} \in X_{1}\times X_{2}.$

Case 2.:

When $q \in X_{2}$, $r_{1}r_{2} \in E_{1i}$

$$\begin{aligned} T_{(B_{1i}\times B_{2i})}((r_{1}q)(r_{2}q))&= T_{A_{2}}(q) \wedge T_{B_{1i}}(r_{1}r_{2})\\&\le T_{A_{2}}(q) \wedge [T_{A_{1}}(r_{1}) \wedge T_{A_{1}}(r_{2})]~\\&= [T_{A_{2}}(q) \wedge T_{A_{1}}(r_{1})]~ \wedge [T_{A_{2}}(q) \wedge T_{A_{1}}(r_{2})]~\\&= T_{(A_{1}\times A_{2})}(r_{1}q)\wedge T_{(A_{1}\times A_{2})}(r_{2}q), \end{aligned}$$

$$\begin{aligned} I_{(B_{1i}\times B_{2i})}((r_{1}q)(r_{2}q))&= I_{A_{2}}(q) \wedge I_{B_{1i}}(r_{1}r_{2})\\&\le I_{A_{2}}(q) \wedge [I_{A_{1}}(r_{1}) \wedge I_{A_{1}}(r_{2})]~\\&= [I_{A_{2}}(q) \wedge I_{A_{1}}(r_{1})]~ \wedge [I_{A_{2}}(q) \wedge I_{A_{1}}(r_{2})]~\\&= I_{(A_{1}\times A_{2})}(r_{1}q)\wedge I_{(A_{1}\times A_{2})}(r_{2}q), \end{aligned}$$

$$\begin{aligned} F_{(B_{1i}\times B_{2i})}((r_{1}q)(r_{2}q))&= F_{A_{2}}(q) \vee F_{B_{1i}}(r_{1}r_{2})\\&\le F_{A_{2}}(q) \vee [F_{A_{1}}(r_{1}) \vee F_{A_{1}}(r_{2})]~\\&= [F_{A_{2}}(q) \vee F_{A_{1}}(r_{1})]~ \vee [F_{A_{2}}(q) \vee F_{A_{1}}(r_{2})]~\\&= F_{(A_{1}\times A_{2})}(r_{1}q)\vee F_{(A_{1}\times A_{2})}(r_{2}q), \end{aligned}$$

for $r_{1}q, r_{2}q \in X_{1}\times X_{2}.$

 

Both cases are satisfied $\forall i \in \{1,2,\ldots , n\}$.

Definition 2.33

Let $G_{1}$ = $(A_{1}, B_{11}, B_{12},\ldots , B_{1n})$ and $G_{2}$ = $(A_{2}, B_{21}, B_{22},\ldots , B_{2n})$ be neutrosophic graph structures. The cross product of $G_{1}$ and $G_{2}$, denoted by

$$ G_{1} * G_{2} = (A_{1} * A_{2}, B_{11}* B_{21}, B_{12}* B_{22},\ldots , B_{1n}* B_{2n}), $$

is defined by the following:

• (i)   $\left\{ \begin{array}{ll} T_{(A_{1} * A_{2})}(qr) = (T_{A_{1}} * T_{A_{2}})(qr) = T_{A_{1}}(q) \wedge T_{A_{2}}(r)\\ I_{(A_{1} * A_{2})}(qr) = (I_{A_{1}} * I_{A_{2}})(qr) = I_{A_{1}}(q) \wedge I_{A_{2}}(r)\\ F_{(A_{1} * A_{2})}(qr) = (F_{A_{1}} * F_{A_{2}})(qr) = F_{A_{1}}(q) \vee F_{A_{2}}(r) \end{array}\right.$ for all $qr \in X_{1} \times X_{2}$,

• (ii)  $\left\{ \begin{array}{ll}T_{(B_{1i} * B_{2i})}(q_{1}r_{1})(q_{2}r_{2}) =(T_{B_{1i}} * T_{B_{2i}})(q_{1}r_{1})(q_{2}r_{2}) = T_{B_{1i}}(q_{1}q_{2}) \wedge T_{B_{2i}}(r_{1}r_{2})\\ I_{(B_{1i} * B_{2i})}(q_{1}r_{1})(q_{2}r_{2}) = (I_{B_{1i}} * I_{B_{2i}})(q_{1}r_{1})(q_{2}r_{2}) = I_{B_{1i}}(q_{1}q_{2}) \wedge I_{B_{2i}}(r_{1}r_{2})\\ F_{(B_{1i}* B_{2i})}(q_{1}r_{1})(q_{2}r_{2}) = (F_{B_{1i}}* F_{B_{2i}})(q_{1}r_{1})(q_{2}r_{2}) = F_{B_{1i}}(q_{1}q_{2}) \vee F_{B_{2i}}(r_{1}r_{2}) \end{array}\right.$ for all $q_{1}q_{2} \in E_{1i}$, $r_{1}r_{2} \in E_{2i}$.

Example 2.17

Cross product of two neutrosophic graph structures $G_{1}$ and $G_{2}$ shown in Fig. 2.14 is defined as $G_{1} * G_{2}$ = $\{A_{1}* A_{2}, B_{11} * B_{21}, B_{12} * B_{22}\}$ and is shown in the Fig. 2.16.

Fig. 2.16

Cross product of two neutrosophic graph structures

Theorem 2.4

The cross product $G_{1} * G_{2}$ = $(A_{1} * A_{2}, B_{11} * B_{21}, B_{12} * B_{22}, \ldots , B_{1n} * B_{2n})$ of two neutrosophic graph structures of the graph structures $G^{}*_{1}$ and $G^{*}_{2}$ is a neutrosophic graph structure of $G^{*}_{1} * G^{*}_{2}$.

Proof

For all $q_{1}r_{1}, q_{2}r_{2} \in X_{1}*X_{2}$

$$\begin{aligned} T_{(B_{1i}* B_{2i})}((q_{1}r_{1})(q_{2}r_{2}))&=T_{B_{1i}}(q_{1}q_{2}) \wedge T_{B_{2i}}(r_{1}r_{2})\\&\le [T_{A_{1}}(q_{1})\wedge T_{A_{1}}(q_{2})]~ \wedge [T_{A_{2}}(r_{1}) \wedge T_{A_{2}}(r_{2})]~\\&= [T_{A_{1}}(q_{1}) \wedge T_{A_{2}}(r_{1})]~ \wedge [T_{A_{1}}(q_{2}) \wedge T_{A_{2}}(r_{2})]~\\&= T_{(A_{1}* A_{2})}(q_{1}r_{1})\wedge T_{(A_{1}* A_{2})}(q_{2}r_{2}), \end{aligned}$$

$$\begin{aligned} I_{(B_{1i}* B_{2i})}((q_{1}r_{1})(q_{2}r_{2}))&=I_{B_{1i}}(q_{1}q_{2}) \wedge I_{B_{2i}}(r_{1}r_{2})\\&\le [I_{A_{1}}(q_{1})\wedge I_{A_{1}}(q_{2})]~ \wedge [I_{A_{2}}(r_{1}) \wedge I_{A_{2}}(r_{2})]~\\&= [I_{A_{1}}(q_{1}) \wedge I_{A_{2}}(r_{1})]~ \wedge [I_{A_{1}}(q_{2}) \wedge I_{A_{2}}(r_{2})]~\\&= I_{(A_{1}* A_{2})}(q_{1}r_{1})\wedge I_{(A_{1}* A_{2})}(q_{2}r_{2}), \end{aligned}$$

$$\begin{aligned} F_{(B_{1i}*B_{2i})}((q_{1}r_{1})(q_{2}r_{2}))&=F_{B_{1i}}(q_{1}q_{2}) \vee F_{B_{2i}}(r_{1}r_{2})\\&\le [F_{A_{1}}(q_{1})\vee F_{A_{1}}(q_{2})]~ \vee [F_{A_{2}}(r_{1}) \vee F_{A_{2}}(r_{2})]~\\&=[F_{A_{1}}(q_{1}) \vee F_{A_{2}}(r_{1})]~ \vee [F_{A_{1}}(q_{2}) \vee F_{A_{2}}(r_{2})]~\\&=F_{(A_{1}* A_{2})}(q_{1}r_{1})\vee F_{(A_{1}* A_{2})}(q_{2}r_{2}), \end{aligned}$$

for $i \in \{1,2,\ldots , n\}$.

Definition 2.34

Let $G_{1}$ = $(A_{1}, B_{11}, B_{12},\ldots , B_{1n})$ and $G_{2}$ = $(A_{2}, B_{21}, B_{22},\ldots , B_{2n})$ be neutrosophic graph structures. The lexicographic product of $G_{1}$ and $G_{2}$, denoted by

$$ G_{1} \bullet G_{2} = (A_{1}\bullet A_{2}, B_{11}\bullet B_{21}, B_{12}\bullet B_{22},\ldots , B_{1n}\bullet B_{2n}), $$

is defined by the following:

• (i)   $\left\{ \begin{array}{ll} T_{(A_{1}\bullet A_{2})}(qr) = (T_{A_{1}}\bullet T_{A_{2}})(qr) = T_{A_{1}}(q) \wedge T_{A_{2}}(r)\\ I_{(A_{1}\bullet A_{2})}(qr) = (I_{A_{1}}\bullet I_{A_{2}})(qr) = I_{A_{1}}(q) \wedge I_{A_{2}}(r)\\ F_{(A_{1}\bullet A_{2})}(qr) = (F_{A_{1}}\bullet F_{A_{2}})(qr) = F_{A_{1}}(q) \vee F_{A_{2}}(r) \end{array}\right.$ for all $qr \in X_{1} \times X_{2}$,

• (ii)   $\left\{ \begin{array}{ll} T_{(B_{1i}\bullet B_{2i})}(qr_{1})(qr_{2}) = (T_{B_{1i}}\bullet T_{B_{2i}})(qr_{1})(qr_{2}) = T_{A_{1}}(q) \wedge T_{B_{2i}}(r_{1}r_{2})\\ I_{(B_{1i}\bullet B_{2i})}(qr_{1})(qr_{2}) = (I_{B_{1i}}\bullet I_{B_{2i}})(qr_{1})(qr_{2}) = I_{A_{1}}(q) \wedge I_{B_{2i}}(r_{1}r_{2})\\ F_{(B_{1i}\bullet B_{2i})}(qr_{1})(qr_{2}) = (F_{B_{1i}}\bullet F_{B_{2i}})(qr_{1})(qr_{2}) = F_{A_{1}}(q) \vee F_{B_{2i}}(r_{1}r_{2})\\ \end{array}\right.$ for all $q \in X_{1}$, $r_{1}r_{2} \in E_{2i}$,

• (iii)   $\left\{ \begin{array}{ll} T_{(B_{1i} \bullet B_{2i})}(q_{1}r_{1})(q_{2}r_{2}) =(T_{B_{1i}} \bullet T_{B_{2i}})(q_{1}r_{1})(q_{2}r_{2}) = T_{B_{1i}}(q_{1}q_{2}) \wedge T_{B_{2i}}(r_{1}r_{2})\\ I_{(B_{1i} \bullet B_{2i})}(q_{1}r_{1})(q_{2}r_{2}) = (I_{B_{1i}} \bullet I_{B_{2i}})(q_{1}r_{1})(q_{2}r_{2}) = I_{B_{1i}}(q_{1}q_{2}) \wedge I_{B_{2i}}(r_{1}r_{2})\\ F_{(B_{1i}\bullet B_{2i})}(q_{1}r_{1})(q_{2}r_{2}) = (F_{B_{1i}}\bullet F_{B_{2i}})(q_{1}r_{1})(q_{2}r_{2}) = F_{B_{1i}}(q_{1}q_{2}) \vee F_{B_{2i}}(r_{1}r_{2}) \end{array}\right.$ for all $q_{1}q_{2} \in E_{1i}$, $r_{1}r_{2} \in E_{2i}$.

