Neutrosophic Soft Rough Graphs

Abstract

Neutrosophic soft rough set model is a hybrid model by combining neutrosophic soft sets with rough sets. We apply neutrosophic soft rough sets to graphs. We present the concept of neutrosophic soft rough graphs and describe different methods of their construction. We develop an efficient algorithm of our method to solve decision-making problems.

Neutrosophic soft rough set model is a hybrid model by combining neutrosophic soft sets with rough sets. We apply neutrosophic soft rough sets to graphs. We present the concept of neutrosophic soft rough graphs and describe different methods of their construction. We develop an efficient algorithm of our method to solve decision-making problems. This chapter is due to [17].

8.1 Introduction

Pawlak [142] introduced the concept of rough set. He was a Polish mathematician (citizen of Poland) and computer scientist. Rough means approximate or inexact. Rough set theory expresses vagueness in terms of a boundary region of a set not in terms of membership function as in fuzzy set. The idea of rough set theory is a generalization of classical set theory to study the intelligence systems containing inexact, uncertain or incomplete information. It is an effective drive for bestowal with uncertain or incomplete information. Rough set theory is a novel mathematical approach to imprecise knowledge. Rough set theory expresses vagueness by means of a boundary region of a set. The emptiness of boundary region of a set shows that this is a crisp set, and nonemptiness shows that this is a rough set. Nonemptiness of boundary region also describes the deficiency of our knowledge about a set. A subset of a universe in rough set theory is expressed by two approximations which are known as lower and upper approximations. Equivalence classes are the basic building blocks in rough set theory, for upper and lower approximations. Dubois and Prade [74] investigated rough sets and fuzzy sets and concluded that these two theories are different approaches to handle vagueness. They reported that these are not opposite theories and to obtain beneficial results, both theories can be combined. Following this idea, Broumi et al. [61] introduced the concept of rough neutrosophic sets. Yang et al. [177] proposed single-valued neutrosophic rough sets by combining single-valued neutrosophic sets and rough sets, and established an algorithm for decision-making problem based on single-valued neutrosophic rough sets on two universes. Zhang et al. [203] presented the notion of intuitionistic fuzzy rough sets. The notions of soft rough neutrosophic sets and neutrosophic soft rough sets as hybrid models are described in [26]. We give a list of notations in Table 8.1.

Table 8.1 List of notations

Definition 8.1

Let X be an initial universal set, P a universal set of parameters and $M \subseteq P$. For an arbitrary neutrosophic soft relation $\mathbb {R}$ over $X\times M$, $(X, M, \mathbb {R})$ is called neutrosophic soft approximation space.

For any neutrosophic set $A \in \mathcal {N}(M),$ we define the upper neutrosophic soft rough approximation and the lower neutrosophic soft rough approximation operators of A with respect to $(X, M, \mathbb {R})$ denoted by $\overline{\mathbb {R}}(A)$ and $\underline{\mathbb {R}}(A),$ respectively, as follows:

$$\begin{aligned} \overline{\mathbb {R}}(A)=&\{(x, T_{\overline{\mathbb {R}}(A)}(x), I_{\overline{\mathbb {R}}(A)}(x), F_{\overline{\mathbb {R}}(A)}(x))~|~x \in X\},\\ \underline{\mathbb {R}}(A)=&\{(x, T_{\underline{\mathbb {R}}(A)}(x), I_{\underline{\mathbb {R}}(A)}(x), F_{\underline{\mathbb {R}}(A)}(x))~|~x \in X\}, \end{aligned}$$

where

The pair $(\underline{\mathbb {R}}(A),\overline{\mathbb {R}}(A))$ is called neutrosophic soft rough set of A w.r.t. $(X, M, \mathbb {R}), and$ $\underline{\mathbb {R}}$ and $\overline{\mathbb {R}}$ are referred to as the lower neutrosophic soft rough approximation and the upper neutrosophic soft rough approximation operators, respectively.

Example 8.1

Suppose that $X=\{x_1, x_2, x_3, x_4\}$ is the set of careers under consideration, Mr. X wants to select best suitable career. $M=\{m_1, m_2, m_3\}$ be a set of decision parameters. Mr. X describe the “most suitable career” by defining a neutrosophic soft set $\mathbb {R}=(F, M)$ on X which is a neutrosophic relation from X to M as shown in Table 8.2.

Table 8.2 Neutrosophic soft relation $\mathbb {R}$

Now, Mr. X gives the most favourable decision object A which is a neutrosophic set on M defined as follows: $A$ $=$ $\{$ $($ $m_1$ $,$ $0.5$ $,$ $0.2$ $,$ $0.4$ $)$,$($ $m_2$ $,$ $0.2$ $,$ $0.3$ $,$ $0.1$ $)$,$(m_3$ $,$ $0.2$ $,$ $0.4$ $,$ $0.6$ $)$ $\}$. By Definition 8.1, we have

$$\begin{aligned} T_{\overline{\mathbb {R}}(A)}(x_1)= 0.3, ~~~ I_{\overline{\mathbb {R}}(A)}(x_1)= 0.4,~~~ F_{\overline{\mathbb {R}}(A)}(x_1)=0.4, \end{aligned}$$

$$\begin{aligned} T_{\overline{\mathbb {R}}(A)}(x_2)=0.4,~~~ I_{\overline{\mathbb {R}}(A)}(x_2)= 0.2,~~~F_{\overline{\mathbb {R}}(A)}(x_2)= 0.4, \end{aligned}$$

$$\begin{aligned} T_{\overline{\mathbb {R}}(A)}(x_3)=0.2,~~~ I_{\overline{\mathbb {R}}(A)}(x_3)= 0.4,~~~F_{\overline{\mathbb {R}}(A)}(x_3)= 0.3, \end{aligned}$$

$$\begin{aligned} T_{\overline{\mathbb {R}}(A)}(x_4)=0.2,~~~ I_{\overline{\mathbb {R}}(A)}(x_4)= 0.3,~~~F_{\overline{\mathbb {R}}(A)}(x_4)= 0.4. \end{aligned}$$

Similarly,

$$\begin{aligned} T_{\underline{\mathbb {R}}(A)}(x_1)=0.4, ~~~ I_{\underline{\mathbb {R}}(A)}(x_1)=0.4,~~~ F_{\underline{\mathbb {R}}(A)}(x_1)=0.3, \end{aligned}$$

$$\begin{aligned} T_{\underline{\mathbb {R}}(A)}(x_2)=0.5,~~~ I_{\underline{\mathbb {R}}(A)}(x_2)= 0.4,~~~F_{\underline{\mathbb {R}}(A)}(x_2)= 0.4, \end{aligned}$$

$$\begin{aligned} T_{\underline{\mathbb {R}}(A)}(x_3)=0.4,~~~ I_{\underline{\mathbb {R}}(A)}(x_3)= 0.4,~~~F_{\underline{\mathbb {R}}(A)}(x_3)= 0.3, \end{aligned}$$

$$\begin{aligned} T_{\underline{\mathbb {R}}(A)}(x_4)=0.5,~~~ I_{\underline{\mathbb {R}}(A)}(x_4)= 0.4,~~~F_{\underline{\mathbb {R}}(A)}(x_4)= 0.5. \end{aligned}$$

Thus, we obtain

$$\begin{aligned} \overline{\mathbb {R}}(A)=&\{(x_1,0.3,0.4,0.4),(x_2,0.4,0.2,0.4),(x_3,0.2,0.4,0.3),(x_4,0.2,0.3,0.4)\},\\ \underline{\mathbb {R}}(A)=&\{(x_1,0.4,0.4,0.3),(x_2,0.5,0.4,0.4),(x_3,0.4,0.4,0.3),(x_4,0.5,0.4,0.5)\}. \end{aligned}$$

Hence $(\underline{\mathbb {R}}(A), \overline{\mathbb {R}}(A))$ is a neutrosophic soft rough set of A.

The conventional neutrosophic soft set is a mapping from a parameter to the neutrosophic subset of universe, and let $\mathbb {R}$ $=$ $($ $F$ $,$ $M$ $)$ be neutrosophic soft set. Now, we present the constructive definition of neutrosophic soft rough relation by using a neutrosophic soft relation $\mathbb {S}$ from $M$ $\times$ $M$ $=$ $\acute{M}$ to $\mathcal {N}$ $($ $X$ $\times$ $X$ $=$ $\acute{X}$ $)$, where X be a universal set and M be a set of parameter.

Definition 8.2

A neutrosophic soft rough relation $(\underline{\mathbb {\mathbb {S}}}(B), \overline{\mathbb {S}}(B))$ on X is a neutrosophic soft rough set, and $\mathbb {S}: \acute{M}\rightarrow \mathcal {N}(\acute{X})$ is a neutrosophic soft relation on X defined by $\mathbb {S}(m_im_j)=\{x_ix_j~|~\exists x_i \in \mathbb {R}(m_i), x_j\in \mathbb {R}(m_j)\},~x_ix_j \in \acute{X}$, such that

$$\begin{aligned} T_{\mathbb {\mathbb {S}}} (x_ix_j,m_im_j) \le&\min \{T_{\mathbb {R}} (x_i,m_i), T_{\mathbb {R}} (x_j,m_j)\} \\ I_{\mathbb {\mathbb {S}}} (x_ix_j,m_im_j) \le&\max \{I_{\mathbb {R}} (x_i,m_i), I_{\mathbb {R}} (x_j,m_j)\} \\ F_{\mathbb {\mathbb {S}}} (x_ix_j,m_im_j) \le&\max \{F_{\mathbb {R}} (x_i,m_i), F_{\mathbb {R}} (x_j, m_j)\}. \end{aligned}$$

For any $B$ $\in$ $\mathcal {N}$ $($ $\acute{M}$ $)$, $B$ $=$ $\{$ $\big ($ $m_i$ $m_j$ $,$ $T_{B}$ $($ $m_i$ $m_j$ $)$ $,$ $I_{B}$ $($ $m_i$ $m_j$ $)$ $,$ $F_{B}$ $($ $m_i$ $m_j$ $)$ $\big )$ $m_i$ $m_j$ $\in$ $\acute{M}$ $\}$,

$$\begin{aligned} T_{B}(m_im_j) \le&\min \{T_A(m_i), T_A(m_j)\}, \\ I_{B}(m_im_j) \le&\max \{I_A(m_i), I_A(m_j)\}, \\ F_{B}(m_im_j) \le&\max \{F_A(m_i), F_A(m_j)\}. \end{aligned}$$

The upper neutrosophic soft approximation and the lower neutrosophic soft approximation of B w.r.t. $(\acute{X}, \acute{M}, \mathbb {S})$ are defined as follows:

$$\begin{aligned} \overline{\mathbb {S}}(B)=\{(x_ix_j, T_{\overline{\mathbb {S}}(B)}(x_ix_j), I_{\overline{\mathbb {S}}(B)}(x_ix_j), F_{\overline{\mathbb {S}}(B)}(x_ix_j))~|~x_ix_j \in \acute{X}\}, \end{aligned}$$

$$\begin{aligned} \underline{\mathbb {S}}(B)=\{(x_ix_j, T_{\underline{\mathbb {S}}(B)}(x_ix_j), I_{\underline{\mathbb {S}}(B)}(x_ix_j), F_{\underline{\mathbb {S}}(B)}(x_ix_j))~|~x_ix_j \in \acute{X}\}, \end{aligned}$$

where

$$\begin{aligned} T_{\overline{\mathbb {S}}(B)}(x_ix_j)=&\bigvee \limits _{m_im_j \in \acute{M}} \big (T_{\mathbb {S}}(x_ix_j,m_im_j)\wedge T_{B}(m_im_j)\big ),\\ I_{\overline{\mathbb {S}}(B)}(x_ix_j)=&\bigwedge \limits _{m_im_j \in \acute{M}} \big (I_{\mathbb {S}}(x_ix_j,m_im_j)\vee I_{B}(m_im_j)\big ),\\F_{\overline{\mathbb {S}}(B)}(x_ix_j)=&\bigwedge \limits _{m_im_j \in \acute{M}}\big (F_{\mathbb {S}}(x_ix_j,m_im_j)\vee F_{B}(m_im_j)\big ),\\ T_{\underline{\mathbb {S}}(B)}(x_ix_j)=&\bigwedge \limits _{m_im_j \in \acute{M}} \big (F_{\mathbb {S}}(x_ix_j, m_im_j)\vee T_{B}(m_im_j)\big ),\\I_{\underline{\mathbb {S}}(B)}(x_ix_j)=&\bigvee \limits _{m_im_j \in \acute{M}} \big ((1-I_{\mathbb {S}}(x_ix_j,m_im_j))\wedge I_{B}(m_im_j)\big ),\\F_{\underline{\mathbb {S}}(B)}(x_ix_j)=&\bigvee \limits _{m_im_j \in \acute{M}}\big (T_{\mathbb {S}}(x_ix_j, m_im_j)\wedge F_{B}(m_im_j)\big ). \end{aligned}$$

The pair $(\underline{\mathbb {S}}(B),\overline{\mathbb {S}}(B))$ is called neutrosophic soft rough relation , and $\underline{\mathbb {S}},\overline{\mathbb {S}}:\mathcal {N}(\acute{M})\rightarrow \mathcal {N}(\acute{X})$ are called the lower neutrosophic soft rough approximation and the upper neutrosophic soft rough approximation operators, respectively.

Remark 8.1

Consider a neutrosophic set B on $\acute{M}$ and a neutrosophic set A on M; according to the definition of neutrosophic soft rough relation, we get

$$\begin{aligned} T_{\overline{\mathbb {S}}(B)}(x_ix_j) \le&\min \{T_{\overline{\mathbb {R}}(A)}(x_i), T_{\overline{\mathbb {R}}(A)}(x_j)\}, \\ I_{\overline{\mathbb {S}}(B)}(x_ix_j) \le&\max \{I_{\overline{\mathbb {R}}(A)}(x_i), I_{\overline{\mathbb {R}}(A)}(x_j)\}, \\ F_{\overline{\mathbb {S}}(B)}(x_ix_j) \le&\max \{F_{\overline{\mathbb {R}}(A)}(x_i), F_{\overline{\mathbb {R}}(A)}(x_j)\}. \end{aligned}$$

Similarly, for lower neutrosophic soft rough approximation operator $\underline{\mathbb {S}}(B)$,

$$\begin{aligned} T_{\underline{\mathbb {S}}(B)}(x_ix_j) \le&\min \{T_{\underline{\mathbb {R}}(A)}(x_i), T_{\underline{\mathbb {R}}(A)}(x_j)\}, \\ I_{\underline{\mathbb {S}}(B)}(x_ix_j) \le&\max \{I_{\underline{\mathbb {R}}(A)}(x_i), I_{\underline{\mathbb {R}}(A)}(x_j)\}, \\ F_{\underline{\mathbb {S}}(B)}(x_ix_j) \le&\max \{F_{\underline{\mathbb {R}}(A)}(x_i), F_{\underline{\mathbb {R}}(A)}(x_j)\}. \end{aligned}$$

Example 8.2

Let $X=\{x_1,x_2,x_3\}$ be a universal set and $M=\{m_1, m_2, m_3\}$ a set of parameters. A neutrosophic soft set $\mathbb {R}=(F, M)$ on X can be defined in Table 8.3 as follows.

Table 8.3 Neutrosophic soft set $\mathbb {R}=(F, M)$

Let $E=\{x_{1}x_{2}, x_{2}x_{3}, x_{2}x_{2}, x_{3}x_{2}\} \subseteq \acute{X}$ and $L=\{m_{1}m_{3}, m_{2}m_{1}, m_{3}m_{2}\} \subseteq \acute{M}$.

Then a soft relation $\mathbb {S}$ on E (from L to E) can be defined in Table 8.4 as follows.

Table 8.4 Neutrosophic soft relation $\mathbb {S}$

Let $A=\{(m_1,0.2,0.4,0.6),(m_2,0.4,0.5,0.2),(m_3,0.1,0.2,0.4)\}$ be a neutrosophic set on M, then

$\overline{\mathbb {R}}(A)=\{(x_1, 0.4,0.2,0.4), (x_2, 0.3, 0.4,0.3), (x_3, 0.4, 0.2, 0.3)\}$

$\underline{\mathbb {R}}(A)=\{(x_1, 0.3,0.5,0.4), (x_2, 0.2, 0.5,0.6), (x_3, 0.4, 0.5, 0.6)\}.$

Let $B=\{(m_1m_3,0.1,0.3,0.5),(m_2m_1,0.2,0.4,0.3),(m_3m_2,0.1,0.2,0.3)\}$ be a neutrosophic set on L,  then

$\overline{\mathbb {S}}(B)=\{(x_1x_2, 0.2,0.3,0.3), (x_2x_3, 0.2, 0.3,0.3), (x_2x_2, 0.2, 0.4, 0.3), (x_3x_2, 0.2, 0.4, 0.3)\}$

$\underline{\mathbb {S}}(B)=\{(x_1x_2, 0.2,0.4,0.4), (x_2x_3, 0.2, 0.4,0.5), (x_2x_2, 0.3, 0.4, 0.5), (x_3x_2, 0.2, 0.4, 0.5)\}$

Hence $\mathbb {S}B=(\underline{\mathbb {S}}(B), \overline{\mathbb {S}}(B))$ is neutrosophic soft rough relation.

8.2 Neutrosophic Soft Rough Information

Definition 8.3

A neutrosophic soft rough graph on a nonempty X is a four-ordered tuple $(X, M, \mathbb {R}A, \mathbb {S}B)$ such that

• (i)   M is a set of parameters.

• (ii)   $\mathbb {R}$ is an arbitrary neutrosophic soft relation over $X\times M$.

• (iii)   $\mathbb {S}$ is an arbitrary neutrosophic soft relation over $\acute{X}\times \acute{M}$.

• (vi)   $\mathbb {R}A=(\underline{\mathbb {R}}(A), \overline{\mathbb {R}}(A))$ is a neutrosophic soft rough set of X.

• (v)   $\mathbb {S}B=(\underline{\mathbb {S}}(B), \overline{\mathbb {S}}(B))$ is a neutrosophic soft rough relation on $\acute{X} \subseteq X \times X$.

$G=(\mathbb {R}A, \mathbb {S}B)$ is a neutrosophic soft rough graph, where $\underline{G}=(\underline{\mathbb {R}}(A), \underline{\mathbb {S}}(B))$ and $\overline{G}=(\overline{\mathbb {R}}(A), \overline{\mathbb {S}}(B))$ are lower neutrosophic approximate graph and upper neutrosophic approximate graph, respectively, of neutrosophic soft rough graph $G=(\mathbb {R}A, \mathbb {S}B)$.

Example 8.3

Let $X=\{x_1,x_2,x_3, x_4, x_5, x_6\}$ be a vertex set and $M=\{m_1, m_2, m_3\}$ a set of parameters. A neutrosophic soft relation over $X \times M$ can be defined in Table 8.5 as follows.