Example 2.18

Lexicographic product of two neutrosophic graph structures $G_{1}$ and $G_{2}$ shown in Fig. 2.14 is defined as

$G_{1} \bullet G_{2}$ = $\{A_{1}\bullet A_{2}, B_{11} \bullet B_{21}, B_{12} \bullet B_{22}\}$ and is shown in the Fig. 2.17.

Fig. 2.17

Lexicographic product of two neutrosophic graph structures

Theorem 2.5

The lexicographic product $G_{1} \bullet G_{2}$ = $(A_{1}\bullet A_{2}, B_{11}\bullet B_{21}, B_{12}\bullet B_{22},\ldots , B_{1n}\bullet B_{2n})$ of two neutrosophic graph structures of the graph structures $G^{*}_{1}$ and $G^{*}_{2}$ is a neutrosophic graph structure of $G^{*}_{1} \bullet G^{*}_{2}$.

Proof

According to the definition of lexicographic product, there are two cases:

 

Case 1.:

When $q \in X_{1}$, $r_{1}r_{2} \in E_{2i}$

$$\begin{aligned} T_{(B_{1i}\bullet B_{2i})}((qr_{1})(qr_{2}))&= T_{A_{1}}(q) \wedge T_{B_{2i}}(r_{1}r_{2})\\&\le T_{A_{1}}(q) \wedge [T_{A_{2}}(r_{1}) \wedge T_{A_{2}}(r_{2})]~\\&= [T_{A_{1}}(q) \wedge T_{A_{2}}(r_{1})]~ \wedge [T_{A_{1}}(q) \wedge T_{A_{2}}(r_{2})]~\\&= T_{(A_{1}\bullet A_{2})}(qr_{1})\wedge T_{(A_{1}\bullet A_{2})}(qr_{2}), \end{aligned}$$

$$\begin{aligned} I_{(B_{1i}\bullet B_{2i})}((qr_{1})(qr_{2}))&= I_{A_{1}}(q) \wedge I_{B_{2i}}(r_{1}r_{2})\\&\le I_{A_{1}}(q) \wedge [I_{A_{2}}(r_{1}) \wedge I_{A_{2}}(r_{2})]~\\&= [I_{A_{1}}(q) \wedge I_{A_{2}}(r_{1})]~ \wedge [I_{A_{1}}(q) \wedge I_{A_{2}}(r_{2})]~\\&= I_{(A_{1}\bullet A_{2})}(qr_{1})\wedge I_{(A_{1}\bullet A_{2})}(qr_{2}), \end{aligned}$$

$$\begin{aligned} F_{(B_{1i}\bullet B_{2i})}((qr_{1})(qr_{2}))&= F_{A_{1}}(q) \vee F_{B_{2i}}(r_{1}r_{2})\\&\le F_{A_{1}}(q) \vee [F_{A_{2}}(r_{1}) \vee F_{A_{2}}(r_{2})]~\\&= [F_{A_{1}}(q) \vee F_{A_{2}}(r_{1})]~ \vee [F_{A_{1}}(q) \vee F_{A_{2}}(r_{2})]~\\&= F_{(A_{1}\bullet A_{2})}(qr_{1})\vee F_{(A_{1}\bullet A_{2})}(qr_{2}), \end{aligned}$$

for $qr_{1}, qr_{2} \in X_{1}\bullet X_{2}.$

Case 2.:

When $q_{1}q_{2} \in E_{1i}, r_{1}r_{2} \in E_{2i}$

$$\begin{aligned} T_{(B_{1i}\bullet B_{2i})}((q_{1}r_{1})(q_{2}r_{2}))&=T_{B_{1i}}(q_{1}q_{2}) \wedge T_{B_{2i}}(r_{1}r_{2})\\&\le [T_{A_{1}}(q_{1})\wedge T_{A_{1}}(q_{2})]~ \wedge [T_{A_{2}}(r_{1}) \wedge T_{A_{2}}(r_{2})]~\\&= [T_{A_{1}}(q_{1}) \wedge T_{A_{2}}(r_{1})]~ \wedge [T_{A_{1}}(q_{2}) \wedge T_{A_{2}}(r_{2})]~\\&= T_{(A_{1}\bullet A_{2})}(q_{1}r_{1})\wedge T_{(A_{1} \bullet A_{2})}(q_{2}r_{2}), \end{aligned}$$

$$\begin{aligned} I_{(B_{1i}\bullet B_{2i})}((q_{1}r_{1})(q_{2}r_{2}))&=I_{B_{1i}}(q_{1}q_{2}) \wedge I_{B_{2i}}(r_{1}r_{2})\\&\le [I_{A_{1}}(q_{1})\wedge I_{A_{1}}(q_{2})]~ \wedge [I_{A_{2}}(r_{1}) \wedge I_{A_{2}}(r_{2})]~\\&= [I_{A_{1}}(q_{1}) \wedge I_{A_{2}}(r_{1})]~ \wedge [I_{A_{1}}(q_{2}) \wedge I_{A_{2}}(r_{2})]~\\&= I_{(A_{1}\bullet A_{2})}(q_{1}r_{1})\wedge I_{(A_{1}\bullet A_{2})}(q_{2}r_{2}), \end{aligned}$$

$$\begin{aligned} F_{(B_{1i}\bullet B_{2i})}((q_{1}r_{1})(q_{2}r_{2}))&=F_{B_{1i}}(q_{1}q_{2}) \vee F_{B_{2i}}(r_{1}r_{2})\\&\le [F_{A_{1}}(q_{1})\vee F_{A_{1}}(q_{2})]~ \vee [F_{A_{2}}(r_{1}) \vee F_{A_{2}}(r_{2})]~\\&=[F_{A_{1}}(q_{1}) \vee F_{A_{2}}(r_{1})]~ \vee [F_{A_{1}}(q_{2}) \vee F_{A_{2}}(r_{2})]~\\&=F_{(A_{1}\bullet A_{2})}(q_{1}r_{1})\vee F_{(A_{1} \bullet A_{2})}(q_{2}r_{2}), \end{aligned}$$

for $q_{1}r_{1}, q_{2}r_{2} \in X_{1} \bullet X_{2}$. Both cases are satisfied for $i \in \{1,2,\ldots , n\}$.

 

Definition 2.35

Let $G_{1}$ = $(A_{1}, B_{11}, B_{12},\ldots , B_{1n})$ and $G_{2}$ = $(A_{2}, B_{21}, B_{22},\ldots , B_{2n})$ be neutrosophic graph structures. The strong product of $G_{1}$ and $G_{2}$, denoted by

$$ G_{1} \boxtimes G_{2} = (A_{1}\boxtimes A_{2}, B_{11}\boxtimes B_{21}, B_{12}\boxtimes B_{22},\ldots , B_{1n}\boxtimes B_{2n}), $$

is defined by the following:

• (i)   $\left\{ \begin{array}{ll} T_{(A_{1}\boxtimes A_{2})}(qr) = (T_{A_{1}}\boxtimes T_{A_{2}})(qr) = T_{A_{1}}(q) \wedge T_{A_{2}}(r)\\ I_{(A_{1}\boxtimes A_{2})}(qr) = (I_{A_{1}}\boxtimes I_{A_{2}})(qr) = I_{A_{1}}(q) \wedge I_{A_{2}}(r)\\ F_{(A_{1}\boxtimes A_{2})}(qr) = (F_{A_{1}}\boxtimes F_{A_{2}})(qr) = F_{A_{1}}(q) \vee F_{A_{2}}(r)\\ \end{array}\right.$ for all $qr \in X_{1} \times X_{2}$,

• (ii)   $\left\{ \begin{array}{ll} T_{(B_{1i}\boxtimes B_{2i})}(qr_{1})(qr_{2}) = (T_{B_{1i}}\boxtimes T_{B_{2i}})(qr_{1})(qr_{2}) = T_{A_{1}}(q) \wedge T_{B_{2i}}(r_{1}r_{2})\\ I_{(B_{1i}\boxtimes B_{2i})}(qr_{1})(qr_{2}) = (I_{B_{1i}}\boxtimes I_{B_{2i}})(qr_{1})(qr_{2}) = I_{A_{1}}(q) \wedge I_{B_{2i}}(r_{1}r_{2})\\ F_{(B_{1i}\boxtimes B_{2i})}(qr_{1})(qr_{2}) = (F_{B_{1i}}\boxtimes F_{B_{2i}})(qr_{1})(qr_{2}) = F_{A_{1}}(q) \vee F_{B_{2i}}(r_{1}r_{2})\\ \end{array}\right.$ for all $q \in X_{1}$, $r_{1}r_{2} \in E_{2i}$,

• (iii)   $\left\{ \begin{array}{ll} T_{(B_{1i}\boxtimes B_{2i})}(q_{1}r)(q_{2}r) = (T_{B_{1i}}\boxtimes T_{B_{2i}})(q_{1}r)(q_{2}r) = T_{A_{2}}(r) \wedge T_{B_{1i}}(q_{1}q_{2})\\ I_{(B_{1i}\boxtimes B_{2i})}(q_{1}r)(q_{2}r) = (I_{B_{1i}}\boxtimes I_{B_{2i}})(q_{1}r)(q_{2}r) = I_{A_{2}}(r) \wedge I_{B_{1i}}(q_{1}q_{2})\\ F_{(B_{1i}\boxtimes B_{2i})}(q_{1}r)(q_{2}r) = (F_{B_{1i}}\boxtimes F_{B_{2i}})(q_{1}r)(q_{2}r) = F_{A_{2}}(r) \vee F_{B_{1i}}(q_{1}q_{2})\\ \end{array}\right.$ for all $r \in X_{2}$, $q_{1}q_{2} \in E_{1i}$,

• (iv)   $\left\{ \!\begin{array}{ll} T_{(B_{1i} \boxtimes B_{2i})}(q_{1}r_{1})(q_{2}r_{2}) {=}(T_{B_{1i}} \boxtimes T_{B_{2i}})(q_{1}r_{1})(q_{2}r_{2}) {=} T_{B_{1i}}(q_{1}q_{2}) \wedge T_{B_{2i}}(r_{1}r_{2})\\ I_{(B_{1i} \boxtimes B_{2i})}(q_{1}r_{1})(q_{2}r_{2}) = (I_{B_{1i}} \boxtimes I_{B_{2i}})(q_{1}r_{1})(q_{2}r_{2}) = I_{B_{1i}}(q_{1}q_{2}) \wedge I_{B_{2i}}(r_{1}r_{2})\\ F_{(B_{1i}\boxtimes B_{2i})}(q_{1}r_{1})(q_{2}r_{2}) {=} (F_{B_{1i}}\boxtimes F_{B_{2i}})(q_{1}r_{1})(q_{2}r_{2}) {=} F_{B_{1i}}(q_{1}q_{2}) \vee F_{B_{2i}}(r_{1}r_{2})\\ \end{array}\right.$ for all $q_{1}q_{2} \in E_{1i}$, $r_{1}r_{2} \in E_{2i}$.

Example 2.19

Strong product of two neutrosophic graph structures $G_{1}$ and $G_{2}$ shown in Fig. 2.14 is defined as $G_{1} \boxtimes G_{2}$ = $\{A_{1}\boxtimes A_{2}, B_{11} \boxtimes B_{21}, B_{12} \boxtimes B_{22}\}$ and is shown in the Fig. 2.18.

Fig. 2.18

Strong product of two neutrosophic graph structures

Theorem 2.6

The strong product $G_{1} \boxtimes G_{2}$ = $(A_{1} \boxtimes A_{2}, B_{11} \boxtimes B_{21}, B_{12} \boxtimes B_{22}, \ldots , B_{1n} \boxtimes B_{2n})$ of two neutrosophic graph structures of the graph structures $G^{*}_{1}$ and $G^{*}_{2}$ is a neutrosophic graph structure of $G^{*}_{1} \boxtimes G^{*}_{2}$.

Proof

According to the definition of strong product, there are three cases:

 

Case 1.:

When $q \in X_{1}$, $r_{1}r_{2} \in E_{2i}$

$$\begin{aligned} T_{(B_{1i}\boxtimes B_{2i})}((qr_{1})(qr_{2}))&= T_{A_{1}}(q) \wedge T_{B_{2i}}(r_{1}r_{2})\\&\le T_{A_{1}}(q) \wedge [T_{A_{2}}(r_{1}) \wedge T_{A_{2}}(r_{2})]~\\&= [T_{A_{1}}(q) \wedge T_{A_{2}}(r_{1})]~ \wedge [T_{A_{1}}(q) \wedge T_{A_{2}}(r_{2})]~\\&= T_{(A_{1}\boxtimes A_{2})}(qr_{1})\wedge T_{(A_{1}\boxtimes A_{2})}(qr_{2}), \end{aligned}$$

$$\begin{aligned} I_{(B_{1i}\boxtimes B_{2i})}((qr_{1})(qr_{2}))&= I_{A_{ 1}}(q) \wedge I_{B_{2i}}(r_{1}r_{2})\\&\le I_{A_{1}}(q) \wedge [I_{A_{2}}(r_{1}) \wedge I_{A_{2}}(r_{2})]~\\&= [I_{A_{1}}(q) \wedge I_{A_{2}}(r_{1})]~ \wedge [I_{A_{1}}(q) \wedge I_{A_{2}}(r_{2})]~\\&= I_{(A_{1}\boxtimes A_{2})}(qr_{1})\wedge I_{(A_{1}\boxtimes A_{2})}(qr_{2}), \end{aligned}$$

$$\begin{aligned} F_{(B_{1i}\boxtimes B_{2i})}((qr_{1})(qr_{2}))&= F_{A_{1}}(q) \vee F_{B_{2i}}(r_{1}r_{2})\\&\le F_{A_{1}}(q) \vee [F_{A_{2}}(r_{1}) \vee F_{A_{2}}(r_{2})]~\\&= [F_{A_{1}}(q) \vee F_{A_{2}}(r_{1})]~ \vee [F_{A_{1}}(q) \vee F_{A_{2}}(r_{2})]~\\&= F_{(A_{1}\boxtimes A_{2})}(qr_{1})\vee F_{(A_{1}\boxtimes A_{2})}(qr_{2}), \end{aligned}$$

for $qr_{1}, qr_{2} \in X_{1}\boxtimes X_{2}.$

Case 2.:

When $q \in X_{2}$, $r_{1}r_{2} \in E_{1i}$

$$\begin{aligned} T_{(B_{1i}\boxtimes B_{2i})}((r_{1}q)(r_{2}q))&= T_{A_{2}}(q) \wedge T_{B_{1i}}(r_{1}r_{2})\\&\le T_{A_{2}}(q) \wedge [T_{A_{1}}(r_{1}) \wedge T_{A_{1}}(r_{2})]~\\&= [T_{A_{2}}(q) \wedge T_{A_{1}}(r_{1})]~ \wedge [T_{A_{2}}(q) \wedge T_{A_{1}}(r_{2})]~\\&= T_{(A_{1}\boxtimes A_{2})}(r_{1}q)\wedge T_{(A_{1}\boxtimes A_{2})}(r_{2}q), \end{aligned}$$

$$\begin{aligned} I_{(B_{1i}\boxtimes B_{2i})}((r_{1}q)(r_{2}q))&= I_{A_{2}}(q) \wedge I_{B_{1i}}(r_{1}r_{2})\\&\le I_{A_{2}}(q) \wedge [I_{A_{1}}(r_{1}) \wedge I_{A_{1}}(r_{2})]~\\&= [I_{A_{2}}(q) \wedge I_{A_{1}}(r_{1})]~ \wedge [I_{A_{2}}(q) \wedge I_{A_{1}}(r_{2})]~\\&= I_{(A_{1}\boxtimes A_{2})}(r_{1}q)\wedge I_{(A_{1}\boxtimes A_{2})}(r_{2}q), \end{aligned}$$

$$\begin{aligned} F_{(B_{1i}\boxtimes B_{2i})}((r_{1}q)(r_{2}q))&= F_{A_{2}}(q) \vee F_{B_{1i}}(r_{1}r_{2})\\&\le F_{A_{2}}(q) \vee [F_{A_{1}}(r_{1}) \vee F_{A_{1}}(r_{2})]~\\&= [F_{A_{2}}(q) \vee F_{A_{1}}(r_{1})]~ \vee [F_{A_{2}}(q) \vee F_{A_{1}}(r_{2})]~\\&= F_{(A_{1}\boxtimes A_{2})}(r_{1}q)\vee F_{(A_{1}\boxtimes A_{2})}(r_{2}q), \end{aligned}$$

for $r_{1}q, r_{2}q \in X_{1}\boxtimes X_{2}.$

Case 3.:

For all $q_{1}q_{2} \in E_{1i}, r_{1}r_{2} \in E_{2i}$

$$\begin{aligned} T_{(B_{1i}\boxtimes B_{2i})}((q_{1}r_{1})(q_{2}r_{2}))&=T_{B_{1i}}(q_{1}q_{2}) \wedge T_{B_{2i}}(r_{1}r_{2})\\&\le [T_{A_{1}}(q_{1})\wedge T_{A_{1}}(q_{2})]~ \wedge [T_{A_{2}}(r_{1}) \wedge T_{A_{2}}(r_{2})]~\\&= [T_{A_{1}}(q_{1}) \wedge T_{A_{2}}(r_{1})]~ \wedge [T_{A_{1}}(q_{2}) \wedge T_{A_{2}}(r_{2})]~\\&= T_{(A_{1}\boxtimes A_{2})}(q_{1}r_{1})\wedge T_{(A_{1}\boxtimes A_{2})}(q_{2}r_{2}), \end{aligned}$$

$$\begin{aligned} I_{(B_{1i}\boxtimes B_{2i})}((q_{1}r_{1})(q_{2}r_{2}))&=I_{B_{1i}}(q_{1}q_{2}) \wedge I_{B_{2i}}(r_{1}r_{2})\\&\le [I_{A_{1}}(q_{1})\wedge I_{A_{1}}(q_{2})]~ \wedge [I_{A_{2}}(r_{1}) \wedge I_{A_{2}}(r_{2})]~\\&= [I_{A_{1}}(q_{1}) \wedge I_{A_{2}}(r_{1})]~ \wedge [I_{A_{1}}(q_{2}) \wedge I_{A_{2}}(r_{2})]~\\&= I_{(A_{1}\boxtimes A_{2})}(q_{1}r_{1})\wedge I_{(A_{1}\boxtimes A_{2})}(q_{2}r_{2}), \end{aligned}$$

$$\begin{aligned} F_{(B_{1i}\boxtimes B_{2i})}((q_{1}r_{1})(q_{2}r_{2}))&=F_{B_{1i}}(q_{1}q_{2}) \vee F_{B_{2i}}(r_{1}r_{2})\\&\le [F_{A_{1}}(q_{1})\vee F_{A_{1}}(q_{2})]~ \vee [F_{A_{2}}(r_{1}) \vee F_{A_{2}}(r_{2})]~\\&=[F_{A_{1}}(q_{1}) \vee F_{A_{2}}(r_{1})]~ \vee [F_{A_{1}}(q_{2}) \vee F_{A_{2}}(r_{2})]~\\&=F_{(A_{1}\boxtimes A_{2})}(q_{1}r_{1})\vee F_{(A_{1}\boxtimes A_{2})}(q_{2}r_{2}), \end{aligned}$$

 

for $q_{1}r_{1}, q_{2}r_{2} \in X_{1}\boxtimes X_{2}.$

All cases are satisfied for $i = 1,2,\ldots , n$.

Definition 2.36

Let $G_{1}$ = $(A_{1}, B_{11}, B_{12},\ldots , B_{1n})$ and $G_{2}$ = $(A_{2}, B_{21}, B_{22},\ldots , B_{2n})$ be neutrosophic graph structures. The composition of $G_{1}$ and $G_{2}$, denoted by

$$ G_{1} \circ G_{2} = (A_{1}\circ A_{2}, B_{11}\circ B_{21}, B_{12}\circ B_{22},\ldots , B_{1n}\circ B_{2n}), $$

is defined by the following:

• (i)   $\left\{ \begin{array}{ll} T_{(A_{1}\circ A_{2})}(qr) = (T_{A_{1}}\circ T_{A_{2}})(qr) = T_{A_{1}}(q) \wedge T_{A_{2}}(r)\\ I_{(A_{1}\circ A_{2})}(qr) = (I_{A_{1}}\circ I_{A_{2}})(qr) = I_{A_{1}}(q) \wedge I_{A_{2}}(r)\\ F_{(A_{1}\circ A_{2})}(qr) = (F_{A_{1}}\circ F_{A_{2}})(qr) = F_{A_{1}}(q) \vee F_{A_{2}}(r)\\ \end{array}\right.$ for all $qr \in X_{1} \times X_{2}$,

• (ii)   $\left\{ \begin{array}{ll} T_{(B_{1i}\circ B_{2i})}(qr_{1})(qr_{2}) = (T_{B_{1i}}\circ T_{B_{2i}})(qr_{1})(qr_{2}) = T_{A_{1}}(q) \wedge T_{B_{2i}}(r_{1}r_{2})\\ I_{(B_{1i}\circ B_{2i})}(qr_{1})(qr_{2}) = (I_{B_{1i}}\circ I_{B_{2i}})(qr_{1})(qr_{2}) = I_{A_{1}}(q) \wedge I_{B_{2i}}(r_{1}r_{2})\\ F_{(B_{1i}\circ B_{2i})}(qr_{1})(qr_{2}) = (F_{B_{1i}}\circ F_{B_{2i}})(qr_{1})(qr_{2}) = F_{A_{1}}(q) \vee F_{B_{2i}}(r_{1}r_{2})\\ \end{array}\right.$ for all $q \in X_{1}$, $r_{1}r_{2} \in E_{2i}$,

• (iii)   $\left\{ \begin{array}{ll} T_{(B_{1i}\circ B_{2i})}(q_{1}r)(q_{2}r) = (T_{B_{1i}}\circ T_{B_{2i}})(q_{1}r)(q_{2}r) = T_{A_{2}}(r) \wedge T_{B_{1i}}(q_{1}q_{2})\\ I_{(B_{1i}\circ B_{2i})}(q_{1}r)(q_{2}r) = (I_{B_{1i}}\circ I_{B_{2i}})(q_{1}r)(q_{2}r) = I_{A_{2}}(r) \wedge I_{B_{1i}}(q_{1}q_{2})\\ F_{(B_{1i}\circ B_{2i})}(q_{1}r)(q_{2}r) = (F_{B_{1i}}\circ F_{B_{2i}})(q_{1}r)(q_{2}r) = F_{A_{2}}(r) \vee F_{B_{1i}}(q_{1}q_{2})\\ \end{array}\right.$ for all $r \in X_{2}$, $q_{1}q_{2} \in E_{1i}$,

• (iv)   $\left\{ \begin{array}{ll} T_{(B_{1i} \circ B_{2i})}(q_{1}r_{1})(q_{2}r_{2}) =(T_{B_{1i}} \circ T_{B_{2i}})(q_{1}r_{1})(q_{2}r_{2}) = T_{B_{1i}}(q_{1}q_{2}) \wedge T_{A_{2}}(r_{1})\wedge T_{A_{2}}(r_{2})\\ I_{(B_{1i} \circ B_{ 2i})}(q_{1}r_{1})(q_{2}r_{2}) = (I_{B_{1i}} \circ I_{B_{2i}})(q_{1}r_{1})(q_{2}r_{2}) = I_{B_{1i}}(q_{1}q_{2}) \wedge I_{A_{2}}(r_{1})\wedge I_{A_{2}}(r_{2})\\ F_{(B_{1i}\circ B_{2i})}(q_{1}r_{1})(q_{2}r_{2}) = (F_{B_{1i}}\circ F_{B_{2i}})(q_{1}r_{1})(q_{2}r_{2}) = F_{B_{1i}}(q_{1}q_{2}) \vee F_{A_{2}}(r_{1})\vee F_{A_{2}}(r_{2})\\ \end{array}\right.$ for all $q_{1}q_{2} \in E_{1i}$, $r_{1}r_{2} \in E_{2i}$ such that $r_{1} \ne r_{2}$.