Table 8.5 Neutrosophic soft relation $\mathbb {R}$

Let $A=\{(m_1,0.5,0.4,0.6),(m_2,0.7,0.4,0.5),(m_3,0.6,0.2,0.5)\}$ be a neutrosophic set on M, then

$$\begin{aligned} \overline{\mathbb {R}}(A)=&\{(x_1, 0.5,0.4,0.5), (x_2, 0.6, 0.3,0.5), (x_3, 0.7, 0.4, 0.5), (x_4, 0.6, 0.2, 0.5),(x_5, 0.5,\\ {}&0.4, 0.5),(x_6, 0.6, 0.3, 0.5)\},\\ \underline{\mathbb {R}}(A)=&\{(x_1, 0.6,0.4,0.5), (x_2, 0.5, 0.4,0.6), (x_3, 0.5, 0.4, 0.6), (x_4, 0.5, 0.4, 0.5),(x_5, 0.6,\\ {}&0.4, 0.5),(x_6, 0.6, 0.4, 0.5)\}. \end{aligned}$$

Let $E=\{x_{1}x_{1}, x_{1}x_{2}, x_{2}x_{1}, x_{2}x_{3}, x_{4}x_{5}, x_{3}x_{4}, x_{5}x_{2}, x_{5}x_{6}\} \subseteq \acute{X}$ and $L=\{m_{1}m_{3}, m_{2}m_{1}, m_{3}m_{2}\} \subseteq \acute{M}$. Then a neutrosophic soft relation $\mathbb {S}$ on E (from L to E) can be defined in Tables 8.6 and 8.7 as follows.

Table 8.6 Neutrosophic soft relation $\mathbb {S}$

Table 8.7 Neutrosophic soft relation $\mathbb {S}$

Let $B=\{(m_1m_2, 0.4, 0.4, 0.5),(m_2m_3, 0.5, 0.4, 0.5),(m_1m_3, 0.5, 0.2, 0.5)\}$ be a neutrosophic set on L, then

$$\begin{aligned} \overline{\mathbb {S}}(B) = \,&\{(x_1x_1, 0.5,0.4,0.5), (x_1x_2, 0.4,0.2, 0.5),(x_2x_1, 0.4, 0.2, 0.5), (x_2x_3, 0.5, 0.3, 0.5), \\ {}&(x_3x_4, 0.5, 0.2, 0.5),(x_4x_5, 0.4, 0.3, 0.5), (x_5x_2, 0.5, 0.3, 0.5), (x_5x_6, 0.5, 0.3, 0.5)\},\\ \underline{\mathbb {S}}(B) = \,&\{(x_1x_1, 0.4, 0.4, 0.5)(x_1x_2, 0.5, 0.4, 0.4),(x_2x_1, 0.5, 0.4, 0.4), (x_2x_3, 0.4, 0.4, 0.5), \\ {}&(x_3x_4, 0.4, 0.4, 0.5),(x_4x_5, 0.4, 0.4, 0.4), (x_5x_2, 0.4, 0.4, 0.5), (x_5x_6, 0.4, 0.4, 0.5)\}. \end{aligned}$$

Hence $\mathbb {S}B=(\underline{\mathbb {S}}(B), \overline{\mathbb {S}}(B))$ is neutrosophic soft rough relation on $\acute{X}$. Thus, $\underline{G}=(\underline{\mathbb {R}}(A), \underline{\mathbb {S}}(B))$ and $\overline{G}=(\overline{\mathbb {R}}(A), \overline{\mathbb {S}}(B))$ are lower neutrosophic approximate graph and upper neutrosophic approximate graph, respectively, as shown in Fig. 8.1. Hence, $G=(\underline{G}, \overline{G})$ is neutrosophic soft rough graph.

Fig. 8.1

Neutrosophic soft rough graph $G=( \underline{G}, \overline{G})$

Definition 8.4

Let $G_{1}=(\underline{G}_{1},\overline{G}_{1})$ and $G_{2}=(\underline{G}_{2},\overline{G}_{2})$ be two neutrosophic soft rough graphs on X. The union of $G_{1}$ and $G_{2}$ is a neutrosophic soft rough graph $G=G_{1}\cup G_{2}=(\underline{ G}_{1}\cup \underline{G}_{2},\overline{G}_{1}\cup \overline{G}_{2})$, where $\underline{G}_{1}\cup \underline{G}_{2}=(\underline{\mathbb {R}}(A_{1})\cup \underline{\mathbb {R}}(A_{2}), \underline{\mathbb {S}}(B_{1})\cup \underline{\mathbb {S}}(B_{2}))$ and $\overline{G}_{1}\cup \overline{G}_{2}=(\overline{\mathbb {R}}(A_{1})\cup \overline{\mathbb {R}}(A_{2}), \overline{\mathbb {S}}(B_{1})\cup \overline{\mathbb {S}}(B_{2}))$ are neutrosophic graphs, such that

(i):

  ${\forall x\in {{{\mathbb {R}}}A_{1}}}$ but $x\notin {{{\mathbb {R}}}A_{2}}.$

$$\begin{aligned} T_{{{\overline{\mathbb {R}}}(A_{1})}\cup {{\overline{\mathbb {R}}}(A_{2})}}(x)=&T_{{\overline{\mathbb {R}}}(A_{1})}(x), T_{{{\underline{\mathbb {R}}}(A_{1})}\cup {{\underline{\mathbb {R}}}(A_{2})}}(x)= T_{{\underline{\mathbb {R}}}(A_{1})}(x),&\\ I_{{{\overline{\mathbb {R}}}(A_{1})}\cup {{\overline{\mathbb {R}}}(A_{2})}}(x)=&I_{{\overline{\mathbb {R}}}(A_{1})}(x),\; I_{{{\underline{\mathbb {R}}}(A_{1})}\cup {{\underline{\mathbb {R}}}(A_{2})}}(x)=I_{{\underline{\mathbb {R}}}(A_{1})}(x),&\\ F_{{{\overline{\mathbb {R}}}(A_{1})}\cup {{\overline{\mathbb {R}}}(A_{2})}}(x)=&F_{{\overline{\mathbb {R}}}(A_{1})}(x), F_{{{\underline{\mathbb {R}}}(A_{1})}\cup {{\underline{\mathbb {R}}}(A_{2})}}(x)=F_{{\underline{\mathbb {R}}}(A_{1})}(x).&\end{aligned}$$

(ii):

  $\forall x\notin {{{\mathbb {R}}}A_{1}}$ but $x\in {{{\mathbb {R}}}A_{2}}.$

$$\begin{aligned} T_{{\overline{\mathbb {R}}}(A_{1})\cup {{\overline{\mathbb {R}}}(A_{2})}}(x)=&T_{{\overline{\mathbb {R}}}(A_{2})}(x), T_{ {{\underline{\mathbb {R}}}A_{1}}\cup {{\underline{\mathbb {R}}}(A_{2})}}(x)= T_{{\underline{\mathbb {R}}}(A_{2})}(x),&\\ I_{{\overline{\mathbb {R}}}(A_{1})\cup {{\overline{\mathbb {R}}}(A_{2})}}(x)=&I_{{\overline{\mathbb {R}}}(A_{2})}(x),\; I_{{{\underline{\mathbb {R}}}(A_{1})}\cup {{\underline{\mathbb {R}}}(A_{2})}}(x)=I_{{\underline{\mathbb {R}}}(A_{2})}(x),&\\ F_{{\overline{\mathbb {R}}}(A_{1})\cup {{\overline{\mathbb {R}}}(A_{2})}}(x)=&F_{{\overline{\mathbb {R}}}(A_{2})}(x), F_{{{\underline{\mathbb {R}}}(A_{1})}\cup {{\underline{\mathbb {R}}}(A_{2})}}(x)=F_{{\underline{\mathbb {R}}}(A_{2})}(x) .&\end{aligned}$$

(iii):

  ${\forall x\in {{{\mathbb {R}}}A_{1}}\cap {{\mathbb {R}}A_{2}}}$

$$\begin{aligned} T_{{{\overline{\mathbb {R}}}(A_{1})}\cup {{\overline{\mathbb {R}}}(A_{2})}}(x)=&\max \{T_{{\overline{\mathbb {R}}}(A_{1})}(x), T_{{\overline{\mathbb {R}}}(A_{2})}(x)\},&\\ T_{{{\underline{\mathbb {R}}}(A_{1})}\cup {{\underline{\mathbb {R}}}(A_{2})}}(x)=&\max \{T_{{\underline{\mathbb {R}}}(A_{1})}(x), T_{{\underline{\mathbb {R}}}(A_{2})}(x)\},&\\ I_{{{\overline{\mathbb {R}}}(A_{1})}\cup {{\overline{\mathbb {R}}}(A_{2})}}(x)=&\min \{I_{{\overline{\mathbb {R}}}(A_{1})}(x), I_{{\overline{\mathbb {R}}}(A_{2})}(x)\},&\\ I_{{{\underline{\mathbb {R}}}(A_{1})}\cup {{\underline{\mathbb {R}}}}(A_{2})}(x)=&\min \{I_{{\underline{\mathbb {R}}}(A_{1})}(x), I_{{\underline{\mathbb {R}}}(A_{2})}(x)\},&\\ F_{{{\overline{\mathbb {R}}}(A_{1})}\cup {{\overline{\mathbb {R}}}(A_{2})}}(x)=&\min \{F_{{\overline{\mathbb {R}}}(A_{1})}(x), F_{{\overline{\mathbb {R}}}(A_{2})}(x)\},&\\ F_{{{\underline{\mathbb {R}}}(A_{1})}\cup {{\underline{\mathbb {R}}}}(A_{2})}(x)=&\min \{F_{{\underline{\mathbb {R}}}(A_{1})}(x), F_{{\underline{\mathbb {R}}}(A_{2})}(x)\}.&\end{aligned}$$

(iv):

  ${\forall xy\in {{{\mathbb {S}}}B_{1}}}$ but $xy\notin {{{\mathbb {S}}}B_{2}}.$

$$\begin{aligned} T_{{{\overline{\mathbb {S}}}(B_{1})}\cup {{\overline{\mathbb {S}}}(B_{2})}}(xy)=&T_{{\overline{\mathbb {S}}}(B_{1})}(xy), T_{{{\underline{\mathbb {S}}}(B_{1})}\cup {{\underline{\mathbb {S}}}(B_{2})}}(xy)= T_{{\underline{\mathbb {S}}}(B_{1})}(xy),&\\ I_{{{\overline{\mathbb {S}}}(B_{1})}\cup {{\overline{\mathbb {S}}}(B_{2})}}(xy)=&I_{{\overline{\mathbb {S}}}(B_{1})}(xy),\; I_{{{\underline{\mathbb {S}}}(B_{1})}\cup {{\underline{\mathbb {S}}}(B_{2})}}(xy)=I_{{\underline{\mathbb {S}}}(B_{1})}(xy),&\\ F_{{{\overline{\mathbb {S}}}(B_{1})}\cup {{\overline{\mathbb {S}}}(B_{2})}}(xy)=&F_{{\overline{\mathbb {S}}}(B_{1})}(xy), F_{{{\underline{\mathbb {S}}}(B_{1})}\cup {{\underline{\mathbb {S}}}(B_{2})}}(xy)=F_{{\underline{\mathbb {S}}}(B_{1})}(xy).&\end{aligned}$$

(v):

  ${\forall xy\notin {{{\mathbb {S}}}B_{1}}}$ but $xy\in {{{\mathbb {S}}}B_{2}}$

$$\begin{aligned} T_{{{\overline{\mathbb {S}}}(B_{1})}\cup {{\overline{\mathbb {S}}}(B_{2})}}(xy)=&T_{{\overline{\mathbb {S}}}(B_{2})}(xy), T_{ {{\underline{\mathbb {S}}}(B_{1})}\cup {{\underline{\mathbb {S}}}(B_{2})}}(xy)= T_{{\underline{\mathbb {S}}}(B_{2})}(xy),&\\ I_{{{\overline{\mathbb {S}}}(B_{1})}\cup {{\overline{\mathbb {S}}}(B_{2})}}(xy)=&I_{{\overline{\mathbb {S}}}(B_{2})}(xy),\; I_{{{\underline{\mathbb {S}}}(B_{1})}\cup {{\underline{\mathbb {S}}}(B_{2})}}(xy)=I_{{\underline{\mathbb {S}}}(B_{2})}(xy),&\\ F_{{{\overline{\mathbb {S}}}(B_{1})}\cup {{\overline{\mathbb {S}}}(B_{2})}}(xy)=&F_{{\overline{\mathbb {S}}}(B_{2})}(xy), F_{{{\underline{\mathbb {S}}}(B_{1})}\cup {{\underline{\mathbb {S}}}(B_{2})}}(xy)=F_{{\underline{\mathbb {S}}}(B_{2})}(xy).&\end{aligned}$$

(vi):

  ${\forall xy\in {{{\mathbb {S}}}B_{1}}\cap {{\mathbb {S}}}(B_{2})}$

$$\begin{aligned} T_{{{\overline{\mathbb {S}}}(B_{1})}\cup {{\overline{\mathbb {S}}}(B_{2})}}(xy)=&\max \{T_{\overline{\mathbb {S}}(B_{1})}(xy), T_{{\overline{\mathbb {S}}}(B_{2})}(xy)\},&\\ T_{{{\underline{\mathbb {S}}}(B_{1})}\cup {{\underline{\mathbb {S}}}(B_{2})}}(xy)=&\max \{T_{\underline{\mathbb {S}}(B_{1})}(xy), T_{{\underline{\mathbb {S}}}(B_{2})}(xy)\},&\\ I_{{{\overline{\mathbb {S}}}(B_{1})}\cup {{\overline{\mathbb {S}}}(B_{2})}}(xy)=&\min \{I_{{\overline{\mathbb {S}}}(B_{1})}(xy), I_{{\overline{\mathbb {S}}}(B_{2})}(xy)\},&\\ I_{{{\underline{\mathbb {S}}}(B_{1})}\cup {{\underline{\mathbb {S}}}(B_{2})}}(xy)=&\min \{I_{{\underline{\mathbb {S}}}(B_{1})}(xy), I_{{\underline{\mathbb {S}}}(B_{2})}(xy)\},&\\ F_{{{\overline{\mathbb {S}}}(B_{1})}\cup {{\overline{\mathbb {S}}}(B_{2})}}(xy)=&\min \{F_{{\overline{\mathbb {S}}}(B_{1})}(xy), F_{{\overline{\mathbb {S}}}(B_{2})}(xy)\},&\\ F_{{{\underline{\mathbb {S}}}(B_{1})}\cup {{\underline{\mathbb {S}}}(B_{2})}}(xy)=&\min \{F_{{\underline{\mathbb {S}}}(B_{1})}(xy), F_{{\underline{\mathbb {S}}}(B_{2})}(xy)\}.&\\ \end{aligned}$$

Example 8.4

Let $X=\{x_1, x_2, x_3, x_4\}$ be a set of universe and $M=\{m_1,m_2,m_3\}$ a set of parameters. Then a neutrosophic soft relation over $X\times \mathbb { M}$ can be written as in Table 8.8.

Table 8.8 Neutrosophic soft relation $\mathbb {R}$

Let $A_1=\{(m_1,0.5,0.7,0.8), (m_2,0.7,0.5,0.3), (m_3,0.4,0.5,0.3)\},$ and

$A_2=\{(m_1,0.6,0.3,0.5),(m_2,0.5,0.8,0.2), (m_3,0.5,0.7,0.2)\}$ be two neutrosophic sets on M, Then $\mathbb {R}A_1=(\underline{\mathbb {R}}(A_1), \overline{\mathbb {R}}(A_1))$ and $\mathbb {R}A_2=(\underline{\mathbb {R}}(A_2), \overline{\mathbb {R}}(A_2))$ are neutrosophic soft rough sets, where

$$\begin{aligned} \underline{\mathbb {R}}(A_1)=&\{(x_1, 0.5, 0.6, 0.5),(x_2, 0.5, 0.5, 0.7) (x_3, 0.5, 0.5, 0.7),(x_40.4, 0.5, 0.5)\}, \\ \overline{\mathbb {R}}(A_1)=&\{(x_1,0.5, 0.5, 0.6),(x_2, 0.5, 0.5, 0.3),(x_3, 0.5, 0.5, 0.5),(x_4 0.5, 0.5, 0.3)\};\\ \underline{\mathbb {R}}(A_2)=&\{(x_1,0.6, 0.5, 0.5), (x_2,0.5, 0.7, 0.5), (x_3,0.5, 0.7, 0.5), (x_4,0.5, 0.6, 0.5)\},\\ \overline{\mathbb {R}}(A_2)=&\{ (x_1, 0.5, 0.4, 0.5), (x_2, 0.6, 0.6, 0.2), (x_3, 0.6, 0.6, 0.5), (x_4, 0.5, 0.7, 0.2) \}. \end{aligned}$$

Let $E=\{x_1x_2,x_1x_4,x_2x_2,x_2x_3,x_3x_3,x_3x_4\}\subseteq X\times X$, and $L=\{m_1m_2, m_1m_3, m_2m_3\}\subset \mathbb {\acute{M}}$. Then a neutrosophic soft relation on E can be written as in Table 8.9

Table 8.9 Neutrosophic soft relation $\mathbb {S}$

Let $B_1=\{(m_1m_2,0.5,0.4,0.5), (m_1m_3,0.3,0.4,0.5), (m_2m_3,0.4,0.4,0.3)\},$ and $B_2=\{(m_1m_2,0.5,0.3,0.2), (m_1m_3,0.4,0.3,0.3), (m_2m_3,0.4,0.6,0.2)\}$ be two neutrosophic sets on L. Then $\mathbb {S}B_1=(\underline{\mathbb {S}}(B_1), \overline{\mathbb {S}}(B_1))$ and $\mathbb {S}B_2=(\underline{\mathbb {S}}(B_2), \overline{\mathbb {S}}(B_2))$ are neutrosophic soft rough relations, where

$$\begin{aligned} \underline{\mathbb {S}}(B_1)=&\{(x_1x_2, 0.3, 0.4, 0.3), (x_1x_4, 0.3, 0.4, 0.4), (x_2x_2, 0.4, 0.4, 0.4), (x_2x_3, 0.3, 0.4, 0.4),\\ {}&(x_3x_3, 0.3, 0.4, 0.5), (x_3x_4, 0.3, 0.4, 0.5)\}, \\ \overline{\mathbb {S}}(B_1)=&\{ (x_1x_2,0.3, 0.4, 0.5), (x_1x_4,0.4, 0.4, 0.3), (x_2x_2,0.4, 0.4, 0.3), (x_2x_3,0.4, 0.4, 0.3),\\ {}&(x_3x_3,0.3, 0.4, 0.5), (x_3x_4,0.4, 0.4, 0.3)\};\\ \underline{\mathbb {S}}(B_2)=&\{ (x_1x_2,0.4, 0.6, 0.2), (x_1x_4,0.4, 0.6, 0.3), (x_2x_2,0.4, 0.6, 0.2), (x_2x_3,0.4, 0.6, 0.3),\\ {}&(x_3x_3,0.4, 0.6, 0.3), (x_3x_4,0.4, 0.6, 0.3) \},\\ \overline{\mathbb {S}}(B_2)=&\{(x_1x_2, 0.3,0.3,0.2), (x_1x_4,0.4,0.3,0.2), (x_2x_2,0.4,0.3,0.2), (x_2x_3,0.4,0.3,0.2),\\ {}&(x_3x_3,0.4,0.3,0.3), (x_3x_4, 0.4,0.4,0.2) \}. \end{aligned}$$

Thus $G_1=(\underline{G}_1, \overline{G}_1)$ and $G_2=(\underline{G}_2, \overline{G}_2)$ are neutrosophic soft rough graphs, where $\underline{G}_1=(\underline{\mathbb {R}}(A_1),\underline{\mathbb {S}}(B_1))$, $\overline{G}_1=(\overline{\mathbb {R}}(A_1),\overline{\mathbb {S}}(B_1))$ as shown in Fig. 8.2

Fig. 8.2

Neutrosophic soft rough graph $G_1=( \underline{G}_1, \overline{G}_1)$

$\underline{G}_2=(\underline{\mathbb {R}}(A_2),\underline{\mathbb {S}}(B_2))$, $\overline{G}_2=(\overline{\mathbb {R}}(A_2),\overline{\mathbb {S}}(B_2))$ as shown in Fig. 8.3.