Example 2.20

Composition of two neutrosophic graph structures $G_{1}$ and $G_{2}$ shown in Fig. 2.14 is defined as $G_{1} \circ G_{2}$ = $\{A_{1}\circ A_{2}, B_{11} \circ B_{21}, B_{12} \circ B_{22}\}$ and is shown in the Fig. 2.19.

Fig. 2.19

Composition of two neutrosophic graph structures

Theorem 2.7

The composition $G_{1} \circ G_{2}$ = $(A_{1} \circ A_{2}, B_{11} \circ B_{21}, B_{12} \circ B_{22},\ldots , B_{1n} \circ B_{2n})$ of two neutrosophic graph structures of the graph structures $G^{*}_{1}$ and $G^{*}_{2}$ is a neutrosophic graph structure of $G^{*}_{1} \circ G^{*}_{2}$.

Proof

According to the definition of composition, there are three cases:

 

Case 1.:

When $q \in X_{1}$, $r_{1}r_{2} \in E_{2i}$

$$\begin{aligned} T_{(B_{1i}\circ B_{2i})}((qr_{1})(qr_{2}))&= T_{A_{1}}(q) \wedge T_{B_{2i}}(r_{1}r_{2})\\&\le T_{A_{1}}(q) \wedge [T_{A_{2}}(r_{1}) \wedge T_{A_{2}}(r_{2})]~\\&= [T_{A_{1}}(q) \wedge T_{A_{2}}(r_{1})]~ \wedge [T_{A_{1}}(q) \wedge T_{A_{2}}(r_{2})]~\\&= T_{(A_{1}\circ A_{2})}(qr_{1})\wedge T_{(A_{1}\circ A_{2})}(qr_{2}), \end{aligned}$$

$$\begin{aligned} I_{(B_{1i}\circ B_{2i})}((qr_{1})(qr_{2}))&= I_{A_{1}}(q) \wedge I_{B_{2i}}(r_{1}r_{2})\\&\le I_{A_{1}}(q) \wedge [I_{A_{2}}(r_{1}) \wedge I_{A_{2}}(r_{2})]~\\&= [I_{A_{1}}(q) \wedge I_{A_{2}}(r_{1})]~ \wedge [I_{A_{1}}(q) \wedge I_{A_{2}}(r_{2})]~\\&= I_{(A_{1}\circ A_{2})}(qr_{1})\wedge I_{(A_{1}\circ A_{2})}(qr_{2}), \end{aligned}$$

$$\begin{aligned} F_{(B_{1i}\circ B_{2i})}((qr_{1})(qr_{2}))&= F_{A_{1}}(q) \vee F_{B_{2i}}(r_{1}r_{2})\\&\le F_{A_{1}}(q) \vee [F_{A_{2}}(r_{1}) \vee F_{A_{2}}(r_{2})]~\\&= [F_{A_{1}}(q) \vee F_{A_{2}}(r_{1})]~ \vee [F_{A_{1}}(q) \vee F_{A_{2}}(r_{2})]~\\&= F_{(A_{1}\circ A_{2})}(qr_{1})\vee F_{(A_{1}\circ A_{2})}(qr_{2}), \end{aligned}$$

for $qr_{1}, qr_{2} \in X_{1}\circ X_{2}.$

Case 2.:

When $q \in X_{2}$, $r_{1}r_{2} \in E_{1i}$

$$\begin{aligned} T_{(B_{1i}\circ B_{2i})}((r_{1}q)(r_{2}q))&= T_{A_{2}}(q) \wedge T_{B_{1i}}(r_{1}r_{2})\\&\le T_{A_{2}}(q) \wedge [T_{A_{1}}(r_{1}) \wedge T_{A_{1}}(r_{2})]~\\&= [T_{A_{2}}(q) \wedge T_{A_{1}}(r_{1})]~ \wedge [T_{A_{2}}(q) \wedge T_{A_{1}}(r_{2})]~\\&= T_{(A_{1}\circ A_{2})}(r_{1}q)\wedge T_{(A_{1}\circ A_{2})}(r_{2}q), \end{aligned}$$

$$\begin{aligned} I_{(B_{1i}\circ B_{2i})}((r_{1}q)(r_{2}q))&= I_{A_{2}}(q) \wedge I_{B_{1i}}(r_{1}r_{2})\\&\le I_{A_{2}}(q) \wedge [I_{A_{1}}(r_{1}) \wedge I_{A_{1}}(r_{2})]~\\&= [I_{A_{2}}(q) \wedge I_{A_{1}}(r_{1})]~ \wedge [I_{A_{2}}(q) \wedge I_{A_{1}}(r_{2})]~\\&= I_{(A_{1}\circ A_{2})}(r_{1}q)\wedge I_{(A_{1}\circ A_{2})}(r_{2}q), \end{aligned}$$

$$\begin{aligned} F_{(B_{1i}\circ B_{2i})}((r_{1}q)(r_{2}q))&= F_{A_{2}}(q) \vee F_{B_{1i}}(r_{1}r_{2})\\&\le F_{A_{2}}(q) \vee [F_{A_{1}}(r_{1}) \vee F_{A_{1}}(r_{2})]~\\&= [F_{A_{2}}(q) \vee F_{A_{1}}(r_{1})]~ \vee [F_{A_{2}}(q) \vee F_{A_{1}}(r_{2})]~\\&= F_{(A_{1}\circ A_{2})}(r_{1}q)\vee F_{(A_{1}\circ A_{2})}(r_{2}q), \end{aligned}$$

for $r_{1}q, r_{2}q \in X_{1}\circ X_{2}.$

Case 3.:

For all $q_{1}q_{2} \in E_{1i}, r_{1}, r_{2} \in X_{2}$ such that $r_{1}\ne r_{2}$

$$\begin{aligned} T_{(B_{1i}\circ B_{2i})}((q_{1}r_{1})(q_{2}r_{2}))&=T_{B_{1i}}(q_{1}q_{2}) \wedge T_{A_{2}}(r_{1})\wedge T_{A_{2}}(r_{2}) \\&\le [T_{A_{1}}(q_{1})\wedge T_{A_{1}}(q_{2})]~ \wedge T_{A_{2}}(r_{1}) \wedge T_{A_{2}}(r_{2})\\&= [T_{A_{1}}(q_{1}) \wedge T_{A_{2}}(r_{1})]~ \wedge [T_{A_{1}}(q_{2}) \wedge T_{A_{2}}(r_{2})]~\\&= T_{(A_{1}\circ A_{2})}(q_{1}r_{1})\wedge T_{(A_{1}\circ A_{2})}(q_{2}r_{2}), \end{aligned}$$

$$\begin{aligned} I_{(B_{1i}\circ B_{2i})}((q_{1}r_{1})(q_{2}r_{2}))&=I_{B_{1i}}(q_{1}q_{2}) \wedge I_{A_{2}}(r_{1})\wedge I_{A_{2}}(r_{2})\\&\le [I_{A_{1}}(q_{1})\wedge I_{A_{1}}(q_{2})]~ \wedge [I_{A_{2}}(r_{1}) \wedge I_{A_{2}}(r_{2})]~\\&= [I_{A_{1}}(q_{1}) \wedge I_{A_{2}}(r_{1})]~ \wedge [I_{A_{1}}(q_{2}) \wedge I_{A_{2}}(r_{2})]~\\&= I_{(A_{1}\circ A_{2})}(q_{1}r_{1})\wedge I_{(A_{1}\circ A_{2})}(q_{2}r_{2}), \end{aligned}$$

$$\begin{aligned} F_{(B_{1i}\circ B_{2i})}((q_{1}r_{1})(q_{2}r_{2}))&=F_{B_{1i}}(q_{1}q_{2}) \vee F_{A_{2}}(r_{1})\vee F_{A_{2}}(r_{2})\\&\le [F_{A_{1}}(q_{1})\vee F_{A_{1}}(q_{2})]~ \vee [F_{A_{2}}(r_{1}) \vee F_{A_{2}}(r_{2})]~\\&=[F_{A_{1}}(q_{1}) \vee F_{A_{2}}(r_{1})]~ \vee [F_{A_{1}}(q_{2}) \vee F_{A_{2}}(r_{2})]~\\&=F_{(A_{1}\circ A_{2})}(q_{1}r_{1})\vee F_{(A_{1}\circ A_{2})}(q_{2}r_{2}), \end{aligned}$$

 

for $q_{1}r_{1}, q_{2}r_{2} \in X_{1}\circ X_{2}.$

All cases are satisfied for $i = 1,2,\ldots , n$.

Definition 2.37

Let $G_{1}$ = $(A_{1}, B_{11}, B_{12},\ldots , B_{1n})$ and $G_{2}$ = $(A_{2}, B_{21}, B_{22},\ldots , B_{2n})$ be neutrosophic graph structures. The union of $G_{1}$ and $G_{2}$, denoted by

$$ G_{1} \cup G_{2} = (A_{1}\cup A_{2}, B_{11}\cup B_{21}, B_{12}\cup B_{22},\ldots , B_{1n}\cup B_{2n}), $$

is defined by following:

• (i)   $\left\{ \begin{array}{ll} T_{(A_{1}\cup A_{2})}(q) = (T_{A_{1}}\cup T_{A_{2}})(q) = T_{A_{1}}(q) \vee T_{A_{2}}(q)\\ I_{(A_{1}\cup A_{2})}(q) = (I_{A_{1}}\cup I_{A_{2}})(q) = I_{A_{1}}(q) \vee I_{A_{2}}(q)\\ F_{(A_{1}\cup A_{2})}(q) = (F_{A_{1}}\cup F_{A_{2}})(q) = F_{A_{1}}(q) \wedge F_{A_{2}}(q)\\ \end{array}\right.$ for all $q \in X_{1} \cup X_{2}$,

• (ii)   $\left\{ \begin{array}{ll} T_{(B_{1i}\cup B_{2i})}(qr) = (T_{B_{1i}}\cup T_{B_{2i}})(qr) = T_{B_{1i}}(qr) \vee T_{B_{2i}}(qr)\\ I_{(B_{1i}\cup B_{2i})}(qr) = (I_{B_{1i}}\cup I_{B_{2i}})(qr) = I_{B_{1i}}(qr) \vee I_{B_{2i}}(qr)\\ F_{(B_{1i}\cup B_{2i})}(qr) = (F_{B_{1i}}\cup F_{B_{2i}})(qr) = F_{B_{1i}}(qr) \wedge F_{B_{2i}}(qr)\\ \end{array}\right.$ for all $qr \in E_{1i} \cup E_{2i}$.

Example 2.21

Union of two neutrosophic graph structures $G_{1}$ and $G_{2}$ shown in Fig. 2.14 is defined as $G_{1} \cup G_{2}$ = $\{A_{1}\cup A_{2}, B_{11} \cup B_{21}, B_{12} \cup B_{22}\}$ and is shown in the Fig. 2.20.

Fig. 2.20

Union of two neutrosophic graph structures

Theorem 2.8

The union $G_{1} \cup G_{2}$ = $(A_{1}\cup A_{2}, B_{11}\cup B_{21}, B_{12}\cup B_{22},\ldots , B_{1n}\cup B_{2n})$ of two neutrosophic graph structures of the graph structures $G^{*}_{1}$ and $G^{*}_{2}$ is a neutrosophic graph structure of $G^{*}_{1} \cup G^{*}_{2}$.

Proof

Let $q_{1}q_{2} \in E_{1i} \cup E_{2i}$. Here we consider two cases:

 

Case 1.:

When $q_{1}, q_{2} \in X_{1}$, then according to Definition 2.37, $T_{A_{2}}(q_{1}) = T_{A_{2}}(q_{2}) = T_{B_{2i}}(q_{1}q_{2}) = 0$, $I_{A_{2}}(q_{1}) = I_{A_{2}}(q_{2}) = I_{B_{2i}}(q_{1}q_{2}) = 0$, $F_{A_{2}}(q_{1}) = F_{A_{2}}(q_{2}) = F_{B_{2i}}(q_{1}q_{2}) = 0$, so

$$\begin{aligned} T_{(B_{1i}\cup B_{2i})}(q_{1}q_{2})&= T_{B_{1i}}(q_{1}q_{2}) \vee T_{B_{2i}}(q_{1}q_{2})\\&= T_{B_{1i}}(q_{1}q_{2}) \vee 0\\&\le [T_{A_{1}}(q_{1}) \wedge T_{A_{1}}(q_{2})]~ \vee 0\\&=[T_{A_{1}}(q_{1}) \vee 0]\wedge [T_{A_{1}}(q_{2}) \vee 0]\\&=[T_{A_{1}}(q_{1}) \vee T_{A_{2}}(q_{1}) ]\wedge [T_{A_{1}}(q_{2}) \vee T_{A_{2}}(q_{2})]~\\&= T_{(A_{1}\cup A_{2})}(q_{1})\wedge T_{(A_{1}\cup A_{2})}(q_{2}), \end{aligned}$$

$$\begin{aligned} I_{(B_{1i}\cup B_{2i})}(q_{1}q_{2})&= I_{B_{1i}}(q_{1}q_{2}) \vee I_{B_{2i}}(q_{1}q_{2})\\&= I_{B_{1i}}(q_{1}q_{2}) \vee 0\\&\le [I_{A_{1}}(q_{1}) \wedge I_{A_{1}}(q_{2})]~ \vee 0\\&=[I_{A_{1}}(q_{1}) \vee 0]\wedge [I_{A_{1}}(q_{2}) \vee 0]\\&=[I_{A_{1}}(q_{1}) \vee I_{A_{2}}(q_{1}) ]\wedge [I_{A_{1}}(q_{2}) \vee I_{A_{2}}(q_{2})]~\\&= I_{(A_{1}\cup A_{2})}(q_{1})\wedge I_{(A_{1}\cup A_{2})}(q_{2}), \end{aligned}$$

$$\begin{aligned} F_{(B_{1i}\cup B_{2i})}(q_{1}q_{2})&= F_{B_{1i}}(q_{1}q_{2}) \wedge F_{B_{2i}}(q_{1}q_{2})\\&= F_{B_{1i}}(q_{1}q_{2}) \wedge 0\\&\le [F_{A_{1}}(q_{1}) \vee F_{A_{1}}(q_{2})]~ \wedge 0\\&=[F_{A_{1}}(q_{1}) \wedge 0]\vee [F_{A_{1}}(q_{2}) \wedge 0]\\&=[F_{A_{1}}(q_{1}) \wedge F_{A_{2}}(q_{1}) ]\vee [F_{A_{1}}(q_{2}) \wedge F_{A_{2}}(q_{2})]~\\&= F_{(A_{1}\cup A_{2})}(q_{1})\vee F_{(A_{1}\cup A_{2})}(q_{2}), \end{aligned}$$

for $q_{1}, q_{2} \in X_{1} \cup X_{2}$.

Case 2.:

When $q_{1}, q_{2} \in X_{2}$, then according to Definition 2.37, $T_{A_{1}}(q_{1}) = T_{A_{1}}(q_{2}) = T_{B_{1i}}(q_{1}q_{2}) = 0$, $I_{A_{1}}(q_{1}) = I_{A_{1}}(q_{2}) = I_{B_{1i}}(q_{1}q_{2}) = 0$, $F_{A_{1}}(q_{1}) = F_{A_{1}}(q_{2}) = F_{B_{1i}}(q_{1}q_{2}) = 0$, so

$$\begin{aligned} T_{(B_{1i}\cup B_{2i})}(q_{1}q_{2})&= T_{B_{1i}}(q_{1}q_{2}) \vee T_{B_{2i}}(q_{1}q_{2})\\&= T_{B_{2i}}(q_{1}q_{2}) \vee 0\\&\le [T_{A_{2}}(q_{1}) \wedge T_{A_{2}}(q_{2})]~ \vee 0\\&=[T_{A_{2}}(q_{1}) \vee 0]\wedge [T_{A_{2}}(q_{2}) \vee 0]\\&=[T_{A_{1}}(q_{1}) \vee T_{A_{2}}(q_{1}) ]\wedge [T_{A_{1}}(q_{2}) \vee T_{A_{2}}(q_{2})]~\\&= T_{(A_{1}\cup A_{2})}(q_{1})\wedge T_{(A_{1}\cup A_{2})}(q_{2}), \end{aligned}$$

$$\begin{aligned} I_{(B_{1i}\cup B_{2i})}(q_{1}q_{2})&= I_{B_{1i}}(q_{1}q_{2}) \vee I_{B_{2i}}(q_{1}q_{2})\\&= I_{B_{2i}}(q_{1}q_{2}) \vee 0\\&\le [I_{A_{2}}(q_{1}) \wedge I_{A_{2}}(q_{2})]~ \vee 0\\&=[I_{A_{2}}(q_{1}) \vee 0]\wedge [I_{A_{2}}(q_{2}) \vee 0]\\&=[I_{A_{1}}(q_{1}) \vee I_{A_{2}}(q_{1}) ]\wedge [I_{A_{1}}(q_{2}) \vee I_{A_{2}}(q_{2})]~\\&= I_{(A_{1}\cup A_{2})}(q_{1})\wedge I_{(A_{1}\cup A_{2})}(q_{2}), \end{aligned}$$

$$\begin{aligned} F_{(B_{1i}\cup B_{2i})}(q_{1}q_{2})&= F_{B_{1i}}(q_{1}q_{2}) \wedge F_{B_{2i}}(q_{1}q_{2})\\&= F_{B_{2i}}(q_{1}q_{2}) \wedge 0\\&\le [F_{A_{2}}(q_{1}) \vee F_{A_{2}}(q_{2})]~ \wedge 0\\&=[F_{A_{2}}(q_{1}) \wedge 0]\vee [F_{A_{2}}(q_{2}) \wedge 0]\\&=[F_{A_{1}}(q_{1}) \wedge F_{A_{2}}(q_{1}) ]\vee [F_{A_{1}}(q_{2}) \wedge F_{A_{2}}(q_{2})]~\\&= F_{(A_{1}\cup A_{2})}(q_{1})\vee F_{(A_{1}\cup A_{2})}(q_{2}), \end{aligned}$$

for $q_{1}, q_{2} \in X_{1} \cup X_{2}$.

  Both cases are satisfied $\forall i \in \{1,2,\ldots , n\}$. This completes the proof.

Theorem 2.9

Let $G^{*}$ = $(X_{1}\cup X_{2}, E_{11}\cup E_{21}, E_{12}\cup E_{22},\ldots , E_{1n}\cup E_{2n})$ be the union of two graph structures $G^{*}_{1}$ = $(X_{1}, E_{11}, E_{12},\ldots , E_{1n})$ and $G^{*}_{2}$ = $(X_{2}, E_{21}, E_{22},\ldots , E_{2n})$. Then every neutrosophic graph structure G = $(A, B_{1}, B_{2},\ldots , B_{n})$ of $G^{*}$ is union of two neutrosophic graph structures $G_{1}= (A_{1}, B_{11}, B_{12},\ldots , B_{1n})$ and $G_{2}$ = $(A_{2}, B_{21}, B_{22},\ldots , B_{2n})$ of graph structures $G^{*}_{1}$ and $G^{*}_{2}$, respectively.

Proof

First we define $A_{1}, A_{2}, B_{1i}$ and $B_{2i}$ for $i \in \{1,2,\ldots , n\}$ as:

$T_{A_{1}}(q) = T_{A}(q), I_{A_{1}}(q) = I_{A}(q), F_{A_{1}}(q) = F_{A}(q),$ if $q \in X_{1}$

$T_{A_{2}}(q) = T_{A}(q), I_{A_{2}}(q) = I_{A}(q), F_{A_{2}}(q) = F_{A}(q),$ if $q \in X_{2}$

$T_{B_{1i}}(q_{1}q_{2}) = T_{B_{i}}(q_{1}q_{2}), I_{B_{1i}}(q_{1}q_{2}) = I_{B_{i}}(q_{1}q_{2}), F_{B_{1i}}(q_{1}q_{2}) = F_{B_{i}}(q_{1}q_{2})$, if $q_{1}q_{2} \in E_{1i}$, $T_{B_{2i}}(q_{1}q_{2}) = T_{B_{i}}(q_{1}q_{2}), I_{B_{2i}}(q_{1}q_{2}) = I_{B_{i}}(q_{1}q_{2}), F_{B_{2i}}(q_{1}q_{2}) = F_{B_{i}}(q_{1}q_{2})$, if $q_{1}q_{2} \in E_{2i}$. Then $A = A_{1} \cup A_{2}$ and $B_{i} = B_{1i} \cup B_{2i}$, $i \in \{1,2,\ldots , n\}$.

Now for $q_{1}q_{2} \in E_{ki}$, $k = 1,2$, $i = {1,2,\ldots , n}$

$T_{B_{ki}}(q_{1}q_{2}) = T_{B_{i}}(q_{1}q_{2}) \le T_{A}(q_{1}) \wedge T_{A}(q_{2}) = T_{A_{ k}}(q_{1}) \wedge T_{A_{k}}(q_{2})$,

$I_{B_{ki}}(q_{1}q_{2}) = I_{B_{i}}(q_{1}q_{2}) \le I_{A}(q_{1}) \wedge I_{A}(q_{2}) = I_{A_{k}}(q_{1}) \wedge I_{A_{k}}(q_{2})$,

$F_{B_{ki}}(q_{1}q_{2}) = F_{B_{i}}(q_{1}q_{2}) \le F_{A}(q_{1}) \vee F_{A}(q_{2}) = F_{A_{k}}(q_{1}) \vee F_{A_{k}}(q_{2})$,

i.e.

${G}_{k}$ = $(A_{k}, B_{k1}, B_{k2},\ldots , B_{kn})$ is a neutrosophic graph structure of $G^{*}_{k}$, $k = 1,2$.

Thus G = $(A, B_{1}, B_{2},\ldots , B_{n})$, a neutrosophic graph structure of $G^{*}$ = $G^{*}_{1} \cup G^{*}_{2}$, is union of two neutrosophic graph structures $G_{1}$ and $G_{2}$.