Fig. 8.3

Neutrosophic soft rough graph $G_2=( \underline{G}_2, \overline{G}_2)$

The union of $G_1=(\underline{G}_1,\overline{G}_1)$ and $G_2=(\underline{G}_2,\overline{G}_2)$ is neutrosophic soft rough graph $G=G_1\cup G_2=(\underline{G}_1\cup \underline{G}_2, \overline{G}_1\cup \overline{G}_2)$ as shown in Fig. 8.4.

Fig. 8.4

Neutrosophic soft rough graph $G_1\cup G_2=( \underline{G}_1\cup \underline{G}_2, \overline{G}_1\cup \overline{G}_2)$

Definition 8.5

Let $G_{1}=(\underline{G}_{1},\overline{G}_{1})$ and $G_{2}=(\underline{G}_{2},\overline{G}_{2})$ be two neutrosophic soft rough graphs on X. The intersection of $G_{1}$ and $G_{2}$ is a neutrosophic soft rough graph $G=G_{1}\cap G_{2}=(\underline{G}_{1}\cap \underline{G}_{2}, \overline{G}_{1}\cap \overline{G}_{2})$, where $\underline{G}_{1}\cap \underline{G}_{2}\,{=}\,(\underline{\mathbb {R}}(A_{1})\cap \underline{\mathbb {R}}(A_{2}), \underline{\mathbb {S}}(B_{1}){\cap }\underline{\mathbb {S}}(B_{2}))$ and $\overline{G}_{1}\cap \overline{G}_{2}=(\overline{\mathbb {R}}(A_{1})\cap \overline{\mathbb {R}}(A_{2}), \overline{\mathbb {S}}(B_{1})\cap \overline{\mathbb {S}}(B_{2}))$ are neutrosophic graphs, respectively, such that

(i):

  ${\forall x\in {{{\mathbb {R}}}A_{1}}}$ but $x\notin {{{\mathbb {R}}}A_{2}}.$

$$\begin{aligned} \begin{aligned} T_{{{\overline{\mathbb {R}}}(A_{1})}\cap {{\overline{\mathbb {R}}}(A_{2})}}(x)=&T_{{\overline{\mathbb {R}}}(A_{1})}(x), T_{{{\underline{\mathbb {R}}}(A_{1})}\cap {{\underline{\mathbb {R}}}(A_{2})}}(x)= T_{{\underline{\mathbb {R}}}(A_{1})}(x),&\\ I_{{{\overline{\mathbb {R}}}(A_{1})}\cap {{\overline{\mathbb {R}}}(A_{2})}}(x)=&I_{{\overline{\mathbb {R}}}(A_{1})}(x),\; I_{{{\underline{\mathbb {R}}}(A_{1})}\cap {{\underline{\mathbb {R}}}(A_{2})}}(x)=I_{{\underline{\mathbb {R}}}(A_{1})}(x),&\\ F_{{{\overline{\mathbb {R}}}(A_{1})}\cap {{\overline{\mathbb {R}}}(A_{2})}}(x)=&F_{{\overline{\mathbb {R}}}(A_{1})}(x), F_{{{\underline{\mathbb {R}}}(A_{1})}\cap {{\underline{\mathbb {R}}}(A_{2})}}(x)=F_{{\underline{\mathbb {R}}}(A_{1})}(x).&\end{aligned} \end{aligned}$$

(ii):

  $\forall x\notin {{{\mathbb {R}}}A_{1}}$ but $x\in {{{\mathbb {R}}}A_{2}}.$

$$\begin{aligned} \begin{aligned} T_{{\overline{\mathbb {R}}}(A_{1})\cap {{\overline{\mathbb {R}}}(A_{2})}}(x)=&T_{{\overline{\mathbb {R}}}(A_{2})}(x), T_{ {{\underline{\mathbb {R}}}(A_{1})}\cap {{\underline{\mathbb {R}}}(A_{2})}}(x)= T_{{\underline{\mathbb {R}}}(A_{2})}(x),&\\ I_{{\overline{\mathbb {R}}}(A_{1})\cap {{\overline{\mathbb {R}}}(A_{2})}}(x)=&I_{{\overline{\mathbb {R}}}(A_{2})}(x),\; I_{{{\underline{\mathbb {R}}}(A_{1})}\cap {{\underline{\mathbb {R}}}(A_{2})}}(x)=I_{{\underline{\mathbb {R}}}(A_{2})}(x),&\\ F_{{\overline{\mathbb {R}}}(A_{1})\cap {{\overline{\mathbb {R}}}(A_{2})}}(x)=&F_{{\overline{\mathbb {R}}}(A_{2})}(x), F_{{{\underline{\mathbb {R}}}(A_{1})}\cap {{\underline{\mathbb {R}}}(A_{2})}}(x)=F_{{\underline{\mathbb {R}}}(A_{2})}(x). \end{aligned} \end{aligned}$$

(iii):

  ${\forall x\in {{{\mathbb {R}}}A_{1}}\cap {{\mathbb {R}}A_{2}}}$

$$\begin{aligned} \begin{aligned} T_{{{\overline{\mathbb {R}}}(A_{1})}\cap {{\overline{\mathbb {R}}}(A_{2})}}(x)=&\min \{T_{{\overline{\mathbb {R}}}(A_{1})}(x), T_{{\overline{\mathbb {R}}}(A_{2})}(x)\},&\\ T_{{{\underline{\mathbb {R}}}(A_{1})}\cap {{\underline{\mathbb {R}}}(A_{2})}}(x)=&\min \{T_{{\underline{\mathbb {R}}}(A_{1})}(x), T_{{\underline{\mathbb {R}}}(A_{2})}(x)\},&\\ I_{{{\overline{\mathbb {R}}}(A_{1})}\cap {{\overline{\mathbb {R}}}(A_{2})}}(x)=&\max \{I_{{\overline{\mathbb {R}}}(A_{1})}(x), I_{{\overline{\mathbb {R}}}(A_{2})}(x)\},&\\ I_{ {{\underline{\mathbb {R}}}(A_{1})}\cap {{\underline{\mathbb {R}}}}(A_{2})}(v)=&\max \{I_{{\underline{\mathbb {R}}}(A_{1})}(x), I_{{\underline{\mathbb {R}}}(A_{2})}(x)\},&\\ F_{{{\overline{\mathbb {R}}}(A_{1})}\cap {{\overline{\mathbb {R}}}(A_{2})}}(x)=&\max \{F_{{\overline{\mathbb {R}}}(A_{1})}(x), F_{{\overline{\mathbb {R}}}(A_{2})}(x)\},&\\ F_{ {{\underline{\mathbb {R}}}(A_{1})}\cap {{\underline{\mathbb {R}}}}(A_{2})}(v)=&\max \{F_{{\underline{\mathbb {R}}}(A_{1})}(x), F_{{\underline{\mathbb {R}}}(A_{2})}(x)\}.&\end{aligned} \end{aligned}$$

(iv):

  ${\forall xy\in {{{\mathbb {S}}}B_{1}}}$ but $xy\notin {{{\mathbb {S}}}B_{2}}.$

$$\begin{aligned} \begin{aligned} T_{{{\overline{\mathbb {S}}}(B_{1})}\cap {{\overline{\mathbb {S}}}(B_{2})}}(xy)=&T_{{\overline{\mathbb {S}}}(B_{1})}(xy), T_{{{\underline{\mathbb {S}}}(B_{1})}\cap {{\underline{\mathbb {S}}}(B_{2})}}(xy)= T_{{\underline{\mathbb {S}}}(B_{1})}(xy),&\\ I_{{{\overline{\mathbb {S}}}(B_{1})}\cap {{\overline{\mathbb {S}}}(B_{2})}}(xy)=&I_{{\overline{\mathbb {S}}}(B_{1})}(xy),\; I_{{{\underline{\mathbb {S}}}(B_{1})}\cap {{\underline{\mathbb {S}}}(B_{2})}}(xy)=I_{{\underline{\mathbb {S}}}(B_{1})}(xy),&\\ F_{{{\overline{\mathbb {S}}}(B_{1})}\cap {{\overline{\mathbb {S}}}(B_{2})}}(xy)=&F_{{\overline{\mathbb {S}}}(B_{1})}(xy), F_{{{\underline{\mathbb {S}}}(B_{1})}\cap {{\underline{\mathbb {S}}}(B_{2})}}(xy)=F_{{\underline{\mathbb {S}}}(B_{1})}(xy).&\end{aligned} \end{aligned}$$

(v):

  ${\forall xy\notin {{{\mathbb {S}}}B_{1}}}$ but $xy\in {{{\mathbb {S}}}B_{2}}$

$$\begin{aligned} \begin{aligned} T_{{{\overline{\mathbb {S}}}(B_{1})}\cap {{\overline{\mathbb {S}}}(B_{2})}}(xy)=&T_{\overline{\mathbb {S}}(B_{2})}(xy), T_{ {{\underline{\mathbb {S}}}(B_{1})}\cap {{\underline{\mathbb {S}}}(B_{2})}}(xy)= T_{{\underline{\mathbb {S}}}(B_{2})}(xy),&\\ I_{{{\overline{\mathbb {S}}}(B_{1})}\cap {{\overline{\mathbb {S}}}(B_{2})}}(xy)=&I_{{\overline{\mathbb {S}}}(B_{2})}(xy),\; I_{{{\underline{\mathbb {S}}}(B_{1})}\cap {{\underline{\mathbb {S}}}(B_{2})}}(xy)=I_{{\underline{\mathbb {S}}}(B_{2})}(xy),&\\ F_{{{\overline{\mathbb {S}}}(B_{1})}\cap {{\overline{\mathbb {S}}}(B_{2})}}(xy)=&F_{{\overline{\mathbb {S}}}(B_{2})}(xy), F_{{{\underline{\mathbb {S}}}(B_{1})}\cap {{\underline{\mathbb {S}}}(B_{2})}}(xy)=F_{{\underline{\mathbb {S}}}(B_{2})}(xy).&\end{aligned} \end{aligned}$$

(vi):

  ${\forall xy\in {{{\mathbb {S}}}B_{1}}\cap {{\mathbb {S}}(B_{2})}}$

$$\begin{aligned} \begin{aligned} T_{{{\overline{\mathbb {S}}}(B_{1})}\cap {{\overline{\mathbb {S}}}(B_{2})}}(xy)=&\min \{{{T_{\overline{\mathbb {S}}(B_{1})}}}(xy), T_{{\overline{\mathbb {S}}}(B_{2})}(xy)\},&\\ T_{{{\underline{\mathbb {S}}}(B_{1})}\cap {{\underline{\mathbb {S}}}(B_{2})}}(xy)=&\min \{{T_{\underline{\mathbb {S}}(B_{1})}}(xy), T_{{\underline{\mathbb {S}}}(B_{2})}(xy)\},&\\ I_{{{\overline{\mathbb {S}}}(B_{1})}\cap {{\overline{\mathbb {S}}}(B_{2})}}(xy)=&\max \{I_{{\overline{\mathbb {S}}}(B_{1})}(xy), I_{{\overline{\mathbb {S}}}(B_{2})}(xy)\},&\\ I_{{{\underline{\mathbb {S}}}(B_{1})}\cap {{\underline{\mathbb {S}}}(B_{2})}}(xy)=&\max \{I_{{\underline{\mathbb {S}}}(B_{1})}(xy), I_{{\underline{\mathbb {S}}}(B_{2})}(xy)\},&\\ F_{{{\overline{\mathbb {S}}}(B_{1})}\cap {{\overline{\mathbb {S}}}(B_{2})}}(xy)=&\max \{F_{{\overline{\mathbb {S}}}(B_{1})}(xy), F_{{\overline{\mathbb {S}}}(B_{2})}(xy)\},&\\ F_{{{\underline{\mathbb {S}}}(B_{1})}\cap {{\underline{\mathbb {S}}}(B_{2})}}(xy)=&\max \{F_{{\underline{\mathbb {S}}}(B_{1})}(xy), F_{{\underline{\mathbb {S}}}(B_{2})}(xy)\}.&\end{aligned} \end{aligned}$$

Definition 8.6

Let $G_{1}=(\underline{G}_{1},\overline{G}_{1})$ and $G_{2}=(\underline{G}_{2},\overline{G}_{2})$ be two neutrosophic soft rough graphs on X. The join of $G_{1}$ and $G_{2}$ is a neutrosophic soft rough graph $G=G_{1}+ G_{2}=(\underline{G}_{1}+\underline{G}_{2}, \overline{G}_{1}+\overline{G}_{2})$, where $\underline{G}_{1}+\underline{G}_{2}=(\underline{\mathbb {R}}(A_{1})+\underline{\mathbb {R}}(A_{2}), \underline{\mathbb {S}}(B_{1})+\underline{\mathbb {S}}(B_{2}))$ and $\overline{G}_{1}+\overline{G}_{2}=(\overline{\mathbb {R}}(A_{1})+\overline{\mathbb {R}}(A_{2}), \overline{\mathbb {S}}(B_{1})+\overline{\mathbb {S}}(B_{2}))$ are neutrosophic graphs, respectively, such that

(i):

  ${\forall x\in {{{\mathbb {R}}}A_{1}}}$ but $x\notin {{{\mathbb {R}}}A_{2}}.$

$$\begin{aligned} \begin{aligned} T_{{{\overline{\mathbb {R}}}(A_{1})}+ {{\overline{\mathbb {R}}}(A_{2})}}(x)=&T_{{\overline{\mathbb {R}}}(A_{1})}(x), T_{{{\underline{\mathbb {R}}}(A_{1})}+ {{\underline{\mathbb {R}}}(A_{2})}}(x)= T_{{\underline{\mathbb {R}}}(A_{1})}(x),&\\ I_{{{\overline{\mathbb {R}}}(A_{1})}+ {{\overline{\mathbb {R}}}(A_{2})}}(x)=&I_{{\overline{\mathbb {R}}}(A_{1})}(x),\; I_{{{\underline{\mathbb {R}}}(A_{1})}+ {{\underline{\mathbb {R}}}(A_{2})}}(x)=I_{{\underline{\mathbb {R}}}(A_{1})}(x),&\\ F_{{{\overline{\mathbb {R}}}(A_{1})}+ {{\overline{\mathbb {R}}}(A_{2})}}(x)=&F_{{\overline{\mathbb {R}}}(A_{1})}(x), F_{{{\underline{\mathbb {R}}}(A_{1})}+ {{\underline{\mathbb {R}}}(A_{2})}}(x)=F_{{\underline{\mathbb {R}}}(A_{1})}(x).&\end{aligned} \end{aligned}$$

(ii):

  $\forall x\notin {{{\mathbb {R}}}A_{1}}$ but $x\in {{{\mathbb {R}}}A_{2}}.$

$$\begin{aligned} \begin{aligned} T_{{\overline{\mathbb {R}}}(A_{1})+ {{\overline{\mathbb {R}}}(A_{2})}}(x)=&T_{{\overline{\mathbb {R}}}(A_{2})}(x), T_{ {{\underline{\mathbb {R}}}(A_{1})}+ {{\underline{\mathbb {R}}}(A_{2})}}(x)= T_{{\underline{\mathbb {R}}}(A_{2})}(x),&\\ I_{{\overline{\mathbb {R}}}(A_{1})+ {{\overline{\mathbb {R}}}(A_{2})}}(x)=&I_{{\overline{\mathbb {R}}}(A_{2})}(x),\; I_{{{\underline{\mathbb {R}}}(A_{1})}+ {{\underline{\mathbb {R}}}(A_{2})}}(x)=I_{{\underline{\mathbb {R}}}(A_{2})}(x),&\\ F_{{\overline{\mathbb {R}}}(A_{1})+ {{\overline{\mathbb {R}}}(A_{2})}}(x)=&F_{{\overline{\mathbb {R}}}(A_{2})}(x), F_{{{\underline{\mathbb {R}}}(A_{1})}+ {{\underline{\mathbb {R}}}(A_{2})}}(x)=F_{{\underline{\mathbb {R}}}(A_{2})}(x) .&\end{aligned} \end{aligned}$$

(iii):

  ${\forall x\in {{{\mathbb {R}}}A_{1}}\cap {{\mathbb {R}}A_{2}}}$

$$\begin{aligned} \begin{aligned} T_{{{\overline{\mathbb {R}}}(A_{1})}+ {{\overline{\mathbb {R}}}(A_{2})}}(x)=&\max \{T_{{\overline{\mathbb {R}}}(A_{1})}(x), T_{{\overline{\mathbb {R}}}(A_{2})}(x)\},&\\ T_{{{\underline{\mathbb {R}}}(A_{1})}+ {{\underline{\mathbb {R}}}(A_{2})}}(x)=&\max \{T_{{\underline{\mathbb {R}}}(A_{1})}(x), T_{{\underline{\mathbb {R}}}(A_{2})}(x)\},&\\ I_{{{\overline{\mathbb {R}}}(A_{1})}+ {{\overline{\mathbb {R}}}(A_{2})}}(x)=&\min \{I_{{\overline{\mathbb {R}}}(A_{1})}(x), I_{{\overline{\mathbb {R}}}(A_{2})}(x)\},&\\ I_{{{\underline{\mathbb {R}}}(A_{1})}+{{\underline{\mathbb {R}}}}(A_{2})}(x)=&\min \{I_{{\underline{\mathbb {R}}}(A_{1})}(x), I_{{\underline{\mathbb {R}}}(A_{2})}(x)\},&\\ F_{{{\overline{\mathbb {R}}}(A_{1})}+ {{\overline{\mathbb {R}}}(A_{2})}}(x)=&\min \{F_{{\overline{\mathbb {R}}}(A_{1})}(x), F_{{\overline{\mathbb {R}}}(A_{2})}(x)\},&\\ F_{{{\underline{\mathbb {R}}}(A_{1})}+{{\underline{\mathbb {R}}}}(A_{2})}(x)=&\min \{F_{{\underline{\mathbb {R}}}(A_{1})}(x), F_{{\underline{\mathbb {R}}}(A_{2})}(x)\}.&\end{aligned} \end{aligned}$$

(iv):

  ${\forall xy\in {{{\mathbb {S}}}B_{1}}}$ but $xy\notin {{{\mathbb {S}}}B_{2}}.$

$$\begin{aligned} \begin{aligned} T_{{{\overline{\mathbb {S}}}(B_{1})}+ {{\overline{\mathbb {S}}}(B_{2})}}(xy)=&T_{{\overline{\mathbb {S}}}(B_{1})}(xy), T_{{{\underline{\mathbb {S}}}(B_{1})}+ {{\underline{\mathbb {S}}}(B_{2})}}(xy)= T_{{\underline{\mathbb {S}}}(B_{1})}(xy),&\\ I_{{{\overline{\mathbb {S}}}(B_{1})}+ {{\overline{\mathbb {S}}}(B_{2})}}(xy)=&I_{{\overline{\mathbb {S}}}(B_{1})}(xy),\; I_{{{\underline{\mathbb {S}}}(B_{1})}+ {{\underline{\mathbb {S}}}(B_{2})}}(xy)=I_{{\underline{\mathbb {S}}}(B_{1})}(xy),&\\ F_{{{\overline{\mathbb {S}}}(B_{1})}+ {{\overline{\mathbb {S}}}(B_{2})}}(xy)=&F_{{\overline{\mathbb {S}}}(B_{1})}(xy), F_{{{\underline{\mathbb {S}}}(B_{1})}+ {{\underline{\mathbb {S}}}(B_{2})}}(xy)=F_{{\underline{\mathbb {S}}}(B_{1})}(xy).&\end{aligned} \end{aligned}$$