Definition 2.38

Let $G_{1}$ = $(A_{1}, B_{11}, B_{12},\ldots , B_{1n})$ and $G_{2}$ = $(A_{2}, B_{21}, B_{22},\ldots , B_{2n})$ be neutrosophic graph structures and let $X_{1} \cap X_{2}$ = $\emptyset$. The join of $G_{1}$ and $G_{2}$, denoted by

$$ G_{1} + G_{2} = (A_{1}+ A_{2}, B_{11}+ B_{21}, B_{12}+ B_{22},\ldots , B_{1n}+ B_{2n}), $$

is defined by the following:

• (i)   $\left\{ \begin{array}{ll} T_{(A_{1}+ A_{2})}(q)= T_{(A_{1}\cup A_{2})}(q)\\ I_{(A_{1}+ A_{2})}(q) = I_{(A_{1}\cup A_{2})}(q)\\ F_{(A_{1}+ A_{2})}(q)= F_{(A_{1}\cup A_{2})}(q)\\ \end{array}\right.$ for all $q \in X_{1} \cup X_{2}$,

• (ii)   $\left\{ \begin{array}{ll} T_{(B_{1i}+ B_{2i})}(qr) = T_{(B_{1i}\cup B_{2i})}(qr)\\ I_{(B_{1i}+ B_{2i})}(qr) = I_{(B_{1i}\cup B_{2i})}(qr)\\ F_{(B_{1i}+ B_{2i})}(qr) = F_{(B_{1i}\cup B_{2i})}(qr)\\ \end{array}\right.$ for all $qr \in E_{1i} \cup E_{2i}$,

• (iii)   $\left\{ \begin{array}{ll} T_{(B_{1i}+ B_{2i})}(qr) = (T_{B_{1i}}+ T_{B_{2i}})(qr) = T_{A_{ 1}}(q) \wedge T_{A_{2}}(r)\\ I_{(B_{1i} + B_{2i})}(qr) = (I_{B_{1i}} + I_{B_{2i}})(qr) = I_{A_{1}}(q) \wedge I_{A_{2}}(r)\\ F_{(B_{1i}+ B_{2i})}(qr) = (F_{B_{1i}}+ F_{B_{2i}})(qr) = F_{A_{1}}(q) \vee F_{A_{2}}(r)\\ \end{array}\right.$ for all $q \in X_{1}$, $r \in X_{2}$.

Example 2.22

Join of two neutrosophic graph structures $G_{1}$ and $G_{2}$ shown in Fig. 2.14 is defined as $G_{1} + G_{2}$ = $\{A_{1} + A_{2}, B_{11} + B_{21}, B_{12} + B_{22}\}$ and is shown in the Fig. 2.21.

Fig. 2.21

Join of two neutrosophic graph structures

Theorem 2.10

The join $G_{1} + G_{2}$ = $(A_{1} + A_{2}, B_{11} + B_{21}, B_{12} + B_{22}, \ldots , B_{1n} + B_{2n})$ of two neutrosophic graph structures of the graph structures $G^{*}_{1}$ and $G^{*}_{2}$ is a neutrosophic graph structure of $G^{*}_{1} + G^{*}_{2}$.

Proof

Let $q_{1}q_{2} \in E_{1i} + E_{2i}$. Here we consider three cases:

 

Case 1.:

When $q_{1}, q_{2} \in X_{1}$, then according to Definition 2.38, $T_{A_{2}}(q_{1}) = T_{A_{2}}(q_{2}) = T_{B_{2i}}(q_{1}q_{2}) = 0$, $I_{A_{2}}(q_{1}) = I_{A_{2}}(q_{2}) = I_{B_{2i}}(q_{1}q_{2}) = 0$, $F_{A_{2}}(q_{1}) = F_{A_{2}}(q_{2}) = F_{B_{2i}}(q_{1}q_{2}) = 0$, so,

$$\begin{aligned} T_{(B_{1i}+ B_{2i})}(q_{1}q_{2})&= T_{B_{1i}}(q_{1}q_{2}) \vee T_{B_{2i}}(q_{1}q_{2})\\&= T_{B_{1i}}(q_{1}q_{2}) \vee 0\\&\le [T_{A_{1}}(q_{1}) \wedge T_{A_{1}}(q_{2})]~ \vee 0\\&=[T_{A_{1}}(q_{1}) \vee 0]\wedge [T_{A_{1}}(q_{2}) \vee 0]\\&=[T_{A_{1}}(q_{1}) \vee T_{A_{2}}(q_{1}) ]\wedge [T_{A_{1}}(q_{2}) \vee T_{A_{2}}(q_{2})]~\\&= T_{(A_{1}+ A_{2})}(q_{1})\wedge T_{(A_{1}+ A_{2})}(q_{2}), \end{aligned}$$

$$\begin{aligned} I_{(B_{1i}+ B_{2i})}(q_{1}q_{2})&= I_{B_{1i}}(q_{1}q_{2}) \vee I_{B_{2i}}(q_{1}q_{2})\\&= I_{B_{1i}}(q_{1}q_{2}) \vee 0\\&\le [I_{A_{1}}(q_{1}) \wedge I_{A_{1}}(q_{2})]~ \vee 0\\&=[I_{A_{1}}(q_{1}) \vee 0]\wedge [I_{A_{1}}(q_{2}) \vee 0]\\&=[I_{A_{1}}(q_{1}) \vee I_{A_{2}}(q_{1}) ]\wedge [I_{A_{1}}(q_{2}) \vee I_{A_{2}}(q_{2})]~\\&= I_{(A_{1}+ A_{2})}(q_{1})\wedge I_{(A_{1}+ A_{2})}(q_{2}), \end{aligned}$$

$$\begin{aligned} F_{(B_{1i}+ B_{2i})}(q_{1}q_{2})&= F_{B_{1i}}(q_{1}q_{2}) \wedge F_{B_{2i}}(q_{1}q_{2})\\&= F_{B_{1i}}(q_{1}q_{2}) \wedge 0\\&\le [F_{A_{1}}(q_{1}) \vee F_{A_{1}}(q_{2})]~ \wedge 0\\&=[F_{A_{1}}(q_{1}) \wedge 0]\vee [F_{A_{1}}(q_{2}) \wedge 0]\\&=[F_{A_{1}}(q_{1}) \wedge F_{A_{2}}(q_{1}) ]\vee [F_{A_{1}}(q_{2}) \wedge F_{A_{2}}(q_{2})]~\\&= F_{(A_{1}+ A_{2})}(q_{1})\vee F_{(A_{1}+ A_{2})}(q_{2}), \end{aligned}$$

for $q_{1}, q_{2} \in X_{1} + X_{2}$.

Case 2.:

When $q_{1}, q_{2} \in X_{2}$, then according to Definition 2.38, $T_{A_{1}}(q_{1}) = T_{A_{1}}(q_{2}) = T_{B_{1i}}(q_{1}q_{2}) = 0$, $I_{A_{1}}(q_{1}) = I_{A_{1}}(q_{2}) = I_{B_{1i}}(q_{1}q_{2}) = 0$, $F_{A_{1}}(q_{1}) = F_{A_{1}}(q_{2}) = F_{B_{1i}}(q_{1}q_{2}) = 0$, so

$$\begin{aligned} T_{(B_{1i}+ B_{2i})}(q_{1}q_{2})&= T_{B_{1i}}(q_{1}q_{2}) \vee T_{B_{2i}}(q_{1}q_{2})\\&= T_{B_{2i}}(q_{1}q_{2}) \vee 0\\&\le [T_{A_{2}}(q_{1}) \wedge T_{A_{2}}(q_{2})]~ \vee 0\\&=[T_{A_{2}}(q_{1}) \vee 0]\wedge [T_{A_{2}}(q_{2}) \vee 0]\\&=[T_{A_{1}}(q_{1}) \vee T_{A_{2}}(q_{1}) ]\wedge [T_{A_{1}}(q_{2}) \vee T_{A_{2}}(q_{2})]~\\&= T_{(A_{1}+ A_{2})}(q_{1})\wedge T_{(A_{1}+ A_{2})}(q_{2}), \end{aligned}$$

$$\begin{aligned} I_{(B_{1i}+ B_{2i})}(q_{1}q_{2})&= I_{B_{1i}}(q_{1}q_{2}) \vee I_{B_{2i}}(q_{1}q_{2})\\&= I_{B_{2i}}(q_{1}q_{2}) \vee 0\\&\le [I_{A_{2}}(q_{1}) \wedge I_{A_{2}}(q_{2})]~ \vee 0\\&=[I_{A_{2}}(q_{1}) \vee 0]\wedge [I_{A_{2}}(q_{2}) \vee 0]\\&=[I_{A_{1}}(q_{1}) \vee I_{A_{2}}(q_{1}) ]\wedge [I_{A_{1}}(q_{2}) \vee I_{A_{2}}(q_{2})]~\\&= I_{(A_{1}+ A_{2})}(q_{1})\wedge I_{(A_{1}+ A_{2})}(q_{2}), \end{aligned}$$

$$\begin{aligned} F_{(B_{1i}+ B_{2i})}(q_{1}q_{2})&= F_{B_{1i}}(q_{1}q_{2}) \wedge F_{B_{2i}}(q_{1}q_{2})\\&= F_{B_{2i}}(q_{1}q_{2}) \wedge 0\\&\le [F_{A_{2}}(q_{1}) \vee F_{A_{2}}(q_{2})]~ \wedge 0\\&=[F_{A_{2}}(q_{1}) \wedge 0]\vee [F_{A_{2}}(q_{2}) \wedge 0]\\&=[F_{A_{1}}(q_{1}) \wedge F_{A_{2}}(q_{1}) ]\vee [F_{A_{1}}(q_{2}) \wedge F_{A_{2}}(q_{2})]~\\&= F_{(A_{1}+ A_{2})}(q_{1})\vee F_{(A_{1}+ A_{2})}(q_{2}), \end{aligned}$$

for $q_{1}, q_{2} \in X_{1} + X_{2}$.

Case 3.:

When $q_{1} \in X_{1}$, $q_{2} \in X_{2}$, then according to Definition 2.38, $T_{A_{1}}(q_{2}) = T_{A_{2}}(q_{1}) = 0$, $I_{A_{1}}(q_{2}) = I_{A_{2}}(q_{1}) = 0$, $F_{A_{1}}(q_{2}) = F_{A_{2}}(q_{1}) = 0$, so

$$\begin{aligned} T_{(B_{1i}+ B_{2i})}(q_{1}q_{2})&= T_{A_{1}}(q_{1}) \wedge T_{A_{2}}(q_{2})\\&= [T_{A_{1}}(q_{1}) \vee 0] \wedge [T_{A_{2}}(q_{2}) \vee 0] \\&= [T_{A_{1}}(q_{1}) \vee T_{A_{2}}(q_{1}) ] \wedge [T_{A_{2}}(q_{2}) \vee T_{A_{1}}(q_{2})]~ \\&= T_{(A_{1}+ A_{2})}(q_{1})\wedge T_{(A_{1}+ A_{2})}(q_{2}), \end{aligned}$$

$$\begin{aligned} I_{(B_{1i}+ B_{2i})}(q_{1}q_{2})&= I_{A_{1}}(q_{1}) \wedge I_{A_{2}}(q_{2})\\&= [I_{A_{1}}(q_{1}) \vee 0] \wedge [I_{A_{2}}(q_{2}) \vee 0] \\&= [I_{A_{1}}(q_{1}) \vee I_{A_{2}}(q_{1}) ] \wedge [I_{A_{2}}(q_{2}) \vee I_{A_{1}}(q_{2})]~ \\&= I_{(A_{1}+ A_{2})}(q_{1})\wedge I_{(A_{1}+ A_{2})}(q_{2}), \end{aligned}$$

$$\begin{aligned} F_{(B_{1i}+ B_{2i})}(q_{1}q_{2})&= F_{A_{1}}(q_{1}) \vee F_{A_{2}}(q_{2})\\&= [F_{A_{1}}(q_{1}) \wedge 0] \vee [F_{A_{2}}(q_{2}) \wedge 0] \\&= [F_{A_{1}}(q_{1}) \wedge F_{A_{2}}(q_{1}) ] \vee [F_{A_{2}}(q_{2}) \wedge F_{A_{1}}(q_{2})]~ \\&= F_{(A_{1}+ A_{2})}(q_{1})\vee F_{(A_{1}+ A_{2})}(q_{2}), \end{aligned}$$

for $q_{1}, q_{2} \in X_{1} + X_{2}$.

  All cases are satisfied $\forall i \in \{1,2,\ldots , n\}$.

Theorem 2.11

If $G^{*}$ = $(X_{1}+ X_{2}, E_{11}+ E_{21}, E_{12}+ E_{22},\ldots , E_{1n}+ E_{2n})$ is join of two graph structures $G^{*}_{1}$ = $(X_{1}, E_{11}, E_{12},\ldots , E_{1n})$ and $G^{*}_{2}$ = $(X_{2}, E_{21}, E_{22},\ldots , E_{2n})$. Then every strong neutrosophic graph structure G = $(A, B_{1}, B_{2},\ldots , B_{n})$ of G is join of two strong neutrosophic graph structures $G_{1}= (A_{1}, B_{11}, B_{12},\ldots , B_{1n})$ and $G_{2}$ = $(A_{2}, B_{21}, B_{22},\ldots , B_{2n})$ of graph structures $G^{*}_{1}$ and $G^{*}_{2}$, respectively.

Proof

First we define $A_{k}$ and $B_{ki}$ for $k = 1,2$ and $i = 1,2,\ldots , n$ as:

$T_{A_{k}}(q) = T_{A}(q), I_{A_{k}}(q) = I_{A}(q), F_{A_{k}}(q) = F_{A}(q),$ if $q \in X_{k}$

$T_{B_{ki}}(q_{1}q_{2}) = T_{B_{i}}(q_{1}q_{2}), I_{B_{ki}}(q_{1}q_{2}) = I_{B_{i}}(q_{1}q_{2}), F_{B_{ki}}(q_{1}q_{2}) = F_{B_{i}}(q_{1}q_{2})$, if $q_{1}q_{2} \in E_{ki}$

Now for $q_{1}q_{2} \in E_{ki}$, $k = 1,2$, $i = {1,2,\ldots , n}$

$T_{B_{ki}}(q_{1}q_{2}) = T_{B_{i}}(q_{1}q_{2}) = T_{A}(q_{1}) \wedge T_{A}(q_{2}) = T_{A_{k}}(q_{1}) \wedge T_{A_{k}}(q_{2})$,

$I_{B_{ki}}(q_{1}q_{2}) = I_{B_{i}}(q_{1}q_{2}) = I_{A}(q_{1}) \wedge I_{A}(q_{2}) = I_{A_{k}}(q_{1}) \wedge I_{A_{k}}(q_{2})$,

$F_{B_{ki}}(q_{1}q_{2}) = F_{B_{i}}(q_{1}q_{2}) = F_{A}(q_{1}) \vee T_{A}(q_{2}) = T_{A_{k}}(q_{1}) \vee T_{A_{k}}(q_{2})$,

i.e.

${G}_{k}$ = $(A_{k}, B_{k1}, B_{k2},\ldots , B_{kn})$ is a strong neutrosophic graph structure of $G^{*}_{k}$, k = 1,2.

Moreover, G is join of $G_{1}$ and $G_{2}$ as shown:

Using Definitions 2.37 and 2.38, $A = A_{1}\cup A_{2} = A_{1} + A_{2}$ and $B_{i} = B_{1i}\cup B_{2i} = B_{1i} + B_{2i}$, $\forall q_{1}q_{2} \in E_{1i}\cup E_{2i}$.

When $q_{1}q_{2} \in E_{1i}+ E_{2i} \ (E_{1i}\cup E_{2i})$, i.e. $q_{1} \in X_{1}$ and $q_{2} \in X_{2}$

$T_{B_{i}}(q_{1}q_{2}) = T_{A}(q_{1}) \wedge T_{A}(q_{2}) = T_{A_{k}}(q_{1}) \wedge T_{A_{k}}(q_{2}) = T_{(B_{1i}+ B_{2i})}(q_{1}q_{2})$,

$I_{B_{i}}(q_{1}q_{2}) = I_{A}(q_{1}) \wedge I_{A}(q_{2}) = I_{A_{k}}(q_{1}) \wedge I_{A_{k}}(q_{2})= I_{(B_{1i}+ B_{2i})}(q_{1}q_{2})$,

$F_{B_{i}}(q_{1}q_{2}) = F_{A}(q_{1}) \vee F_{A}(q_{2}) = F_{A_{k}}(q_{1}) \vee F_{A_{k}}(q_{2}) = F_{(B_{1i}+ B_{2i})}(q_{1}q_{2})$,

Calculations are similar when $q_{1} \in X_{2}$, $q_{2} \in X_{1}$. It is true when $i = {1,2,\ldots , n}$. This completes the proof.

2.4 Applications of Neutrosophic Graph Structures

Graph structures are amazing source of graph-theoretical notions to represent the most prominent relations between objects. But these graph structures do not represent all real-world relations. Therefore, fuzzy graph structures are important to represent the relations between objects of uncertain systems existing in nature. However, graph structures and fuzzy graph structures are failed to depict the most prominent relations between objects in many real-world phenomenons due to natural existence of indeterminacy or neutrality. It increases the utility of neutrosophic graph structures.

2.4.1 Detection of Crucial Crimes During Maritime Trade

Waters are very important for trade in whole world but crimes through waters are increasing day by day. Crimes held during maritime trade are in abundance but some are very crucial including human trafficking, illegal carrying of weapons, black money transfer, smuggling of precious metals, drug trafficking and smuggling of rare plants and animals. Using neutrosophic graph structure, we can easily investigate the fact that between any two countries which maritime crime is chronic and increasing rapidly with time. Moreover, we can decide which country is most sensitive for particular type of maritime crimes. We consider a set X consisting of eight countries.

X={Bangladesh, Malaysia, Singapore, United Arab Emirates, Pakistan, India, Kenya, Italy}. Let A be the neutrosophic set on X, defined in Table 2.1.

Table 2.1 Neutrosophic set A of eight countries

In Table 2.1, T depicts the importance of that particular country in the world due to its geographic position, F indicates the degree of its nonimportance in the world, and I expresses, to which extent it is undecided/indeterminate to be beneficial for the world, geographically.

Let Bangladesh = B, Malaysia = M, Singapore = S, United Arab Emirates = UAE, Pakistan = P, India = I, Kenya = K, Italy = IT.

In Tables 2.2, 2.3, 2.4, 2.5, 2.6, 2.7 and 2.8, we have shown the values of T, I and F of different crimes for each pair of countries.

Table 2.2 Neutrosophic set of crimes between Pakistan and other countries during maritime trade

Table 2.3 Neutrosophic set of crimes between UAE and other countries during maritime trade

Table 2.4 Neutrosophic set of crimes between Bangladesh and other countries during maritime trade

Table 2.5 Neutrosophic set of crimes between Malaysia and other countries during maritime trade

Table 2.6 Neutrosophic set of crimes between Singapore and other countries during maritime trade

Table 2.7 Neutrosophic set of crimes between Italy and other countries during maritime trade

Table 2.8 Neutrosophic set of crimes between Kenya and other countries during maritime trade

Many relations on set X can be defined, let we define six relations on X as:

$E_{1}$ = Human trafficking, $E_{2}$ = Illegal carrying of weapons, $E_{3}$ = Black money transfer, $E_{4}$ = Smuggling of precious metals, $E_{5}$ = Drug trafficking, $E_{6}$ = Smuggling of rare plants and animals, such that $(X, E_{1}, E_{2}, E_{3}, E_{4}, E_{5}, E_{6})$ is a graph structure. An element in a relation detects that kind of crime during maritime trade between those two countries.

As $(X, E_{1}, E_{2}, E_{3}, E_{4}, E_{5}, E_{6})$ is a graph structure, an element will not be in more than one relations, so it can appear just once. Therefore, we will consider it an element of that relation for which its percentage of truth is high, and percentage of both falsity and indeterminacy is low as compared to other relations.

According to given data, we write the elements in relation to their truth, falsity and indeterminacy values, resulting sets are neutrosophic sets on $E_{1}$, $E_{2}$, $E_{3}$, $E_{4}$, $E_{5}$, $E_{6}$, respectively. We can name these sets as $B_{1}$, $B_{2}$, $B_{3}$, $B_{4}$, $B_{5}$, $B_{6}$, respectively. Let

$E_{1}$ = $\{(Bangladesh, Pakistan), (Malaysia, Pakistan), (Bangladesh, Singapore)\}$,

$E_{2}$ = $\{(Pakistan, India)\}$,

$E_{3}$ = $\{(Singapore, Pakistan) \}$,

$E_{4}$ = $\{(India, Singapore),$ $(United Arab Emirates, India)\}$,

$E_{5}$ = $\{(Italy, Pakistan), (India, Italy)\}$,

$E_{6}$ = $\{(Kenya, Singapore)\}$.

And corresponding neutrosophic sets are:

$B_{1}$ = $\{((B, P), 0.8, 0.2, 0.2),$ $((M, P), 0.7, 0.4, 0.2),$ $((B, S), 0.8, 0.3, 0.2) \}$,

$B_{2}$ = $\{((P, I), 0.8, 0.2, 0.4) \}$,

$B_{3}$ = $\{((S, P), 0.9, 0.2, 0.2), \}$,

$B_{4}$ = $\{((I, S), 0.8, 0.3, 0.4)$,$((UAE, I), 0.8, 0.3, 0.2) \}$,

$B_{5}$ = $\{((IT, P), 0.9, 0.3, 0.3),$ $((I, IT), 0.8, 0.3, 0.3) \}$,

$B_{6} = \{((K, S), 0.7, 0.2, 0.4) \}$.

Clearly, $(A, B_{1}, B_{2}, B_{3}, B_{4}, B_{5}, B_{6})$ is a neutrosophic graph structure as shown in Fig. 2.22.

Fig. 2.22

Neutrosophic graph structure showing most crucial maritime crime between any two countries

In neutrosophic graph structure shown in Fig. 2.22, every edge detects most frequent crime between adjacent countries during maritime trade. For instance, most frequent maritime crime between Pakistan and Singapore is black money transfer, its strength is 90%, weakness is 20% and indeterminacy is 20%. We can also note that for relation human trafficking, vertex Pakistan has highest vertex degree, it means Pakistan is most sensitive country for human trafficking. Moreover, according to our neutrosophic graph structure, most frequent crime is human trafficking. It means that navy and maritime forces of these eight countries should take action to control human trafficking.

2.4.2 Decision-Making of Prominent Relationships

Among the countries of this world, various types of relationships exist, for example friendship, rival or enemy, religious affection, trade, political and military. Between any two countries, all relationships are not of same strength. Some relationships are comparatively stronger than other relationships. In general, it is difficult and time consuming to judge all relationships among the countries and to decide the most prominent one. But through neutrosophic graph structure, we can represent all these in easiest way and can be judged even in a single glance on graph. Moreover, we can be aware of the status of relationship, that is, what is percentage of its strength, weakness and in how much percentage it is indeterminate. We can also examine which pair of countries are in same kind of relationship. We consider a set X of eight countries.

$X = \{America, Russia, China, Japan, Pakistan, India, Iran, Saudi Arabia\}$. Let A be the neutrosophic set on X, defined in Table 2.9.

Table 2.9 Neutrosophic set A of a few countries on globe

In Table 2.9, T indicates positive impact (strength) of a particular country for whole world, F indicates negative impact (weakness), and I expresses that in what percentage or magnitude that country’s position is undecided or indeterminate for global world. Let we denote the countries with alphabets: A = America, R = Russia, CH = China, J = Japan, P = Pakistan, I = India, IR = Iran, S = Saudi Arabia.