(v):

  ${\forall xy\notin {{{\mathbb {S}}}B_{1}}}$ but $xy\in {{{\mathbb {S}}}B_{2}}$

$$\begin{aligned} \begin{aligned} T_{{{\overline{\mathbb {S}}}(B_{1})}+ {{\overline{\mathbb {S}}}(B_{2})}}(xy)=&T_{{\overline{\mathbb {S}}}(B_{2})}(xy), T_{ {{\underline{\mathbb {S}}}(B_{1})}+ {{\underline{\mathbb {S}}}(B_{2})}}(xy)= T_{{\underline{\mathbb {S}}}(B_{2})}(xy),&\\ I_{{{\overline{\mathbb {S}}}(B_{1})}+ {{\overline{\mathbb {S}}}(B_{2})}}(xy)=&I_{{\overline{\mathbb {S}}}(B_{2})}(xy),\; I_{{{\underline{\mathbb {S}}}(B_{1})}+ {{\underline{\mathbb {S}}}(B_{2})}}(xy)=I_{{\underline{\mathbb {S}}}(B_{2})}(xy),&\\ F_{{{\overline{\mathbb {S}}}(B_{1})}+ {{\overline{\mathbb {S}}}(B_{2})}}(xy)=&F_{{\overline{\mathbb {S}}}(B_{2})}(xy), F_{{{\underline{\mathbb {S}}}(B_{1})}+ {{\underline{\mathbb {S}}}(B_{2})}}(xy)=F_{{\underline{\mathbb {S}}}(B_{2})}(xy).&\end{aligned} \end{aligned}$$

(vi):

  ${\forall xy\in {{{\mathbb {S}}}B_{1}}\cap {{\mathbb {S}}(B_{2})}}$

$$\begin{aligned} \begin{aligned} T_{{{\overline{\mathbb {S}}}(B_{1})}+ {{\overline{\mathbb {S}}}(B_{2})}}(xy)=&\max \{{{T_{\overline{\mathbb {S}}(B_{1})}}}(xy), T_{{\overline{\mathbb {S}}}(B_{2})}(xy)\},&\\ T_{{{\underline{\mathbb {S}}}(B_{1})}+ {{\underline{\mathbb {S}}}(B_{2})}}(xy)=&\max \{T_{\underline{\mathbb {S}}(B_{1})}(xy), T_{{\underline{\mathbb {S}}}(B_{2})}(xy)\},&\\ I_{{{\overline{\mathbb {S}}}(B_{1})}+ {{\overline{\mathbb {S}}}(B_{2})}}(xy)=&\min \{I_{{\overline{\mathbb {S}}}(B_{1})}(xy), I_{{\overline{\mathbb {S}}}(B_{2})}(xy)\},&\\ I_{{{\underline{\mathbb {S}}}(B_{1})}+ {{\underline{\mathbb {S}}}(B_{2})}}(xy)=&\min \{I_{{\underline{\mathbb {S}}}(B_{1})}(xy), I_{{\underline{\mathbb {S}}}(B_{2})}(xy)\},&\\ F_{{{\overline{\mathbb {S}}}(B_{1})}+ {{\overline{\mathbb {S}}}(B_{2})}}(xy)=&\min \{F_{{\overline{\mathbb {S}}}(B_{1})}(xy), F_{{\overline{\mathbb {S}}}(B_{2})}(xy)\},&\\ F_{{{\underline{\mathbb {S}}}(B_{1})}+ {{\underline{\mathbb {S}}}(B_{2})}}(xy)=&\min \{F_{{\underline{\mathbb {S}}}(B_{1})}(xy), F_{{\underline{\mathbb {S}}}(B_{2})}(xy)\}.&\end{aligned} \end{aligned}$$

(vii):

  ${\forall xy\in \tilde{E}}$, where $\tilde{E}$ is the set of edges joining vertices of $\mathbb {R}A_1$ and $\mathbb {R}A_2$.

$$\begin{aligned} \begin{aligned} T_{{{\overline{\mathbb {S}}}(B_{1})}+ {{\overline{\mathbb {S}}}(B_{2})}}(xy)=&\min \{T_{{\overline{\mathbb {R}}}(A_{1})}(x), T_{{\overline{\mathbb {R}}}(A_{2})}(y)\},&\\ T_{{{\underline{\mathbb {S}}}(B_{1})}+ {{\underline{\mathbb {S}}}(B_{2})}}(xy)=&\min \{T_{{\underline{\mathbb {R}}}(A_{1})}(x), T_{{\underline{\mathbb {R}}}(A_{2})}(y)\},&\\ I_{{{\overline{\mathbb {S}}}(B_{1})}`+ {{\overline{\mathbb {S}}}(B_{2})}}(xy)=&\max \{I_{{\overline{\mathbb {R}}}(A_{1})}(x), I_{{\overline{\mathbb {R}}}(A_{2})}(y)\},&\\ I_{{{\underline{\mathbb {S}}}(B_{1})}+ {{\underline{\mathbb {S}}}(B_{2})}}(xy)=&\max \{I_{{\underline{\mathbb {R}}}(A_{1})}(x), I_{{\underline{\mathbb {R}}}(A_{2})}(y)\},&\\ F_{{{\overline{\mathbb {S}}}(B_{1})}+ {{\overline{\mathbb {S}}}(B_{2})}}(xy)=&\max \{F_{{\overline{\mathbb {R}}}(A_{1})}(x), F_{{\overline{\mathbb {R}}}(A_{2})}(y)\},&\\ F_{{{\underline{\mathbb {S}}}(B_{1})}+ {{\underline{\mathbb {S}}}(B_{2})}}(xy)=&\max \{F_{{\underline{\mathbb {R}}}(A_{1})}(x), F_{{\underline{\mathbb {R}}}(A_{2})}(y)\}.&\end{aligned} \end{aligned}$$

Definition 8.7

The Cartesian product of ${G}_{1}$ and ${G}_{2}$ is a ${G} = {G}_{1} \ltimes {G}_{2}= (\underline{{G}}_{1} \ltimes \underline{{G}}_{2}, \overline{{G}}_{1} \ltimes \overline{{G}}_{2}),$ where $\underline{{G}}_{1}\ltimes \underline{{G}}_{2}=(\underline{\mathbb {R}}(A_{1})\ltimes \underline{\mathbb {R}}(A_{2}), \underline{\mathbb {S}}(B_{1})\ltimes \underline{\mathbb {S}}(B_{2}))$ and $\overline{{G}}_{1}\ltimes \overline{{G}}_{2}=(\overline{\mathbb {R}}(A_{1})\ltimes \overline{\mathbb {R}}(A_{2}), \overline{\mathbb {S}}(B_{1})\ltimes \overline{\mathbb {S}}(B_{2}))$ are neutrosophic graphs, such that

• (i)   $\forall ~(x, y)\in {\mathbb {R}}A_{1}\times {\mathbb {R}}A_{2}.$

  $$\begin{aligned} T_{\overline{\mathbb {R}}(A_{1})\ltimes \overline{\mathbb {R}}(A_{2})}(x, y) =&\min \{T_{\overline{\mathbb {R}}(A_{1})}(x), T_{\overline{\mathbb {R}}(A_{2})}(x)\},&\\ T_{\underline{\mathbb {R}}(A_{1})\ltimes \underline{\mathbb {R}}(A_{2})}(x, y) =&\min \{T_{\underline{\mathbb {R}}(A_{1})}(x), T_{\underline{\mathbb {R}}(A_{2})}(x)\},&\\ I_{\overline{\mathbb {R}}(A_{1})\ltimes \overline{\mathbb {R}}(A_{2})}(x, y) =&\max \{I_{\overline{\mathbb {R}}(A_{1})}(x), I_{\overline{\mathbb {R}}(A_{2})}(x)\},&\\ I_{\underline{\mathbb {R}}(A_{1})\ltimes \underline{\mathbb {R}}(A_{2})}(x, y) =&\max \{I_{\underline{\mathbb {R}}(A_{1})}(x), I_{\underline{\mathbb {R}}(A_{2})}(x)\},&\\ F_{\overline{\mathbb {R}}(A_{1})\ltimes \overline{\mathbb {R}}(A_{2})}(x, y) =&\max \{F_{\overline{\mathbb {R}}(A_{1})}(x), F_{\overline{\mathbb {R}}(A_{2})}(x)\},&\\ F_{\underline{\mathbb {R}}(A_{1})\ltimes \underline{\mathbb {R}}(A_{2})}(x, y) =&\max \{F_{\underline{\mathbb {R}}(A_{1})}(x), F_{\underline{\mathbb {R}}(A_{2})}(x_1 )\}.&\end{aligned}$$

• (ii)   $\forall y_1y_2\in {\mathbb {S}}B_2,x\in {\mathbb {R}}A_1$.

  $$\begin{aligned} T_{\overline{\mathbb {S}}(B_{1})\ltimes \overline{\mathbb {S}}(B_{2})}\big ((x, y_1)(x, y_2)\big ) =&\min \{T_{\overline{\mathbb {R}}(A_{1})}(x), T_{\overline{\mathbb {S}}(B_{2})}(y_1y_2)\},&\\ T_{\underline{\mathbb {S}}(B_{1})\ltimes \underline{\mathbb {S}}(B_{2})}\big ((x, y_1)(x, y_2)\big ) =&\min \{T_{\underline{\mathbb {R}}(A_{1})}(x), T_{\underline{\mathbb {S}}(B_{2})}(y_1y_2)\},&\\ I_{\overline{\mathbb {S}}(B_{1})\ltimes \overline{\mathbb {S}}(B_{2})}\big ((x, y_1)(x, y_2)\big ) =&\max \{I_{\overline{\mathbb {R}}(A_{1})}(x), I_{\overline{\mathbb {S}}(B_{2})}(y_1y_2)\},&\\ I_{\underline{\mathbb {S}}(B_{1})\ltimes \underline{\mathbb {S}}(B_{2})}\big ((x, y_1)(x, y_2 )\big ) =&\max \{I_{\underline{\mathbb {R}}(A_{1})}(x), I_{\underline{\mathbb {S}}(B_{2})}(y_1y_2)\},&\\ F_{\overline{\mathbb {S}}(B_{1})\ltimes \overline{\mathbb {S}}(B_{2})}\big ((x, y_1)(x, y_2)\big ) =&\max \{F_{\overline{\mathbb {R}}(A_{1})}(x), F_{\overline{\mathbb {S}}(B_{2})}(y_1y_2)\},&\\ F_{\underline{\mathbb {S}}(B_{1})\ltimes \underline{\mathbb {S}}(B_{2})}\big ((x, y_1)(x, y_2)\big ) =&\max \{F_{\underline{\mathbb {R}}(A_{1})}(x), F_{\underline{\mathbb {S}}(B_{2})}(y_1y_2)\}.&\end{aligned}$$

• (iii)   $\forall x_1x_2\in {\mathbb {S}}B_1,y\in {\mathbb {R}}A_2$.

  $$\begin{aligned} T_{\underline{\mathbb {S}}(B_{1})\ltimes \underline{\mathbb {S}}(B_{2})}\big ((x_1,y)(x_2,y)\big ) =&\min \{T_{\underline{\mathbb {S}}(B_{1})}(x_1x_2), T_{\underline{\mathbb {R}}(A_{2})}(y)\},\\ T_{\overline{\mathbb {S}}(B_{1})\ltimes \overline{\mathbb {S}}(B_{2})}\big ((x_1,y)(x_2,y)\big ) =&\min \{T_{\overline{\mathbb {S}}(B_{1})}(x_1x_2), T_{\overline{\mathbb {R}}(A_{2})}(y)\},\\ I_{\overline{\mathbb {S}}(B_{1})\ltimes \overline{\mathbb {S}}(B_{2})}\big ((x_1,y)(x_2,y)\big ) =&\max \{I_{\overline{\mathbb {S}}(B_{1})}(x_1x_2), I_{\overline{\mathbb {R}}(A_{2})}(y)\},\\ I_{\underline{\mathbb {S}}(B_{1})\ltimes \underline{\mathbb {S}}(B_{2})}\big ((x_1,y)(x_2,y)\big ) =&\max \{I_{\underline{\mathbb {S}}(B_{1})}(x_1x_2), I_{\underline{\mathbb {R}}(A_{2})}(y)\},\\ F_{\overline{\mathbb {S}}(B_{1})\ltimes \overline{\mathbb {S}}(B_{2})}\big ((x_1,y)(x_2,y)\big ) =&\max \{F_{\overline{\mathbb {S}}(B_{1})}(x_1x_2), F_{\overline{\mathbb {R}}(A_{2})}(y)\},\\ F_{\underline{\mathbb {S}}(B_{1})\ltimes \underline{\mathbb {S}}(B_{2})}\big ((x_1,y)(x_2,y)\big ) =&\max \{F_{\underline{\mathbb {S}}(B_{1})}(x_1x_2), F_{\underline{\mathbb {R}}(A_{2})}(y)\}. \end{aligned}$$

Definition 8.8

The cross product of ${G}_{1}$ and ${G}_{2}$ is a neutrosophic soft rough graph $G={G}_{1} \circledcirc {G}_{2} = (\underline{{G}}_{1} \circledcirc \underline{{G}}_{2}, \overline{{G}}_{1} \circledcirc \overline{{G}}_{2}),$ where $\underline{{G}}_{1}\circledcirc \underline{{G}}_{2}=(\underline{\mathbb {R}}(A_{1})\circledcirc \underline{\mathbb {R}}(A_{2}), \underline{\mathbb {S}}(B_{1})\circledcirc \underline{\mathbb {S}}(B_{2}))$ and $\overline{{G}}_{1}\circledcirc \overline{{G}}_{2}=(\overline{\mathbb {R}}(A_{1})\circledcirc \overline{\mathbb {R}}(A_{2}), \overline{\mathbb {S}}(B_{1})\circledcirc \overline{\mathbb {S}}(B_{2}))$ are neutrosophic graphs, respectively, such that

• (i)   $\forall ~(x, y)\in {\mathbb {R}}A_{1}\times {\mathbb {R}}A_{2}$.

  $$\begin{aligned} T_{\overline{\mathbb {R}}(A_{1})\circledcirc \overline{\mathbb {R}}(A_{2})}(x, y) =&\min \{T_{\overline{\mathbb {R}}(A_{1})}(x), T_{\overline{\mathbb {R}}(A_{2})}(x)\},&\\ T_{\underline{\mathbb {R}}(A_{1})\circledcirc \underline{\mathbb {R}}(A_{2})}(x, y) =&\min \{T_{\underline{\mathbb {R}}(A_{1})}(x), T_{\underline{\mathbb {R}}(A_{2})}(x)\},&\\ I_{\overline{\mathbb {R}}(A_{1})\circledcirc \overline{\mathbb {R}}(A_{2})}(x, y) =&\max \{I_{\overline{\mathbb {R}}(A_{1})}(x), I_{\overline{\mathbb {R}}(A_{2})}(x)\},\\\; I_{\underline{\mathbb {R}}(A_{1})\circledcirc \underline{\mathbb {R}}(A_{2})}(x, y) =&\max \{I_{\underline{\mathbb {R}}(A_{1})}(x), I_{\underline{\mathbb {R}}(A_{2})}(x)\},&\\ F_{\overline{\mathbb {R}}(A_{1})\circledcirc \overline{\mathbb {R}}(A_{2})}(x, y) =&\max \{F_{\overline{\mathbb {R}}(A_{1})}(x), F_{\overline{\mathbb {R}}(A_{2})}(x)\},&\\ F_{\underline{\mathbb {R}}(A_{1})\circledcirc \underline{\mathbb {R}}(A_{2})}(x, y) =&\max \{F_{\underline{\mathbb {R}}(A_{1})}(x), F_{\underline{\mathbb {R}}(A_{2})}(x)\}.&\end{aligned}$$

• (ii)   $\forall x_1x_2\in {\mathbb {S}}B_1,y_1y_2\in {\mathbb {S}}B_2$.

  $$\begin{aligned} T_{\overline{\mathbb {S}}(B_{1})\circledcirc \overline{\mathbb {S}}(B_{2})}\big ((x_1,y_1)(x_2,y_2)\big ) =&\min \{T_{\overline{\mathbb {S}}(B_{1})}(x_1x_2), T_{\overline{\mathbb {S}}(B_{2})}(y_1y_2)\},\\ T_{\underline{\mathbb {S}}(B_{1})\circledcirc \underline{\mathbb {S}}(B_{2})}\big ((x_1,y_1)(x_2,y_2)\big )=&\min \{T_{\underline{\mathbb {S}}(B_{1})}(x_1x_2), T_{\underline{\mathbb {S}}(B_{2})}(y_1y_2)\},\\ I_{\overline{\mathbb {S}}(B_{1})\circledcirc \overline{\mathbb {S}}(B_{2})}\big ((x_1,y_1)(x_2,y_2)\big ) =&\max \{I_{\overline{\mathbb {S}}(B_{1})}(x_1x_2), I_{\overline{\mathbb {S}}(B_{2})}(y_1y_2)\},\\ I_{\underline{\mathbb {S}}(B_{1})\circledcirc \underline{\mathbb {S}}(B_{2})}\big ((x_1,y_1)(x_2,y_2)\big )=&\max \{I_{\underline{\mathbb {S}}(B_{1})}(x_1x_2), I_{\underline{\mathbb {S}}(B_{2})}(y_1y_2)\},\\ F_{\overline{\mathbb {S}}(B_{1})\circledcirc \overline{\mathbb {S}}(B_{2})}\big ((x_1,y_1)(x_2,y_2)\big ) =&\max \{F_{\overline{\mathbb {S}}(B_{1})}(x_1x_2), F_{\overline{\mathbb {S}}(B_{2})}(y_1y_2)\},\\ F_{\underline{\mathbb {S}}(B_{1})\circledcirc \underline{\mathbb {S}}(B_{2})}\big ((x_1,y_1)(x_2,y_2)\big ) =&\max \{F_{\underline{\mathbb {S}}(B_{1})}(x_1x_2), F_{\underline{\mathbb {S}}(B_{2})}(y_1y_2)\}. \end{aligned}$$

Definition 8.9

The rejection of $G_{1}$ and $G_{2}$ is a neutrosophic soft rough graph $G =G_{1} | G_{2} =(\underline{G}_{1} | \underline{G}_{2}, \overline{G}_{1} | \overline{G}_{2}),$ where $\underline{G}_{1}|\underline{G}_{2}=(\underline{\mathbb {S}}A_{1}|\underline{\mathbb {S}}A_{2}, \underline{\mathbb {S}}(B_{1})|\underline{\mathbb {S}}(B_{2}))$ and $\overline{G}_{1}|\overline{G}_{2}=(\overline{\mathbb {S}}A_{1}|\overline{\mathbb {S}}A_{2}, \overline{\mathbb {S}}(B_{1})|\overline{\mathbb {S}}(B_{2}))$ are neutrosophic graphs such that

• (i)   $\forall ~(x, y)\in {\mathbb {R}}A_{1}\times {\mathbb {R}}A_{2}$.