In Tables 2.10, 2.11, 2.12, 2.13, 2.14 and 2.15, we have shown the T, I and F values of different relationships for each pair of countries.

Table 2.10 Neutrosophic set of relationships between America and other countries

Table 2.11 Neutrosophic set of relationships between Russia and other countries

Table 2.12 Neutrosophic set of relationships between China and other countries

Table 2.13 Neutrosophic set of relationships between Japan and other countries

Table 2.14 Neutrosophic set of relationships between Saudi Arabia and other countries

Table 2.15 Neutrosophic set of relationships between Pakistan and other countries

We can define many relations on set X, let we define six relations on X as:

$E_{1}$ = Friendship, $E_{2}$ = Rival or Enemy, $E_{3}$ = Religious affection, $E_{4}$ = Trade, $E_{5}$ = Politics, $E_{6}$ = Military, such that $(X, E_{1}, E_{2}, E_{3}, E_{4}, E_{5}, E_{6})$ is a graph structure. An element in a relation indicates that these two countries have a particular relationship. As $(X, E_{1}, E_{2}, E_{3}, E_{4}, E_{5}, E_{6})$ is a graph structure, so an element will not be in more than one relation. So, we will put it in that relation for which percentage of truth is high, percentage of both falsity and indeterminacy is low as compared to other relationships, using above-mentioned data.

We write the elements in relations with their truth, falsity and indeterminacy values according to given data, resulting sets are neutrosophic sets on $E_{1}$, $E_{2}$, $E_{3}$, $E_{4}$, $E_{5}$, $E_{6}$, respectively. We can name these sets as $B_{1}$, $B_{2}$, $B_{3}$, $B_{4}$, $B_{5}$, $B_{6}$, respectively. Let

$B_{1}$ = $\{((P,CH), 0.7,0.1,0.1) \}$,

$B_{2}$ = $\{((P,I), 0.7,0.1,0.1), ((A,R), 0.7,0.1,0.1)$, $((A,CH), 0.8,0.2,0.1), ((I,CH), 0.7,0.2,0.2) \}$,

$B_{3}$ = $\{((P,S), 0.6,0.1,0.1),$ $((P,IR), 0.7,0.4,0.5) \}$,

$B_{4}$ = $\{((P,J), 0.7,0.3,0.2), ((I,J), 0.7,0.2,0.1) \}$,

$B_{5}$ = $\{((P,A), 0.6,0.1,0.1), ((A,I), 0.7,0.3,0.2), ((A,S), 0.6,0.2,0.3)$,

$((A,IR), 0.7,0.3,0.1), ((A,J), 0.8,0.3,0.3) \}$,

$B_{6}$ = $\{((P,R), 0.7,0.1,0.3), ((R,I), 0.7,0.2,0.4) \}$.

Clearly, (A, $B_{1}$, $B_{2}$, $B_{3}$, $B_{4}$, $B_{5}$, $B_{6}$) is a neutrosophic graph structure as shown in Fig. 2.23.

Fig. 2.23

Neutrosophic graph structure showing most prominent relationship between any two vertices(countries)

In neutrosophic graph structure shown in Fig. 2.23, every edge indicates the most prominent relationship of adjacent vertices(countries), for example most prominent relationship between Pakistan and China is friendship, it is 70% strong, 10% weak and 10% indeterminate. It can be noted that for the relation politics, vertex America has highest degree, it shows that America is the most prominent country for having political relationship with other countries in A. Further, we can tell that China and India, America and Russia, Pakistan and India have common relationship, that is, they are rival or enemy of each other. Moreover, according to our neutrosophic graph structure most frequent relation is politics, it means that among these eight countries politics is dominating relationship.

This neutrosophic graph structure depicts most prominent relationships among some elements (countries) of A. By taking large neutrosophic graph structure, most dominating relationships among all the countries of A can be detected. On the similar basis, we can make a neutrosophic graph structure for all countries across the world, in order to find the status and strength of prominent relationships among them. From neutrosophic graph structure, we can also determine that which pair of countries have common relationships. Further, we can find which country is most prominent for having a particular kind of relationship with other countries. Most frequent relationship in the neutrosophic graph structure will indicate that this relationship is prevailing in the world. So, using neutrosophic graph structure, it is quite easy to judge, in which direction this world is moving? whether it is moving towards peace or war/Cold War.

2.4.3 Detection of Most Frequent Smuggling

Smuggling on the seaports are increasing rapidly with time. There are 4,764 seaports on Atlantic ocean, Arctic ocean, Indian ocean, Pacific ocean, etc. These seaports are very useful and advantageous for import and export of different types of goods through out the world. Besides, there are also many disadvantages of these seaports. Crimes held on seaports are in abundance, but Smuggling of different kinds like human smuggling, weapons smuggling, black money smuggling, gold and diamond smuggling, smuggling of ivory and drug smuggling are most alarming. A lot of time and labour is required to collect and manipulate the data from all seaports to judge that which type of smuggling is frequent. But using neutrosophic graph structure, we can easily investigate the fact that between any two seaports which type of smuggling is chronic and increasing violently. Moreover, we can decide which seaport is most sensitive for smuggling, globally and need to be focused by security teams. We consider a set X consisting of eight seaports.

X= {Chalna, Penang, Singapore, Dubai, Karachi, Mumbai, Mombasa, Gioia Tauro}. Let A be the neutrosophic set on X, defined in Table 2.16.

Table 2.16 Neutrosophic set A of eight seaports

In Table 2.16, T depicts the importance of that particular seaport in the world due to its geographic position, F indicates the degree of its nonimportance in the world, and I expresses, to which extent it is undecided/indeterminate to be beneficial for the world, geographically.

Let Chalna = C, Pengang = P, Singapore = S, Dubai = DU, Karachi = K, Mumbai = MU, Mombasa = MO, Gioia Tauro = GT.

In Tables 2.17, 2.18, 2.19, 2.20, 2.21, 2.22 and 2.23, we have shown the values of T, I and F of different smuggling for each pair of seaports.

Table 2.17 Neutrosophic set of smuggling between Karachi and other seaports

Table 2.18 Neutrosophic set of smuggling between Dubai and other seaports

Table 2.19 Neutrosophic set of smuggling between Chalna and other seaports

Table 2.20 Neutrosophic set of smuggling between Penang and other seaports

Table 2.21 Neutrosophic set of smuggling between Singapore and other seaports

Table 2.22 Neutrosophic set of smuggling between Gioia Tauro and other seaports

Table 2.23 Neutrosophic set of smuggling between Mombasa and other seaports

Many relations on set X can be defined, let we define six relations on X as:

$E_{1}$ = Human smuggling, $E_{2}$ = Weapons smuggling, $E_{3}$ = Black money smuggling, $E_{4}$ = Gold and diamond smuggling, $E_{5}$ = Drug smuggling, $E_{6}$ = Smuggling of ivory, such that $(X, E_{1}, E_{2}, E_{3}, E_{4}, E_{5}, E_{6})$ is a graph structure. An element in a relation detects that kind of smuggling between those two seaports.

As $(X, E_{1}, E_{2}, E_{3}, E_{4}, E_{5}, E_{6})$ is a graph structure, an element will not be in more than one relations, so it can appear just once. Therefore, we will consider it an element of that relation for which its percentage of truth is high, and percentage of both falsity and indeterminacy is low as compared to other relations.

According to given data, we write the elements in relations with their truth, falsity and indeterminacy values, so the resulting sets are neutrosophic sets on $E_{1}$, $E_{2}$, $E_{3}$, $E_{4}$, $E_{5}$, $E_{6}$, respectively. We can name these sets as $B_{1}$, $B_{2}$, $B_{3}$, $B_{4}$, $B_{5}$, $B_{6}$, respectively. Let

$E_{1}$ = $\{(Chalna, Karachi), (Penang, Karachi), (Chalna, Singapore)\}$,

$E_{2}$ = $\{(Karachi, Mumbai)\}$,

$E_{3}$ = $\{(Singapore, Karachi) \}$,

$E_{4}$ = $\{(Mumbai, Singapore),$ $(Dubai, Mumbai)\}$,

$E_{5}$ = $\{(Gioia Tauro, Karachi), (Mumbai,$Gioia Tauro$)\}$,

$E_{6}$ = $\{(Mombasa, Singapore)\}$.

And corresponding neutrosophic sets are:

$B_{1}$ = $\{((C, K), 0.7, 0.2, 0.3),$ $((P, K), 0.6, 0.3, 0.1),$ $((C, S), 0.7, 0.2, 0.1) \}$,

$B_{2}$ = $\{((K, MU), 0.7, 0.1, 0.3) \}$,

$B_{3}$ = $\{((S, K), 0.8, 0.1, 0.1), \}$,

$B_{4}$ = $\{((MU, S), 0.7, 0.2, 0.3)$, $((DU, MU), 0.7, 0.2, 0.1) \}$,

$B_{5}$ = $\{((GT, K), 0.8, 0.2, 0.2),$ $((MU, GT), 0.7, 0.2, 0.2) \}$,

$B_{6} = \{((MO, S), 0.6, 0.1, 0.3) \}$.

Clearly, $(A, B_{1}, B_{2}, B_{3}, B_{4}, B_{5}, B_{6})$ is a neutrosophic graph structure as shown in Fig. 2.24.

Fig. 2.24

Neutrosophic graph structure showing most frequent smuggling between any two seaports

In neutrosophic graph structure shown in Fig. 2.24, every edge detects most frequent smuggling between adjacent seaports. For instance, most frequent smuggling between Karachi and Singapore is black money smuggling, its strength is 80%, weakness is 10% and indeterminacy is 10%. We can also note that for relation human smuggling, vertex Karachi has highest vertex degree, it means Karachi is most sensitive seaport for human smuggling. Moreover, according to our neutrosophic graph structure most frequent smuggling is human smuggling. It means that at these eight seaports, security forces should take action to control human smuggling.

This neutrosophic graph structure detects most frequent smuggling between some seaports of set A. By making a neutrosophic graph structure of all seaports, we can examine between any two seaports, which kind of smuggling is most frequent, we can also tell that which seaport is most sensitive for particular kind of smuggling. Further, we may get information about violently increasing smuggling through seaports in the whole world. That is why neutrosophic graph structures can be very helpful for security forces to overcome the smuggling at seaports.

We now elaborate general procedure of our applications in the following Algorithm.

Algorithm 2.4.1

• Step 1. Input the set X = $\{A_{1}, A_{2}, \ldots , A_{n}\}$ of vertices and the neutrosophic vertex set A defined on X.

• Step 2. Input neutrosophic set of relationships or smuggling of a vertex with other vertices and compute T, I and F of each pair of vertices using: $T(A_{i}A_{j}) \le \min (T(A_{i}), T(A_{j}))$,   $I(A_{i}A_{j}) \le \min (I(A_{i}), I(A_{j}))$, $F(A_{i}A_{j}) \le \max (F(A_{i}), F(A_{j}))$.

• Step 3. Repeat Step 2 for all vertices in X.

• Step 4. Define relations $E_{1}, E_{2}, \ldots , E_{n}$ on set X such that $(X, E_{1}, E_{2}, \ldots , E_{n} )$ is a graph structure.

• Step 5. Put an element in that relation for which value of T is high, and values of I and F are low as compared to other relations.

• Step 6. Write all elements of relations with their T, I and F values, resulting relations $B_{1}, B_{2}, \ldots , B_{n}$ are neutrosophic sets on $E_{1}, E_{2}, E_{3}, \ldots , E_{n}$, respectively, and $(A, B_{1}, B_{2}, \ldots , B_{n} )$ is a neutrosophic graph structure.