  $$\begin{aligned} T_{\overline{\mathbb {R}}(A_{1})|\overline{\mathbb {R}}(A_{2})}(x, y) =&\min \{T_{\overline{\mathbb {R}}(A_{1})}(x), T_{\overline{\mathbb {R}}(A_{2})}(y)\},&\\ T_{\underline{\mathbb {R}}(A_{1})|\underline{\mathbb {R}}(A_{2})}(x, y) =&\min \{T_{\underline{\mathbb {R}}(A_{1})}(x), T_{\underline{\mathbb {R}}(A_{2})}(y)\},&\\ I_{\overline{\mathbb {R}}(A_{1})|\overline{\mathbb {R}}(A_{2})}(x, y) =&\max \{I_{\overline{\mathbb {R}}(A_{1})}(x), I_{\overline{\mathbb {R}}(A_{2})}(y)\},\\ I_{\underline{\mathbb {R}}(A_{1})|\underline{\mathbb {R}}(A_{2})}(x, y) =&\max \{I_{\underline{\mathbb {R}}(A_{1})}(x), I_{\underline{\mathbb {R}}(A_{2})}(y)\},&\\ F_{\overline{\mathbb {R}}(A_{1})|\overline{\mathbb {R}}(A_{2})}(x, y) =&\max \{F_{\overline{\mathbb {R}}(A_{1})}(x), F_{\overline{\mathbb {R}}(A_{2})}(y)\},&\\ F_{\underline{\mathbb {R}}(A_{1})|\underline{\mathbb {R}}(A_{2})}(x, y) =&\max \{F_{\underline{\mathbb {R}}(A_{1})}(x), F_{\underline{\mathbb {R}}(A_{2})}(y)\}.&\end{aligned}$$

• (ii)   $\forall y_1y_2\notin {\mathbb {S}}B_2,\;x\in {\mathbb {R}}A_1$.

  $$\begin{aligned} T_{\overline{\mathbb {S}}(B_{1})|\overline{\mathbb {S}}(B_{2})}\big ((x, y_1)(x, y_2)\big )=&\min \{T_{\overline{\mathbb {R}}(A_{1})}(x), T_{\overline{\mathbb {R}}(A_{2})}(y_1), T_{\overline{\mathbb {R}}(A_{2})}(y_2)\},\\ T_{\underline{\mathbb {S}}(B_{1})|\underline{\mathbb {R}}B_{2})}\big ((x, y_1)(x, y_2)\big )=&\min \{T_{\underline{\mathbb {R}}(A_{1})}(x), T_{\underline{\mathbb {R}}(A_{2})}(y_1), T_{\underline{\mathbb {R}}(A_{2})}(y_2)\},\\ (I_{\overline{\mathbb {S}}(B_{1})|\overline{\mathbb {S}}(B_{2})}\big ((x, y_1)(x, y_2)\big )=&\max \{I_{\overline{\mathbb {R}}(A_{1})}(x), I_{\overline{\mathbb {R}}(A_{2})}(y_1), I_{\overline{\mathbb {R}}(A_{2})}(y_2)\},\\ (I_{\underline{\mathbb {S}}(B_{1})|\underline{\mathbb {S}}(B_{2})}\big ((x, y_1)(x, y_2)\big )=&\max \{I_{\underline{\mathbb {R}}(A_{1})}(x), I_{\underline{\mathbb {R}}(A_{2})}(y_1), I_{\underline{\mathbb {R}}(A_{2})}(y_2)\},\\ (F_{\overline{\mathbb {S}}(B_{1})|\overline{\mathbb {S}}(B_{2})}\big ((x, y_1)(x, y_2)\big )=&\max \{F_{\overline{\mathbb {R}}(A_{1})}(x), F_{\overline{\mathbb {R}}(A_{2})}(y_1), F_{\overline{\mathbb {R}}(A_{2})}(y_2)\},\\ (F_{\underline{\mathbb {S}}(B_{1})|\underline{\mathbb {S}}(B_{2})}\big ((x, y_1)(x, y_2)\big )=&\max \{F_{\underline{\mathbb {R}}(A_{1})}(x), F_{\underline{\mathbb {R}}(A_{2})}(y_1), F_{\underline{\mathbb {R}}(A_{2})}(y_2)\}. \end{aligned}$$

• (iii)   $\forall x_1x_2\notin {\mathbb {S}}B_1,\;y\in {\mathbb {R}}A_2$,

  $$\begin{aligned} T_{\underline{\mathbb {S}}(B_{1})|\underline{\mathbb {S}}(B_{2})}\big ((x_1,y)(x_2,y)\big )=&\min \{T_{\underline{\mathbb {R}}(A_{1})}(x_1), T_{\underline{\mathbb {R}}(A)_1}(x_2), T_{\underline{\mathbb {R}}(A_{2})}(y)\},\\ I_{\underline{\mathbb {S}}(B_{1})|\underline{\mathbb {S}}(B_{2})}\big ((x_1,y)(x_2,y)\big )=&\max \{I_{\underline{\mathbb {R}}(A_{1})}(x_1), I_{\underline{\mathbb {R}}(A)_1}(x_2), I_{\underline{\mathbb {R}}(A_{2})}(y)\},\\ F_{\underline{\mathbb {S}}(B_{1})|\underline{\mathbb {S}}(B_{2})}\big ((x_1,y)(x_2,y)\big )=&\max \{F_{\underline{\mathbb {R}}(A_{1})}(x_1), F_{\underline{\mathbb {R}}(A)_1}(x_2), F_{\underline{\mathbb {R}}(A_{2})}(y)\},\\ T_{\overline{\mathbb {S}}(B_{1})|\overline{\mathbb {S}}(B_{2})}\big ((x_1,y)(x_2,y)\big ) =&\min \{T_{\overline{\mathbb {R}}(A_{1})}(x_1), T_{\overline{\mathbb {R}}(A)_1}(x_2), T_{\overline{\mathbb {R}}(A_{2})}(y)\},\\ I_{\overline{\mathbb {S}}(B_{1})|\overline{\mathbb {S}}(B_{2})}\big ((x_1,y)(x_2,y)\big ) =&\max \{I_{\overline{\mathbb {R}}(A_{1})}(x_1), I_{\overline{\mathbb {R}}(A)_1}(x_2), I_{\overline{\mathbb {R}}(A_{2})}(y)\},\\ F_{\overline{\mathbb {S}}(B_{1})|\overline{\mathbb {S}}(B_{2})}\big ((x_1,y)(x_2,y)\big ) =&\max \{F_{\overline{\mathbb {R}}(A_{1})}(x_1), F_{\overline{\mathbb {R}}(A)_1}(x_2), F_{\overline{\mathbb {R}}(A_{2})}(y)\}. \end{aligned}$$

• (iv)   $\forall x_1x_2\notin \mathbb {S}B_1,\;y_1y_2\notin \mathbb {S}B_2,\;x_1\ne x_2, y_1\ne y_2 .$

  $$\begin{aligned} T_{\underline{\mathbb {S}}(B_{1})|\underline{\mathbb {S}}(B_{2})}\big ((x_1,y_1)(x_2,y_2)\big )=&\min \{T_{\underline{\mathbb {R}}(A_{1})}(x_1), T_{\underline{\mathbb {R}}(A)_1}(x_2), T_{\underline{\mathbb {R}}(A_{2})}(y_1), T_{\underline{\mathbb {R}}(A_{2})}(y_2)\},\\ I_{\underline{\mathbb {S}}(B_{1})|\underline{\mathbb {S}}(B_{2})}\big ((x_1,y_1)(x_2,y_2)\big )=&\max \{I_{\underline{\mathbb {R}}(A_{1})}(x_1), I_{\underline{\mathbb {R}}(A)_1}(x_2), I_{\underline{\mathbb {R}}(A_{2})}(y_1), I_{\underline{\mathbb {R}}(A_{2})}(y_2)\},\\ F_{\underline{\mathbb {S}}(B_{1})|\underline{\mathbb {S}}(B_{2})}\big ((x_1,y_1)(x_2,y_2)\big )=&\max \{F_{\underline{\mathbb {R}}(A_{1})}(x_1), F_{\underline{\mathbb {R}}(A)_1}(x_2), F_{\underline{\mathbb {R}}(A_{2})}(y_1), F_{\underline{\mathbb {R}}(A_{2})}(y_2)\},\\ T_{\overline{\mathbb {S}}(B_{1})|\overline{\mathbb {S}}(B_{2})}\big ((x_1,y_1)(x_2,y_2)\big )=&\min \{T_{\overline{\mathbb {R}}(A_{1})}(x_1), T_{\overline{\mathbb {R}}(A)_1}(x_2), T_{\overline{\mathbb {R}}(A_{2})}(y_1), T_{\overline{\mathbb {R}}(A_{2})}(y_2)\},\\ I_{\overline{\mathbb {S}}(B_{1})|\overline{\mathbb {S}}(B_{2})}\big ((x_1,y_1)(x_2,y_2)\big )=&\max \{I_{\overline{\mathbb {R}}(A_{1})}(x_1), I_{\overline{\mathbb {R}}(A)_1}(x_2), I_{\overline{\mathbb {R}}(A_{2})}(y_1), I_{\overline{\mathbb {R}}(A_{2})}(y_2)\},\\ F_{\overline{\mathbb {S}}(B_{1})|\overline{\mathbb {S}}(B_{2})}\big ((x_1,y_1)(x_2,y_2)\big )=&\max \{F_{\overline{\mathbb {R}}(A_{1})}(x_1), F_{\overline{\mathbb {R}}(A)_1}(x_2), F_{\overline{\mathbb {R}}(A_{2})}(y_1), F_{\overline{\mathbb {R}}(A_{2})}(y_2)\},\\ \end{aligned}$$

Example 8.5

Let ${G}_{1}=(\underline{{G}}_{1},\overline{{G}}_{1})$ and ${G}_{2}=(\underline{{G}}_{2},\overline{{G}}_{2})$ be two neutrosophic soft rough graphs on X, where $\underline{{G}}_{1}=(\underline{\mathbb {R}}(A_{1}), \underline{\mathbb {S}}(B_{1}))$ and $\overline{{G}}_{1}=(\overline{\mathbb {R}}(A_{1}), \overline{\mathbb {S}}(B_{1}))$ are neutrosophic graphs as shown in Fig. 8.2 and $\underline{{G}}_{2}=(\underline{\mathbb {R}}(A_{2}), \underline{\mathbb {S}}(B_{2}))$ and $\overline{{G}}_{2}=(\overline{\mathbb {R}}(A_{2}), \overline{\mathbb {S}}(B_{2}))$ are neutrosophic graphs as shown in Fig. 8.3. The Cartesian product of $G_1=(\underline{G}_1,\overline{G}_1)$ and $G_2=(\underline{G}_2,\overline{G}_2)$ is neutrosophic soft rough graph $G=G_1 \times G_2=(\underline{G}_1 \times \underline{G}_2, \overline{G}_1\times \overline{G}_2)$ as shown in Fig. 8.5.

Fig. 8.5

Cartesian product of two neutrosophic soft rough graphs $G_1\times G_2$

Definition 8.10

The symmetric difference of ${G}_{1}$ and ${G}_{2}$ is a neutrosophic soft rough graph ${G}={G}_{1} \oplus {G}_{2} =(\underline{{G}}_{1} \oplus \underline{{G}}_{2}, \overline{{G}}_{1} \oplus \overline{{G}}_{2}),$ where $\underline{{G}}_{1}\oplus \underline{{G}}_{2}=(\underline{\mathbb {R}}(A_{1})\oplus \underline{\mathbb {R}}(A_{2}), \underline{\mathbb {S}}(B_{1})\oplus \underline{\mathbb {S}}(B_{2}))$ and $\overline{{G}}_{1}\oplus \overline{{G}}_{2}=(\overline{\mathbb {R}}(A_{1})\oplus \overline{\mathbb {R}}(A_{2}), \overline{\mathbb {S}}(B_{1})\oplus \overline{\mathbb {S}}(B_{2}))$ are neutrosophic graphs, respectively, such that

• (i)   $\forall ~(x, y)\in {\mathbb {R}}A_{1}\times {\mathbb {R}}A_{2}$.

  $$\begin{aligned} T_{\overline{\mathbb {R}}(A_{1})\oplus \overline{\mathbb {R}}(A_{2})}(x, y) =&\min \{T_{\overline{\mathbb {R}}(A_{1})}(x), T_{\overline{\mathbb {R}}(A_{2})}(y)\},&\\ T_{\underline{\mathbb {R}}(A_{1})\oplus \underline{\mathbb {R}}(A_{2})}(x, y) =&\min \{T_{\underline{\mathbb {R}}(A_{1})}(x), T_{\underline{\mathbb {R}}(A_{2})}(y)\},&\\ I_{\overline{\mathbb {R}}(A_{1})\oplus \overline{\mathbb {R}}(A_{2})}(x, y) =&\max \{I_{\overline{\mathbb {R}}(A_{1})}(x), I_{\overline{\mathbb {R}}(A_{2})}(y)\},&\\ I_{\underline{\mathbb {R}}(A_{1})\oplus \underline{\mathbb {R}}(A_{2})}(x, y) =&\max \{I_{\underline{\mathbb {R}}(A_{1})}(x), I_{\underline{\mathbb {R}}(A_{2})}(y)\},&\\ F_{\overline{\mathbb {R}}(A_{1})\oplus \overline{\mathbb {R}}(A_{2})}(x, y) =&\max \{F_{\overline{\mathbb {R}}(A_{1})}(x), F_{\overline{\mathbb {R}}(A_{2})}(y)\},&\\ F_{\underline{\mathbb {R}}(A_{1})\oplus \underline{\mathbb {R}}(A_{2})}(x, y) =&\max \{F_{\underline{\mathbb {R}}(A_{1})}(x), F_{\underline{\mathbb {R}}(A_{2})}(y)\}.&\end{aligned}$$

• (ii)   $\forall y_1y_2\in {\mathbb {S}}B_2,\;x\in {\mathbb {R}}A_1$.

  $$\begin{aligned} T_{\overline{\mathbb {S}}(B_{1})\oplus \overline{\mathbb {S}}(B_{2})}\big ((x, y_1)(x, y_2)\big ) =&\min \{T_{\overline{\mathbb {R}}(A_{1})}(x), T_{\overline{\mathbb {S}}(B_{2})}(y_1y_2)\},&\\ T_{\underline{\mathbb {S}}(B_{1})\oplus \underline{\mathbb {S}}(B_{2})}\big ((x, y_1)(x, y_2)\big ) =&\min \{T_{\underline{\mathbb {R}}(A_{1})}(x), T_{\underline{\mathbb {S}}(B_{2})}(y_1y_2)\},&\\ I_{\overline{\mathbb {S}}(B_{1})\oplus \overline{\mathbb {S}}(B_{2})}\big ((x, y_1)(x, y_2)\big ) =&\max \{I_{\overline{\mathbb {R}}(A_{1})}(x), I_{\overline{\mathbb {S}}(B_{2})}(y_1y_2)\},&\\ I_{\underline{\mathbb {S}}(B_{1})\oplus \underline{\mathbb {S}}(B_{2})}\big ((x, y_1)(x, y_2)\big ) =&\max \{I_{\underline{\mathbb {R}}(A_{1})}(x), I_{\underline{\mathbb {S}}(B_{2})}(y_1y_2)\},&\\ F_{\overline{\mathbb {S}}(B_{1})\oplus \overline{\mathbb {S}}(B_{2})}\big ((x, y_1)(x, y_2)\big ) =&\max \{F_{\overline{\mathbb {R}}(A_{1})}(x), F_{\overline{\mathbb {S}}(B_{2})}(y_1y_2)\},&\\ F_{\underline{\mathbb {S}}(B_{1})\oplus \underline{\mathbb {S}}(B_{2})}\big ((x, y_1)(x, y_2)\big ) =&\max \{F_{\underline{\mathbb {R}}(A_{1})}(x), F_{\underline{\mathbb {S}}(B_{2})}(y_1y_2)\}. \end{aligned}$$

• (iii)   $\forall x_1x_2\in {\mathbb {S}}B_1,y\in {\mathbb {R}}A_2$.

  $$\begin{aligned} T_{\overline{\mathbb {S}}(B_{1})\oplus \overline{\mathbb {S}}(B_{2})}\big ((x_1,y)(x_2,y)\big ) =&\min \{T_{\overline{\mathbb {S}}(B_{1})}(x_1x_2), T_{\overline{\mathbb {R}}(A_{2})}(y)\},\\ T_{\underline{\mathbb {S}}(B_{1})\oplus \underline{\mathbb {S}}(B_{2})}\big ((x_1,y)(x_2,y)\big ) =&\min \{T_{\underline{\mathbb {S}}(B_{1})}(x_1x_2), T_{\underline{\mathbb {R}}(A_{2})}(y)\},\\ I_{\overline{\mathbb {S}}(B_{1})\oplus \overline{\mathbb {S}}(B_{2})}\big ((x_1,y)(x_2,y)\big ) =&\max \{I_{\overline{\mathbb {S}}(B_{1})}(x_1x_2), I_{\overline{\mathbb {R}}(A_{2})}(y)\},\\ I_{\underline{\mathbb {S}}(B_{1})\oplus \underline{\mathbb {S}}(B_{2})}\big ((x_1,y)(x_2,y)\big ) =&\max \{I_{\underline{\mathbb {S}}(B_{1})}(x_1x_2), I_{\underline{\mathbb {R}}(A_{2})}(y)\},\\ F_{\overline{\mathbb {S}}(B_{1})\oplus \overline{\mathbb {S}}(B_{2})}\big ((x_1,y)(x_2,y)\big ) =&\max \{F_{\overline{\mathbb {S}}(B_{1})}(x_1x_2), F_{\overline{\mathbb {R}}(A_{2})}(y)\},\\ F_{\underline{\mathbb {S}}(B_{1})\oplus \underline{\mathbb {S}}(B_{2})}\big ((x_1,y)(x_2,y)\big ) =&\max \{F_{\underline{\mathbb {S}}(B_{1})}(x_1x_2), F_{\underline{\mathbb {R}}(A_{2})}(y)\}. \end{aligned}$$

• (iv)   $\forall x_1x_2\in {\mathbb {S}}B_1,\;y_1y_2\notin {\mathbb {S}}B_2$.

  $$\begin{aligned} T_{\overline{\mathbb {S}}(B_{1})\oplus \overline{\mathbb {S}}(B_{2})}\big ((x_1,y_{1})(x_2,y_2)\big )=&\min \{T_{\overline{\mathbb {S}}(B_{1})}(x_1x_2), T_{\overline{\mathbb {R}}(A_{2})}(y_1), T_{\overline{\mathbb {R}}(A_{2})}(y_2)\},\\ T_{\underline{\mathbb {S}}(B_{1})\oplus \underline{\mathbb {S}}(B_{2})}\big ((x_1,y_{1})(x_2,y_2)\big )=&\min \{T_{\underline{\mathbb {S}}(B_{1})}(x_1x_2), T_{\underline{\mathbb {R}}(A_{2})}(y_1), T_{\underline{\mathbb {R}}(A_{2})}(y_2)\},\\ I_{\overline{\mathbb {S}}(B_{1})\oplus \overline{\mathbb {S}}(B_{2})}\big ((x_1,y_{1})(x_2,y_2)\big )=&\max \{I_{\overline{\mathbb {S}}(B_{1})}(x_1x_2), I_{\overline{\mathbb {R}}(A_{2})}(y_1), I_{\overline{\mathbb {R}}(A_{2})}(y_2)\},\\ I_{\underline{\mathbb {S}}(B_{1})\oplus \underline{\mathbb {S}}(B_{2})}\big ((x_1,y_{1})(x_2,y_2)\big )=&\max \{I_{\underline{\mathbb {S}}(B_{1})}(x_1x_2), I_{\underline{\mathbb {R}}(A_{2})}(y_1), I_{\underline{\mathbb {R}}(A_{2})}(y_2)\},\\ F_{\overline{\mathbb {S}}(B_{1})\oplus \overline{\mathbb {S}}(B_{2})}\big ((x_1,y_{1})(x_2,y_2)\big )=&\max \{F_{\overline{\mathbb {S}}(B_{1})}(x_1x_2), F_{\overline{\mathbb {R}}(A_{2})}(y_1), F_{\overline{\mathbb {R}}(A_{2})}(y_2)\},\\ F_{\underline{\mathbb {S}}(B_{1})\oplus \underline{\mathbb {S}}(B_{2})}\big ((x_1,y_{1})(x_2,y_2)\big )=&\max \{F_{\underline{\mathbb {S}}(B_{1})}(x_1x_2), F_{\underline{\mathbb {R}}(A_{2})}(y_1), F_{\underline{\mathbb {R}}(A_{2})}(y_2)\}. \end{aligned}$$

• (v)  $\forall x_1x_2\notin {\mathbb {S}}B_1,\;y_1y_2\in {\mathbb {S}}B_2$.

  $$\begin{aligned} T_{\overline{\mathbb {S}}(B_{1})\oplus \overline{\mathbb {S}}(B_{2})}\big ((x_1,y_{1})(x_2,y_2)\big )=&\min \{T_{\overline{\mathbb {R}}(A_{1})}(x_1), T_{\overline{\mathbb {R}}(A_{1})}(x_2), T_{\overline{\mathbb {S}}(B_{2})}(y_1y_2)\},\\ T_{\underline{\mathbb {S}}(B_{1})\oplus \underline{\mathbb {S}}(B_{2})}\big ((x_1,y_{1})(x_2,y_2)\big )=&\min \{T_{\underline{\mathbb {R}}(A_{1})}(x_1), T_{\underline{\mathbb {R}}(A_{1})}(x_2), T_{\underline{\mathbb {S}}(B_{2})}(y_1y_2)\},\\ I_{\overline{\mathbb {S}}(B_{1})\oplus \overline{\mathbb {S}}(B_{2})}\big ((x_1,y_{1})(x_2,y_2)\big )=&\max \{I_{\overline{\mathbb {R}}(A_{1})}(x_1), I_{\overline{\mathbb {R}}(A_{1})}(x_2), I_{\overline{\mathbb {S}}(B_{2})}(y_1y_2)\},\\ I_{\underline{\mathbb {S}}(B_{1})\oplus \underline{\mathbb {S}}(B_{2})}\big ((x_1,y_{1})(x_2,y_2)\big )=&\max \{I_{\underline{\mathbb {R}}(A_{1})}(x_1), I_{\underline{\mathbb {R}}(A_{1})}(x_2), I_{\underline{\mathbb {S}}(B_{2})}(y_1y_2)\},\\ F_{\overline{\mathbb {S}}(B_{1})\oplus \overline{\mathbb {S}}(B_{2})}\big ((x_1,y_{1})(x_2,y_2)\big )=&\max \{F_{\overline{\mathbb {R}}(A_{1})}(x_1), F_{\overline{\mathbb {R}}(A_{1})}(x_2), F_{\overline{\mathbb {S}}(B_{2})}(y_1y_2)\},\\ F_{\underline{\mathbb {S}}(B_{1})\oplus \underline{\mathbb {S}}(B_{2})}\big ((x_1,y_{1})(x_2,y_2)\big )=&\max \{F_{\underline{\mathbb {R}}(A_{1})}(x_1), F_{\underline{\mathbb {R}}(A_{1})}(x_2), F_{\underline{\mathbb {S}}(B_{2})}(y_1y_2)\}. \end{aligned}$$

Example 8.6

Let ${G}_{1}=(\underline{{G}}_{1},\overline{{G}}_{1})$ and ${G}_{2}=(\underline{{G}}_{2},\overline{{G}}_{2})$ be two neutrosophic soft rough graphs on X, where $\underline{{G}}_{1}=(\underline{\mathbb {R}}(A_{1}), \underline{\mathbb {S}}(B_{1}))$ and $\overline{{G}}_{1}=(\overline{\mathbb {R}}(A_{1}), \overline{\mathbb {S}}(B_{1}))$ are neutrosophic graphs as shown in Fig. 8.6 and $\underline{{G}}_{2}=(\underline{\mathbb {R}}(A_{2}), \underline{\mathbb {S}}(B_{2}))$ and $\overline{{G}}_{2}=(\overline{\mathbb {R}}(A_{2}), \overline{\mathbb {S}}(B_{2}))$ are neutrosophic graphs as shown in Fig. 8.7

Fig. 8.6

Neutrosophic soft rough graph ${G}_1 =( \underline{{G}}_1, \overline{{G}}_1)$

Fig. 8.7

Neutrosophic soft rough graph ${G}_2 =( \underline{{G}}_2, \overline{{G}}_2)$

The symmetric difference of ${G}_{1}$ and ${G}_{2}$ is ${G}={G}_{1}\oplus {G}_{2}=(\underline{{G}}_{1}\oplus \underline{{G}}_{2}, \overline{{G}}_{1}\oplus \overline{{G}}_{2})$, where $\underline{{G}}_{1}\oplus \underline{{G}}_{2}=(\underline{\mathbb {R}}(A_{1})\oplus \underline{\mathbb {R}}(A_{2}), \underline{\mathbb {S}}(B_{1})\oplus \underline{\mathbb {S}}(B_{2}))$ and $\overline{{G}}_{1}\oplus \overline{{G}}_{2}=(\overline{\mathbb {R}}(A_{1})\oplus \overline{\mathbb {R}}(A_{2}), \overline{\mathbb {S}}(B_{1})\oplus \overline{\mathbb {S}}(B_{2}))$ are neutrosophic graphs as shown in Fig. 8.8.

Fig. 8.8

Neutrosophic soft rough graph ${G}_1\oplus {G}_2 =( \underline{{G}}_1\oplus {{G}}_2, \overline{{G}}_1\oplus \overline{{G}}_2)$

Definition 8.11

The lexicographic product of ${G}_{1}$ and ${G}_{2}$ is a neutrosophic soft rough graph $G={G}_{1} \odot {G}_{2}{=}({{G}_{1}}_*\odot {{G}_{2}}_*, {{G}}^*_{1} \odot {{G}}^*_{2}),$ where ${{G}_{1}}_*\odot {{G}_{2}}_*{=}( \underline{\mathbb {R}} A_{1}\odot \underline{\mathbb {R}} A_{2}, \underline{\mathbb {S}} B_{1}\odot \underline{\mathbb {S}} B_{2})$ and ${{G}}^*_{1}\odot {{G}}^*_{2}=( \overline{\mathbb {R}} A_{1}\odot \overline{\mathbb {R}} A_{2}, \overline{\mathbb {S}} B_{1}\odot \overline{\mathbb {S}} B_{2})$ are neutrosophic graphs, respectively, such that

• (i)   $\forall ~(x, y)\in {\mathbb {R}}A_{1}\times {\mathbb {R}}A_{2}$.

  $$\begin{aligned} T_{\overline{\mathbb {R}}(A_{1})\odot \overline{\mathbb {R}}(A_{2})}(x, y) =&\min \{T_{\overline{\mathbb {R}}(A_{1})}(x), T_{\overline{\mathbb {R}}(A_{2})}(y)\},&\\ T_{\underline{\mathbb {R}}(A_{1})\odot \underline{\mathbb {R}}(A_{2})}(x, y) =&\min \{T_{\underline{\mathbb {R}}(A_{1})}(x), T_{\underline{\mathbb {R}}(A_{2})}(y)\},&\\ I_{\overline{\mathbb {R}}(A_{1})\odot \overline{\mathbb {R}}(A_{2})}(x, y) =&\max \{I_{\overline{\mathbb {R}}(A_{1})}(x), I_{\overline{\mathbb {R}}(A_{2})}(y)\},&\\ I_{\underline{\mathbb {R}}(A_{1})\odot \underline{\mathbb {R}}(A_{2})}(x, y) =&\max \{I_{\underline{\mathbb {R}}(A_{1})}(x), I_{\underline{\mathbb {R}}(A_{2})}(y)\},&\\ F_{\overline{\mathbb {R}}(A_{1})\odot \overline{\mathbb {R}}(A_{2})}(x, y) =&\max \{F_{\overline{\mathbb {R}}(A_{1})}(x), F_{\overline{\mathbb {R}}(A_{2})}(y)\},&\\ F_{\underline{\mathbb {R}}(A_{1})\odot \underline{\mathbb {R}}(A_{2})}(x, y) =&\max \{F_{\underline{\mathbb {R}}(A_{1})}(x), F_{\underline{\mathbb {R}}(A_{2})}(y)\}.&\end{aligned}$$

• (ii)   $\forall y_1y_2\in {\mathbb {S}}B_2,\;x\in {\mathbb {R}}A_1$.

  $$\begin{aligned} T_{\overline{\mathbb {S}}(B_{1})\odot \overline{\mathbb {S}}(B_{2})}\big ((x, y_1)(x, y_2)\big )=&\min \{T_{\overline{\mathbb {R}}(A_{1})}(x), T_{\overline{\mathbb {S}}(B_{2})}(y_1y_2)\},\\ T_{\underline{\mathbb {S}}(B_{1})\odot \underline{\mathbb {S}}(B_{2})}\big ((x, y_1)(x, y_2)\big )=&\min \{T_{\underline{\mathbb {R}}(A_{1})}(x), T_{\underline{\mathbb {S}}(B_{2})}(y_1y_2)\},\\ I_{\overline{\mathbb {S}}(B_{1})\odot \overline{\mathbb {S}}(B_{2})}\big ((x, y_1)(x, y_2)\big )=&\max \{I_{\overline{\mathbb {R}}(A_{1})}(x), I_{\overline{\mathbb {S}}(B_{2})}(y_1y_2)\},\\ I_{\underline{\mathbb {S}}(B_{1})\odot \underline{\mathbb {S}}(B_{2})}\big ((x, y_1)(x, y_2)\big )=&\max \{I_{\underline{\mathbb {R}}(A_{1})}(x), I_{\underline{\mathbb {S}}(B_{2})}(y_1y_2)\},\\ F_{\overline{\mathbb {S}}(B_{1})\odot \overline{\mathbb {S}}(B_{2})}\big ((x, y_1)(x, y_2)\big )=&\max \{F_{\overline{\mathbb {R}}(A_{1})}(x), F_{\overline{\mathbb {S}}(B_{2})}(y_1y_2)\},\\ F_{\underline{\mathbb {S}}(B_{1})\odot \underline{\mathbb {S}}(B_{2})}\big ((x, y_1)(x, y_2)\big )=&\max \{F_{\underline{\mathbb {R}}(A_{1})}(x), F_{\underline{\mathbb {S}}(B_{2})}(y_1y_2)\}. \end{aligned}$$

• (iii)   $\forall x_1x_2\in {\mathbb {S}}B_1,y_1y_2\in {\mathbb {S}}B_2$.

  $$\begin{aligned} T_{\overline{\mathbb {S}}(B_{1})\odot \overline{\mathbb {S}}(B_{2})}\big ((x_1,y_1)(x_2,y_2)\big )=&\min \{T_{\overline{\mathbb {S}}(B_{1})}(x_1x_2), T_{\overline{\mathbb {S}}(B_{2})}(y_1y_2)\},\\ T_{\underline{\mathbb {S}}(B_{1})\odot \underline{\mathbb {S}}(B_{2})}\big ((x_1,y_1)(x_2,y_2)\big )=&\min \{T_{\underline{\mathbb {S}}(B_{1})}(x_1x_2), T_{\underline{\mathbb {S}}(B_{2})}(y_1y_2)\},\\ I_{\overline{\mathbb {S}}(B_{1})\odot \overline{\mathbb {S}}(B_{2})}\big ((x_1,y_1)(x_2,y_2)\big )=&\max \{I_{\overline{\mathbb {S}}(B_{1})}(x_1x_2), I_{\overline{\mathbb {S}}(B_{2})}(y_1y_2)\},\\ I_{\underline{\mathbb {S}}(B_{1})\odot \underline{\mathbb {S}}(B_{2})}\big ((x_1,y_1)(x_2,y_2)\big )=&\max \{I_{\underline{\mathbb {S}}(B_{1})}(x_1x_2), I_{\underline{\mathbb {S}}(B_{2})}(y_1y_2)\},\\ F_{\overline{\mathbb {S}}(B_{1})\odot \overline{\mathbb {S}}(B_{2})}\big ((x_1,y_1)(x_2,y_2)\big )=&\max \{F_{\overline{\mathbb {S}}(B_{1})}(x_1x_2), F_{\overline{\mathbb {S}}(B_{2})}(y_1y_2)\},\\ F_{\underline{\mathbb {S}}(B_{1})\odot \underline{\mathbb {S}}(B_{2})}\big ((x_1,y_1)(x_2,y_2)\big )=&\max \{F_{\underline{\mathbb {S}}(B_{1})}(x_1x_2), F_{\underline{\mathbb {S}}(B_{2})}(y_1y_2)\}. \end{aligned}$$

Definition 8.12

The strong product of ${G}_{1}$ and ${G}_{2}$ is a neutrosophic soft rough graph ${G}={G}_{1} \otimes {G}_{2} =({{G}_{1}}_*\otimes {{G}_{2}}_*, {{G}}^*_{1} \otimes {{G}}^*_{2}),$ where ${{G}_{1}}_*\otimes {{G}_{2}}_*=( \underline{\mathbb {R}} A_{1}\otimes \underline{\mathbb {R}} A_{2}, \underline{\mathbb {S}} B_{1}\otimes \underline{\mathbb {S}} B_{2})$ and ${{G}}^*_{1}\otimes {{G}}^*_{2}=( \overline{\mathbb {R}} A_{1}\otimes \overline{\mathbb {R}} A_{2}, \overline{\mathbb {S}} B_{1}\otimes \overline{\mathbb {S}} B_{2})$ are neutrosophic graphs, respectively, such that

• (i)   $\forall ~(x, y)\in {\mathbb {R}}A_{1}\times {\mathbb {R}}A_{2}$.

  $$\begin{aligned} T_{\overline{\mathbb {R}}(A_{1})\otimes \overline{\mathbb {R}}(A_{2})}(x, y) =&\min \{T_{\overline{\mathbb {R}}(A_{1})}(x), T_{\overline{\mathbb {R}}(A_{2})}(y)\},&\\ T_{\underline{\mathbb {R}}(A_{1})\otimes \underline{\mathbb {R}}(A_{2})}(x, y) =&\min \{T_{\underline{\mathbb {R}}(A_{1})}(x), T_{\underline{\mathbb {R}}(A_{2})}(y)\},&\\ I_{\overline{\mathbb {R}}(A_{1})\otimes \overline{\mathbb {R}}(A_{2})}(x, y) =&\max \{I_{\overline{\mathbb {R}}(A_{1})}(x), I_{\overline{\mathbb {R}}(A_{2})}(y)\},&\\ I_{\underline{\mathbb {R}}(A_{1})\otimes \underline{\mathbb {R}}(A_{2})}(x, y) =&\max \{I_{\underline{\mathbb {R}}(A_{1})}(x), I_{\underline{\mathbb {R}}(A_{2})}(y)\},&\\ F_{\overline{\mathbb {R}}(A_{1})\otimes \overline{\mathbb {R}}(A_{2})}(x, y) =&\max \{F_{\overline{\mathbb {R}}(A_{1})}(x), F_{\overline{\mathbb {R}}(A_{2})}(y)\},&\\ F_{\underline{\mathbb {R}}(A_{1})\otimes \underline{\mathbb {R}}(A_{2})}(x, y) =&\max \{F_{\underline{\mathbb {R}}(A_{1})}(x), F_{\underline{\mathbb {R}}(A_{2})}(y)\}.&\end{aligned}$$

• (ii)   $\forall y_1y_2\in {\mathbb {S}}B_2,\;x\in {\mathbb {R}}A_1$.

  $$\begin{aligned} T_{\overline{\mathbb {S}}(B_{1})\otimes \overline{\mathbb {S}}(B_{2})}\big ((x, y_1)(x, y_2)\big )=&\min \{T_{\overline{\mathbb {R}}(A_{1})}(x), T_{\overline{\mathbb {S}}(B_{2})}(y_1y_2)\},\\ T_{\underline{\mathbb {S}}(B_{1})\otimes \underline{\mathbb {S}}(B_{2})}\big ((x, y_1)(x, y_2)\big )=&\min \{T_{\underline{\mathbb {R}}(A_{1})}(x), T_{\underline{\mathbb {S}}(B_{2})}(y_1y_2)\},\\ I_{\overline{\mathbb {S}}(B_{1})\otimes \overline{\mathbb {S}}(B_{2})}\big ((x, y_1)(x, y_2)\big )=&\max \{I_{\overline{\mathbb {R}}(A_{1})}(x), I_{\overline{\mathbb {S}}(B_{2})}(y_1y_2)\},\\ I_{\underline{\mathbb {S}}(B_{1})\otimes \underline{\mathbb {S}}(B_{2})}\big ((x, y_1)(x, y_2)\big )=&\max \{I_{\underline{\mathbb {R}}(A_{1})}(x), I_{\underline{\mathbb {S}}(B_{2})}(y_1y_2)\},\\ F_{\overline{\mathbb {S}}(B_{1})\otimes \overline{\mathbb {S}}(B_{2})}\big ((x, y_1)(x, y_2)\big )=&\max \{F_{\overline{\mathbb {R}}(A_{1})}(x), F_{\overline{\mathbb {S}}(B_{2})}(y_1y_2)\},\\ F_{\underline{\mathbb {S}}(B_{1})\otimes \underline{\mathbb {S}}(B_{2})}\big ((x, y_1)(x, y_2)\big )=&\max \{F_{\underline{\mathbb {R}}(A_{1})}(x), F_{\underline{\mathbb {S}}(B_{2})}(y_1y_2)\}. \end{aligned}$$

• (iii)   $\forall x_1x_2\in {\mathbb {S}}B_1,\;y\in {\mathbb {R}}A_2$.

  $$\begin{aligned} T_{\overline{\mathbb {S}}(B_{1})\otimes \overline{\mathbb {S}}(B_{2})}\big ((x_1,y)(x_2,y)\big )=&\min \{T_{\overline{\mathbb {S}}(B_{1})}(x_1x_2), T_{\overline{\mathbb {R}}(A_{2})}(y)\},\\ T_{\underline{\mathbb {S}}(B_{1})\otimes \underline{\mathbb {S}}(B_{2})}\big ((x_1,y)(x_2,y)\big )=&\min \{T_{\underline{\mathbb {S}}(B_{1})}(x_1x_2), T_{\underline{\mathbb {R}}(A_{2})}(y)\},\\ I_{\overline{\mathbb {S}}(B_{1})\otimes \overline{\mathbb {S}}(B_{2})}\big ((x_1,y)(x_2,y)\big )=&\max \{I_{\overline{\mathbb {S}}(B_{1})}(x_1x_2), I_{\overline{\mathbb {R}}(A_{2})}(y)\},\\ I_{\underline{\mathbb {S}}(B_{1})\otimes \underline{\mathbb {S}}(B_{2})}\big ((x_1,y)(x_2,y)\big )=&\max \{I_{\underline{\mathbb {S}}(B_{1})}(x_1x_2), I_{\underline{\mathbb {R}}(A_{2})}(y)\},\\ F_{\overline{\mathbb {S}}(B_{1})\otimes \overline{\mathbb {S}}(B_{2})}\big ((x_1,y)(x_2,y)\big )=&\max \{F_{\overline{\mathbb {S}}(B_{1})}(x_1x_2), F_{\overline{\mathbb {R}}(A_{2})}(y)\},\\ F_{\underline{\mathbb {S}}(B_{1})\otimes \underline{\mathbb {S}}(B_{2})}\big ((x_1,y)(x_2,y)\big )=&\max \{F_{\underline{\mathbb {S}}(B_{1})}(x_1x_2), F_{\underline{\mathbb {R}}(A_{2})}(y)\}. \end{aligned}$$

• (iv)   $\forall x_1x_2\in {\mathbb {S}}B_1,y_1y_2\in {\mathbb {S}}B_2$.

  $$\begin{aligned} T_{\overline{\mathbb {S}}(B_{1})\otimes \overline{\mathbb {S}}(B_{2})}\big ((x_1,y_1)(x_2,y_2)\big )=&\min \{T_{\overline{\mathbb {S}}(B_{1})}(x_1x_2), T_{\overline{\mathbb {S}}(B_{2})}(y_1y_2)\},\\ T_{\underline{\mathbb {S}}(B_{1})\otimes \underline{\mathbb {S}}(B_{2})}\big ((x_1,y_1)(x_2,y_2)\big )=&\min \{T_{\underline{\mathbb {S}}(B_{1})}(x_1x_2), T_{\underline{\mathbb {S}}(B_{2})}(y_1y_2)\},\\ I_{\overline{\mathbb {S}}(B_{1})\otimes \overline{\mathbb {S}}(B_{2})}\big ((x_1,y_1)(x_2,y_2)\big )=&\max \{I_{\overline{\mathbb {S}}(B_{1})}(x_1x_2), I_{\overline{\mathbb {S}}(B_{2})}(y_1y_2)\},\\ I_{\underline{\mathbb {S}}(B_{1})\otimes \underline{\mathbb {S}}(B_{2})}\big ((x_1,y_1)(x_2,y_2)\big )=&\max \{I_{\underline{\mathbb {S}}(B_{1})}(x_1x_2), I_{\underline{\mathbb {S}}(B_{2})}(y_1y_2)\},\\ F_{\overline{\mathbb {S}}(B_{1})\otimes \overline{\mathbb {S}}(B_{2})}\big ((x_1,y_1)(x_1,y_2)\big )=&\max \{F_{\overline{\mathbb {S}}(B_{1})}(x_1x_2), F_{\overline{\mathbb {S}}(B_{2})}(y_1y_2)\},\\ F_{\underline{\mathbb {S}}(B_{1})\otimes \underline{\mathbb {S}}(B_{2})}\big ((x_1,x_1)(x_1,x_2)\big )=&\max \{F_{\underline{\mathbb {S}}(B_{1})}(x_1x_2), F_{\underline{\mathbb {S}}(B_{2})}(y_1y_2)\}. \end{aligned}$$

Definition 8.13

The composition of ${G}_{1}$ and ${G}_{2}$ is a neutrosophic soft rough graph $G={G}_{1}[{G}_{2}] =({{G}_{1}}_*[{{G}_{2}}_*], {{G}}^*_{1}[{{G}}^*_{2}]),$ where ${{G}_{1}}_*[{{G}_{2}}_*]=( \underline{\mathbb {R}} A_{1}[ \underline{\mathbb {R}} A_{2}], \underline{\mathbb {S}} B_{1}[\underline{\mathbb {S}} B_{2}])]~$ and ${{G}}^*_{1}[{{G}}^*_{2}]=( \overline{\mathbb {R}} A_{1}[\overline{\mathbb {R}} A_{2}], \overline{\mathbb {S}} B_{1}[\overline{\mathbb {S}} B_{2}])$ are neutrosophic graphs, respectively, such that

• (i)   $\forall (x, y)\in {\mathbb {R}}A_{1}\times {\mathbb {R}}A_{2}$.

  $$\begin{aligned} T_{\overline{\mathbb {R}}(A_{1})\times \overline{\mathbb {R}}(A_{2})}(x, y) =&\min \{T_{\overline{\mathbb {R}}(A_{1})}(x), T_{\overline{\mathbb {R}}(A_{2})}(y)\},&\\ T_{\underline{\mathbb {R}}(A_{1})\times \underline{\mathbb {R}}(A_{2})}(x, y) =&\min \{T_{\underline{\mathbb {R}}(A_{1})}(x), T_{\underline{\mathbb {R}}(A_{2})}(y)\},&\\ I_{\overline{\mathbb {R}}(A_{1})\times \overline{\mathbb {R}}(A_{2})}(x, y) =&\max \{I_{\overline{\mathbb {R}}(A_{1})}(x), I_{\overline{\mathbb {R}}(A_{2})}(y)\},&\\ I_{\underline{\mathbb {R}}(A_{1})\times \underline{\mathbb {R}}(A_{2})}(x, y) =&\max \{I_{\underline{\mathbb {R}}(A_{1})}(x), I_{\underline{\mathbb {R}}(A_{2})}(y)\},&\\ F_{\overline{\mathbb {R}}(A_{1})\times \overline{\mathbb {R}}(A_{2})}(x, y) =&\max \{F_{\overline{\mathbb {R}}(A_{1})}(x), F_{\overline{\mathbb {R}}(A_{2})}(y)\},&\\ F_{\underline{\mathbb {R}}(A_{1})\times \underline{\mathbb {R}}(A_{2})}(x, y) =&\max \{F_{\underline{\mathbb {R}}(A_{1})}(x), F_{\underline{\mathbb {R}}(A_{2})}(y)\}.&\end{aligned}$$

• (ii)  $\forall y_1y_2\in {\mathbb {S}}B_2,\;x\in {\mathbb {R}}A_1$.

  $$\begin{aligned} T_{\overline{\mathbb {S}}(B_{1})\times \overline{\mathbb {S}}(B_{2})}\big ((x, y_1)(x, y_2)\big )=&\min \{T_{\overline{\mathbb {R}}(A_{1})}(x), T_{\overline{\mathbb {S}}(B_{2})}(y_1y_2)\},\\ T_{\underline{\mathbb {S}}(B_{1})\times \underline{\mathbb {S}}(B_{2})}\big ((x, y_1)(x, y_2)\big )=&\min \{T_{\underline{\mathbb {R}}(A_{1})}(x), T_{\underline{\mathbb {S}}(B_{2})}(y_1y_2)\},\\ I_{\overline{\mathbb {S}}(B_{1})\times \overline{\mathbb {S}}(B_{2})}\big ((x, y_1)(x, y_2)\big )=&\max \{I_{\overline{\mathbb {R}}(A_{1})}(x), I_{\overline{\mathbb {S}}(B_{2})}(y_1y_2)\},\\ I_{\underline{\mathbb {S}}(B_{1})\times \underline{\mathbb {S}}(B_{2})}\big ((x, y_1)(x, y_2)\big )=&\max \{I_{\underline{\mathbb {R}}(A_{1})}(x), I_{\underline{\mathbb {S}}(B_{2})}(y_1y_2)\},\\ F_{\overline{\mathbb {S}}(B_{1})\times \overline{\mathbb {S}}(B_{2})}\big ((x, y_1)(x, y_2)\big )=&\max \{F_{\overline{\mathbb {R}}(A_{1})}(x), F_{\overline{\mathbb {S}}(B_{2})}(y_1y_2)\},\\ F_{\underline{\mathbb {S}}(B_{1})\times \underline{\mathbb {S}}(B_{2})}\big ((x, y_1)(x, y_2)\big )=&\max \{F_{\underline{\mathbb {R}}(A_{1})}(x), F_{\underline{\mathbb {S}}(B_{2})}(y_1y_2)\}. \end{aligned}$$

• (iii)  $\forall x_1x_2\in {\mathbb {S}}B_1,\;y\in {\mathbb {R}}A_2$.

  $$\begin{aligned} T_{\overline{\mathbb {S}}(B_{1})\times \overline{\mathbb {S}}(B_{2})}\big ((x_1,y)(x_2,y)\big )=&\min \{T_{\overline{\mathbb {S}}(B_{1})}(x_1x_2), T_{\overline{\mathbb {R}}(A_{2})}(y)\},\\ T_{\underline{\mathbb {S}}(B_{1})\times \underline{\mathbb {S}}(B_{2})}\big ((x_1,y)(x_2,y)\big )=&\min \{T_{\underline{\mathbb {S}}(B_{1})}(x_1x_2), T_{\underline{\mathbb {R}}(A_{2})}(y)\},\\ I_{\overline{\mathbb {S}}(B_{1})\times \overline{\mathbb {S}}(B_{2})}\big ((x_1,y)(x_2,y)\big )=&\max \{I_{\overline{\mathbb {S}}(B_{1})}(x_1x_2), I_{\overline{\mathbb {R}}(A_{2})}(y)\},\\ I_{\underline{\mathbb {S}}(B_{1})\times \underline{\mathbb {S}}(B_{2})}\big ((x_1,y)(x_2,y)\big )=&\max \{I_{\underline{\mathbb {S}}(B_{1})}(x_1x_2), I_{\underline{\mathbb {R}}(A_{2})}(y)\},\\ F_{\overline{\mathbb {S}}(B_{1})\times \overline{\mathbb {S}}(B_{2})}\big ((x_1,y)(x_2,y)\big )=&\max \{F_{\overline{\mathbb {S}}(B_{1})}(x_1x_2), F_{\overline{\mathbb {R}}(A_{2})}(y)\},\\ F_{\underline{\mathbb {S}}(B_{1})\times \underline{\mathbb {S}}(B_{2})}\big ((x_1,y)(x_2,y)\big )=&\max \{F_{\underline{\mathbb {S}}(B_{1})}(x_1x_2), F_{\underline{\mathbb {R}}(A_{2})}(y)\}. \end{aligned}$$

• (iv)  $\forall x_1x_2\in {\mathbb {S}}B_1,\;y_1\ne y_2\in {\mathbb {R}}A_2$.

  $$\begin{aligned} T_{\overline{\mathbb {S}}(B_{1})\times \overline{\mathbb {S}}(B_{2})}\big ((x_1,y_1)(x_2,y_2)\big )=&\min \{T_{\overline{\mathbb {S}}(B_{1})}(x_1x_1), T_{\overline{\mathbb {R}}(A_{2})}(y_1), T_{\overline{\mathbb {R}}(A_{2})}(y_2)\},\\ T_{\underline{\mathbb {S}}(B_{1})\times \underline{\mathbb {S}}(B_{2})}\big ((x_1,y_1)(x_2,y_2)\big )=&\min \{T_{\underline{\mathbb {S}}(B_{1})}(x_1x_1), T_{\underline{\mathbb {R}}(A_{2})}(y_1), T_{\underline{\mathbb {R}}(A_{2})}(y_2)\},\\ I_{\overline{\mathbb {S}}(B_{1})\times \overline{\mathbb {S}}(B_{2})}\big ((x_1,y_1)(x_2,y_2)\big )=&\max \{I_{\overline{\mathbb {S}}(B_{1})}(x_1x_1), I_{\overline{\mathbb {R}}(A_{2})}(y_1), I_{\overline{\mathbb {R}}(A_{2})}(y_2)\},\\ I_{\underline{\mathbb {S}}(B_{1})\times \underline{\mathbb {S}}(B_{2})}\big ((x_1,y_1)(x_2,y_2)\big )=&\max \{I_{\underline{\mathbb {S}}(B_{1})}(x_1x_1), I_{\underline{\mathbb {R}}(A_{2})}(y_1), I_{\underline{\mathbb {R}}(A_{2})}(y_2)\},\\ F_{\overline{\mathbb {S}}(B_{1})\times \overline{\mathbb {S}}(B_{2})}\big ((x_1,y_1)(x_2,y_2)\big )=&\max \{F_{\overline{\mathbb {S}}(B_{1})}(x_1x_1), F_{\overline{\mathbb {R}}(A_{2})}(y_1), F_{\overline{\mathbb {R}}(A_{2})}(y_2)\},\\ F_{\underline{\mathbb {S}}(B_{1})\times \underline{\mathbb {S}}(B_{2})}\big ((x_1,y_1)(x_2,y_2)\big )=&\max \{F_{\underline{\mathbb {S}}(B_{1})}(x_1x_1), F_{\underline{\mathbb {R}}(A_{2})}(y_1), F_{\underline{\mathbb {R}}(A_{2})}(y_2)\}. \end{aligned}$$

Definition 8.14

Let ${G}=(\underline{{G}}, \overline{{G}})$ be a neutrosophic soft rough graph. The complement of G, denoted by $\acute{\underline{{G}}}=(\acute{\underline{{G}}},\acute{ \overline{{G}}})$, is a neutrosophic soft rough graph, where $\acute{\underline{{G}}}=(\acute{\underline{\mathbb {R}}(A)},\acute{\underline{\mathbb {S}}(B)})$ and $\acute{\overline{{G}}}=(\acute{\overline{\mathbb {R}}(A)},\acute{\overline{\mathbb {S}}(B)})$ are neutrosophic graphs such that

• (i)  $\forall x\in {\mathbb {R}}A$.

  $$\begin{aligned} \acute{T_{\overline{\mathbb {R}}(A)}}(x) =&T_{{\overline{\mathbb {R}}(A)}(x)},~~ \acute{I_{\overline{\mathbb {R}}(A)}}(x) =I_{{\overline{\mathbb {R}}(A)}(x)},~~ \acute{F_{\overline{\mathbb {R}}(A)}}(x) =F_{{\overline{\mathbb {R}}(A)}(x)},\\ \acute{T_{\underline{\mathbb {R}}(A)}}(x) =&T_{{\underline{\mathbb {R}}(A)}(x)},~~ \acute{I_{\underline{\mathbb {R}}(A)}}(x) =I_{{\underline{\mathbb {R}}(A)}(x)},~~ \acute{F_{\underline{\mathbb {R}}(A)}}(x) =F_{{\underline{\mathbb {R}}(A)}(x)}. \end{aligned}$$

• (ii)  $\forall ~ v, u\in {\mathbb {R}}A$.

  $$\begin{aligned} \acute{T_{\overline{\mathbb {S}}(B)}}(xy) =&\min \{T_{\overline{\mathbb {R}}(A)}(x),T_{\overline{\mathbb {R}}(A)}(y)\}-T_{\overline{\mathbb {S}}(B)}(xy),\\ \acute{I_{\overline{\mathbb {S}}(B)}}(xy) =&\max \{I_{\overline{\mathbb {R}}(A)}(x),I_{\overline{\mathbb {R}}(A)}(y)\}-I_{\overline{\mathbb {S}}(B)}(xy),\\ \acute{F_{\overline{\mathbb {S}}(B)}}(xy) =&\max \{F_{\overline{\mathbb {R}}(A)}(x),F_{\overline{\mathbb {R}}(A)}(y)\}-F_{\overline{\mathbb {S}}(B)}(xy),\\ \acute{T_{\underline{\mathbb {S}}(B)}}(xy)=&\min \{T_{\underline{\mathbb {R}}(A)}(x),T_{\underline{\mathbb {R}}(A)}(y)\}-T_{\underline{\mathbb {S}}(B)}(xy),\\ \acute{I_{\underline{\mathbb {S}}(B)}}(xy) =&\max \{I_{\underline{\mathbb {R}}(A)}(x),I_{\underline{\mathbb {R}}(A)}(y)\}-I_{\overline{\mathbb {S}}(B)}(xy),\\ \acute{F_{\underline{\mathbb {S}}(B)}}(xy) =&\max \{F_{\underline{\mathbb {R}}(A)}(x), F_{\underline{\mathbb {R}}(A)}(y)\}-F_{\underline{\mathbb {S}}(B)}(xy). \end{aligned}$$

Example 8.7

Consider a neutrosophic soft rough graphs G as shown in Fig. 8.9. The complement of G is $\acute{{G} }=(\acute{\underline{{G}}}, \acute{\overline{{G}}} )$ obtained by using Definition 8.14, where $\acute{\underline{{G}}}=(\acute{\underline{\mathbb {R}}(A)},\acute{\underline{\mathbb {S}}(B)})$ and $\acute{\overline{{G}}}=(\acute{\overline{\mathbb {R}}(A)},\acute{\overline{\mathbb {S}}(B)})$ are neutrosophic graphs as shown in Fig. 8.10.

Fig. 8.9

Neutrosophic soft rough graph ${G}=({ \underline{G}}, { \overline{G}})$

Fig. 8.10

Neutrosophic soft rough graph $\acute{G}=(\acute{ \underline{G}}, \acute{ \overline{G}})$

Definition 8.15

A graph G is called self-complement if ${G}=\acute{G},$ i.e.

• (i)  $\forall x\in {\mathbb {R}}A$.

  $$\begin{aligned} \acute{T_{\overline{\mathbb {R}}(A)}}(x) =&T_{{\overline{\mathbb {R}}(A)}(x)}, ~~ \acute{I_{\overline{\mathbb {R}}(A)}}(x) =I_{{\overline{\mathbb {R}}(A)}(x)}, \acute{F_{\overline{\mathbb {R}}(A)}}(x) =F_{{\overline{\mathbb {R}}(A)}(x)},\\ \acute{T_{\underline{\mathbb {R}}(A)}}(x) =&T_{{\underline{\mathbb {R}}(A)}(x)}, ~ \acute{I_{\underline{\mathbb {R}}(A)}}(x) =I_{{\underline{\mathbb {R}}(A)}(x)},~ \acute{F_{\underline{\mathbb {R}}(A)}}(x) =F_{{\underline{\mathbb {R}}(A)}(x)}. \end{aligned}$$

• (ii)  $\forall ~ x, y\in {\mathbb {R}}A$.

  $$\begin{aligned} \acute{T_{\overline{\mathbb {S}}(B)}}(xy) =&{T_{\overline{\mathbb {S}}(B)}}(xy),~ \acute{I_{\overline{\mathbb {S}}(B)}}(xy) ={I_{\overline{\mathbb {S}}(B)}}(xy),~ \acute{F_{\overline{\mathbb {S}}(B)}}(xy) ={F_{\overline{\mathbb {S}}(B)}}(xy),\\ \acute{T_{\underline{\mathbb {S}}(B)}}(xy) =&{T_{\underline{\mathbb {S}}(B)}}(xy),~ \acute{I_{\underline{\mathbb {S}}(B)}}(xy) ={I_{\underline{\mathbb {S}}(B)}}(xy),~ \acute{F_{\underline{\mathbb {S}}(B)}}(xy) ={F_{\underline{\mathbb {S}}(B)}}(xy). \end{aligned}$$

Definition 8.16

A neutrosophic soft rough graph G is called strong neutrosophic soft rough graph if $\forall xy\in \mathbb {S}B$,

$$\begin{aligned} {T_{\overline{\mathbb {S}}(B)}}(xy) =&\min \{T_{\overline{\mathbb {R}}(A)}(x),T_{\overline{\mathbb {R}}(A)}(y)\},\\ {I_{\overline{\mathbb {S}}(B)}}(xy) =&\max \{I_{\overline{\mathbb {R}}(A)}(x),I_{\overline{\mathbb {R}}(A)}(y)\}),\\ {F_{\overline{\mathbb {S}}(B)}}(xy) =&\max \{F_{\overline{\mathbb {R}}(A)}(x),F_{\overline{\mathbb {R}}(A)}(y)\},\\ {T_{\underline{\mathbb {S}}(B)}}(xy) =&\min \{T_{\underline{\mathbb {R}}(A)}(x),T_{\underline{\mathbb {R}}(A)}(y)\},\\ {I_{\underline{\mathbb {S}}(B)}}(xy) =&\max \{I_{\underline{\mathbb {R}}(A)}(x),I_{\underline{\mathbb {R}}(A)}(y)\},\\ {F_{\underline{\mathbb {S}}(B)}}(xy) =&\max \{F_{\underline{\mathbb {R}}(A)}(x), F_{\underline{\mathbb {R}}(A)}(y)\}. \end{aligned}$$

Example 8.8

Consider a graph G such that $X =\{u, v, w\}$ and $E= \{uv, vw, wu\}.$ Let $\mathbb {R}A$ be a neutrosophic soft rough set of X, and let $\mathbb {S}B$ be a neutrosophic soft rough set of E defined in Tables 8.10 and 8.11, respectively.

Table 8.10 Neutrosophic soft rough set on X

Table 8.11 Neutrosophic soft rough set on E

Fig. 8.11

Strong neutrosophic soft rough graph $G=( \mathbb {R}A, \mathbb {S}B)$

Hence, $G=( \mathbb {R}A, \mathbb {S}B)$ is a strong neutrosophic soft rough graph as shown in Fig. 8.11.

Definition 8.17

A neutrosophic soft rough graph G is called complete neutrosophic soft rough graph if $\forall ~x, y\in X,$

$$\begin{aligned} {T_{\overline{\mathbb {S}}(B)}}(xy) =&\min \{T_{\overline{\mathbb {R}}(A)}(x),T_{\overline{\mathbb {R}}(A)}(y)\},\\ {I_{\overline{\mathbb {S}}(B)}}(xy) =&\max \{I_{\overline{\mathbb {R}}(A)}(x),I_{\overline{\mathbb {R}}(A)}(y)\},\\ {F_{\overline{\mathbb {S}}(B)}}(xy) =&\max \{F_{\overline{\mathbb {R}}(A)}(x),F_{\overline{\mathbb {R}}(A)}(y)\},\\ {T_{\underline{\mathbb {S}}(B)}}(xy) =&\min \{T_{\underline{\mathbb {R}}(A)}(x),T_{\underline{\mathbb {R}}(A)}(y)\},\\ {I_{\underline{\mathbb {S}}(B)}}(xy) =&\max \{I_{\underline{\mathbb {R}}(A)}(x),I_{\underline{\mathbb {R}}(A)}(y)\},\\ {F_{\underline{\mathbb {S}}(B)}}(xy) =&\max \{F_{\underline{\mathbb {R}}(A)}(x), F_{\underline{\mathbb {R}}(A)}(y)\}. \end{aligned}$$

Remark 8.2

Every complete neutrosophic soft rough graph is a strong neutrosophic soft rough graph. But the converse is not true.

Definition 8.18

A neutrosophic soft rough graph G is isolated if $\forall x, y\in X$.

$$ {T_{\underline{\mathbb {S}}(B)}}(xy) =0,~ {I_{\underline{\mathbb {S}}(B)}}(xy) =0,~ {F_{\underline{\mathbb {S}}(B)}}(xy)= 0,~ {T_{\overline{\mathbb {S}}(B)}}(xy) =0,~ {I_{\overline{\mathbb {S}}(B)}}(xy) =0,~ {F_{\overline{\mathbb {S}}(B)}}(xy) =0,~ $$

Theorem 8.1

The rejection of two neutrosophic soft rough graphs is a neutrosophic soft rough graph.

Proof

Let ${G}_{1} =(\underline{{G}}_1,\overline{{G}}_1)$ and ${G}_{2} =(\underline{{G}}_2,\overline{{G}}_2)$ be two neutrosophic soft rough graphs. Let $G={G}_{1} | {G}_{2} =(\underline{{G}}_1|\underline{{G}}_2, \overline{{G}}_1|\overline{{G}}_2)$ be the rejection of ${G}_{1}$ and ${G}_{2}$, where $\underline{{G}}_1|\underline{{G}}_2 =(\underline{\mathbb {R}}(A_{1})|\underline{\mathbb {R}}(A_{2}), \underline{\mathbb {S}}(B_1)|\underline{\mathbb {S}}(B_2))$ and $\overline{{G}}_1|\overline{{G}}_2 =(\overline{\mathbb {R}}(A_{1})|\overline{\mathbb {R}}(A_{2}), \overline{\mathbb {S}}(B_1)|\underline{\mathbb {S}}(B_2)).$ We claim that $G={G}_{1} | {G}_{2}$ is an neutrosophic soft rough graph. It is enough to show that $\underline{\mathbb {S}}(B_1)|\underline{\mathbb {S}}(B_2)$ and $\overline{\mathbb {S}}(B_1)|\overline{\mathbb {S}}(B_2)$ are neutrosophic relations on $\underline{\mathbb {R}}(A_{1})|\underline{\mathbb {R}}(A_{2})$ and $\overline{\mathbb {R}}(A_{1})|\overline{\mathbb {R}}(A_{2}),$ respectively. First, we show that $\underline{\mathbb {S}}(B_1)|\underline{\mathbb {S}}(B_2)$ is a neutrosophic relation on $\underline{\mathbb {R}}(A_{1})|\underline{\mathbb {R}}(A_{2})$.

If $x\in \underline{\mathbb {R}}(A_1)$, $y_1y_2\notin \underline{\mathbb {S}}(B_2)$, then

$$\begin{aligned} T_{\underline{\mathbb {S}}(B_1)|\underline{\mathbb {S}}(B_2)}((x, y_1)(x, y_2)) =&(T_{\underline{\mathbb {R}}(A_1)}(x)\wedge (T_{\underline{\mathbb {R}}(A_2)}(y_2)\wedge T_{\underline{\mathbb {R}}(A_2)}(y_2))) \\ =&(T_{\underline{\mathbb {R}}(A_1)}(x)\wedge T_{\underline{\mathbb {R}}(A_2)}(y_2))\wedge (T_{\underline{\mathbb {R}}(A_1)}(x)\wedge T_{\underline{\mathbb {R}}(A_2)}(y_2)) \\ =&T_{\underline{\mathbb {R}}(A_1)|\underline{\mathbb {R}}(A_2)}(x, y_1)\wedge T_{\underline{\mathbb {R}}(A_1)|\underline{\mathbb {R}}(A_2)}(x, y_2) \\ T_{\underline{\mathbb {S}}(B_1)|\underline{\mathbb {S}}(B_2)}((x, y_1)(x, y_2)) =&T_{\underline{\mathbb {R}}(A_1)|\underline{\mathbb {R}}(A_2)}(x, y_1)\wedge T_{\underline{\mathbb {R}}(A_1)|\underline{\mathbb {R}}(A_2)}(x, y_2)\\ \text {Similarly, } I_{\underline{\mathbb {S}}(B_1)|\underline{\mathbb {S}}(B_2)}((x, y_1)(x, y_2)) =&I_{\underline{\mathbb {R}}(A_1)|\underline{\mathbb {R}}(A_2)}(x, y_1)\vee I_{\underline{\mathbb {R}}(A_1)|\underline{\mathbb {R}}(A_2)}(x, y_2)\\ F_{\underline{\mathbb {S}}(B_1)|\underline{\mathbb {S}}(B_2)}((x, y_1)(x, y_2)) =&F_{\underline{\mathbb {R}}(A_1)|\underline{\mathbb {R}}(A_2)}(x, y_1)\vee F_{\underline{\mathbb {R}}(A_1)|\underline{\mathbb {R}}(A_2)}(x, y_2) \end{aligned}$$

If $x_1x_2\notin \underline{\mathbb {S}}(B_1)$, $y\in \underline{\mathbb {R}}(A_2)$, then

$$\begin{aligned} T_{\underline{\mathbb {S}}(B_1)|\underline{\mathbb {S}}(B_2)}((x_1,y)(x_2,y)) =&((T_{\underline{\mathbb {R}}(A_1)}(x_1)\wedge T_{\underline{\mathbb {R}}(A_1)}(x_2))\wedge T_{\underline{\mathbb {R}}(A_2)}(y)) \\ =&((T_{\underline{\mathbb {R}}(A_1)}(x_1)\wedge T_{\underline{\mathbb {R}}(A_2)}(y))\wedge (T_{\underline{\mathbb {R}}(A_1)}(x_2)\wedge T_{\underline{\mathbb {R}}(A_2)}(y))) \\ =&T_{\underline{\mathbb {R}}(A_1)|\underline{\mathbb {R}}(A_2)}(x_1,y)\wedge T_{\underline{\mathbb {R}}(A_1)|\underline{\mathbb {R}}(A_2)}(x_2,y) \\ T_{\underline{\mathbb {S}}(B_1)|\underline{\mathbb {S}}(B_2)}((x_1,y)(x_2,y)) =&T_{\underline{\mathbb {R}}(A_1)|\underline{\mathbb {R}}(A_2)}(x_1,y)\wedge T_{\underline{\mathbb {R}}(A_1)|\underline{\mathbb {R}}(A_2)}(x_2,y)\\ \text {Similarly, } I_{\underline{\mathbb {S}}(B_1)|\underline{\mathbb {S}}(B_2)}((x_1,y)(x_2,y)) =&I_{\underline{\mathbb {R}}(A_1)|\underline{\mathbb {R}}(A_2)}(x_1,y)\vee I_{\underline{\mathbb {R}}(A_1)|\underline{\mathbb {R}}(A_2)}(x_2,y)\\ F_{\underline{\mathbb {S}}(B_1)|\underline{\mathbb {S}}(B_2)}((x_1,y)(x_2,y)) =&F_{\underline{\mathbb {R}}(A_1)|\underline{\mathbb {R}}(A_2)}(x_1,y)\vee F_{\underline{\mathbb {R}}(A_1)|\underline{\mathbb {R}}(A_2)}(x_2,y) \end{aligned}$$

If $x_1x_2\notin \underline{\mathbb {S}}(B_1)$, $y_1, y_2\notin \underline{\mathbb {S}}(B_2),$ then

$$\begin{aligned} T_{\underline{\mathbb {S}}(B_1)|\underline{\mathbb {S}}(B_2)}((x_1,y_1)(x_2,y_2)) =&((T_{\underline{\mathbb {R}}(A_1)}(x_1)\wedge T_{\underline{\mathbb {R}}(A_1)}(x_2))\wedge (T_{\underline{\mathbb {R}}(A_2)}(y_1)\wedge T_{\underline{\mathbb {R}}(A_2)}(y_2))) \\ =&( T_{\underline{\mathbb {R}}(A_1)}(x_1)\wedge T_{\underline{\mathbb {R}}(A_2)}(y_1))\wedge ( T_{\underline{\mathbb {R}}(A_1)}(x_2)\wedge T_{\underline{\mathbb {R}}(A_2)}(y_2)) \\ =&T_{\underline{\mathbb {R}}(A_1)|\underline{\mathbb {R}}(A_2)}(x_1,y_1)\wedge T_{\underline{\mathbb {R}}(A_1)|\underline{\mathbb {R}}(A_2)}(x_2,y_2) \\ T_{\underline{\mathbb {S}}(B_1)|\underline{\mathbb {S}}(B_2)}((x_1,y_1)(x_2,y_2))=&T_{\underline{\mathbb {R}}(A_1)|\underline{\mathbb {R}}(A_2)}(x_1,y_1)\wedge T_{\underline{\mathbb {R}}(A_1)|\underline{\mathbb {R}}(A_2)}(x_2,y_2)\\ \text {Similarly, } I_{\underline{\mathbb {S}}(B_1)|\underline{\mathbb {S}}(B_2)}((x_1,y_1)(x_2,y_2)) =&I_{\underline{\mathbb {R}}(A_1)|\underline{\mathbb {R}}(A_2)}(x_1,y_1)\vee I_{\underline{\mathbb {R}}(A_1)|\underline{\mathbb {R}}(A_2)}(=y_1,y_2)\\ F_{\underline{\mathbb {S}}(B_1)|\underline{\mathbb {S}}(B_2)}((x_1,y_1)(x_2,y_2)) =&F_{\underline{\mathbb {R}}(A_1)|\underline{\mathbb {R}}(A_2)}(x_1,y_1)\vee F_{\underline{\mathbb {R}}(A_1)|\underline{\mathbb {R}}(A_2)}(x_2,y_2) \end{aligned}$$

Thus, $\underline{\mathbb {S}}(B_1)|\underline{\mathbb {S}}(B_2)$ is a neutrosophic relation on $\underline{\mathbb {R}}(A_{1})|\underline{\mathbb {R}}(A_{2})$. Similarly, we can show that $\overline{\mathbb {S}}(B_1)|\overline{\mathbb {S}}(B_2)$ is a neutrosophic relation on $\overline{\mathbb {R}}(A_{1})|\overline{\mathbb {R}}(A_{2})$. Hence, G is a neutrosophic soft rough graph.

Theorem 8.2

The Cartesian product of two neutrosophic soft rough graphs is a neutrosophic soft rough graph.

Proof

Let ${G}_{1} =(\underline{{G}}_1,\overline{{G}}_1)$ and ${G}_{2} =(\underline{{G}}_2,\overline{{G}}_2)$ be two neutrosophic soft rough graphs. Let $G={G}_{1} \ltimes {G}_{2} =(\underline{{G}}_1\ltimes \underline{{G}}_2, \overline{{G}}_1\ltimes \overline{{G}}_2)$ be the Cartesian product of ${G}_{1}$ and ${G}_{2}$, where $\underline{{G}}_1\ltimes \underline{{G}}_2 =(\underline{\mathbb {R}}(A_{1})\ltimes \underline{\mathbb {R}}(A_{2}), \underline{\mathbb {S}}(B_1)\ltimes \underline{\mathbb {S}}(B_2))$ and $\overline{{G}}_1\ltimes \overline{{G}}_2 = (\overline{\mathbb {R}}(A_{1})\ltimes \overline{\mathbb {R}}(A_{2}), \overline{\mathbb {S}}(B_1)\ltimes \underline{\mathbb {S}}(B_2)).$ We claim that $G={G}_{1} \ltimes {G}_{2}$ is a neutrosophic soft rough graph. It is enough to show that $\underline{\mathbb {S}}(B_1)\ltimes \underline{\mathbb {S}}(B_2)$ and $\overline{\mathbb {S}}(B_1)\ltimes \overline{\mathbb {S}}(B_2)$ are neutrosophic relations on $\underline{\mathbb {R}}(A_{1})\ltimes \underline{\mathbb {R}}(A_{2})$ and $\overline{\mathbb {R}}(A_{1})\ltimes \overline{\mathbb {R}}(A_{2}),$ respectively. We have to show that $\underline{\mathbb {S}}(B_1)\ltimes \underline{\mathbb {S}}(B_2)$ is a neutrosophic relation on $\underline{\mathbb {R}}(A_{1})\ltimes \underline{\mathbb {R}}(A_{2})$.

If $x\in \underline{\mathbb {R}}(A_1)$, $y_1y_2\in \underline{\mathbb {S}}(B_2)$, then

$$\begin{aligned} T_{\underline{\mathbb {S}}(B_1)\ltimes \underline{\mathbb {S}}(B_2)}((x, y_1)(x, y_2)) =&T_{\underline{\mathbb {R}}(A_1))}(x)\wedge T_{\underline{\mathbb {S}}(B_2)}(y_1y_2) \\ \le&T_{\underline{\mathbb {R}}(A_1))}(x)\wedge (T_{\underline{\mathbb {R}}(A_2)}(y_1)\wedge T_{\underline{\mathbb {R}}(A_2)}(y_2)) \\ =&(T_{\underline{\mathbb {R}}(A_1))}(x)\wedge T_{\underline{\mathbb {R}}(A_2)}(y_1))\wedge (T_{\underline{\mathbb {R}}(A_1))}(x)\wedge T_{\underline{\mathbb {R}}(A_2)}(y_2)) \\ =&T_{\underline{\mathbb {R}}(A_1)\ltimes \underline{\mathbb {R}}(A_2)}(x, y_1)\wedge T_{\underline{\mathbb {R}}(A_1)\ltimes \underline{\mathbb {R}}(A_2)}(x, y_2) \\ T_{\underline{\mathbb {S}}(B_1)\ltimes \underline{\mathbb {S}}(B_2)}((x, y_1)(x, y_2)) \le&T_{\underline{\mathbb {R}}(A_1)\ltimes \underline{\mathbb {R}}(A_2)}(x, y_1)\wedge T_{\underline{\mathbb {R}}(A_1)\ltimes \underline{\mathbb {R}}(A_2)}(x, y_2)\\ \text {Similarly, } I_{\underline{\mathbb {S}}(B_1)\ltimes \underline{\mathbb {S}}(B_2)}((x, y_1)(x, y_2)) \le&I_{\underline{\mathbb {R}}(A_1)\ltimes \underline{\mathbb {R}}(A_2)}(x, y_1)\vee I_{\underline{\mathbb {R}}(A_1)\ltimes \underline{\mathbb {R}}(A_2)}(x, y_2)\\ F_{\underline{\mathbb {S}}(B_1)\ltimes \underline{\mathbb {S}}(B_2)}((x, y_1)(x, y_2)) \le&F_{\underline{\mathbb {R}}(A_1)\ltimes \underline{\mathbb {R}}(A_2)}(x, y_1)\vee F_{\underline{\mathbb {R}}(A_1)\ltimes \underline{\mathbb {R}}(A_2)}(x, y_2) \end{aligned}$$

If $x_1x_2\in \underline{\mathbb {S}}(B_1)$, $z\in \underline{\mathbb {R}}(A_2)$, then

$$\begin{aligned} T_{\underline{\mathbb {S}}(B_1)\ltimes \underline{\mathbb {S}}(B_2)}((x_1,z)(x_2,z)) =&T_{\underline{\mathbb {S}}(B_1))}(x_1x_2)\wedge T_{\underline{\mathbb {R}}(A_2)}(z)\\ \le&(T_{\underline{\mathbb {R}}(A_1))(x_1)\wedge \underline{\mathbb {R}}(A_1))}(x_2))\wedge T_{\underline{\mathbb {R}}(A_2)}(z)\\ =&T_{\underline{\mathbb {R}}(A_1)\ltimes \underline{\mathbb {R}}(A_2)}(x_1,z)\wedge T_{\underline{\mathbb {R}}(A_1)\ltimes \underline{\mathbb {R}}(A_2)}(x_2,z) \\ T_{\underline{\mathbb {S}}(B_1)\ltimes \underline{\mathbb {S}}(B_2)}((x_1,z)(x_2,z)) \le&T_{\underline{\mathbb {R}}(A_1)\ltimes \underline{\mathbb {R}}(A_2)}(x_1,z)\wedge T_{\underline{\mathbb {R}}(A_1)\ltimes \underline{\mathbb {R}}(A_2)}(x_2,z)\\ \text {Similarly, } I_{\underline{\mathbb {S}}(B_1)\ltimes \underline{\mathbb {S}}(B_2)}((x_1,z)(x_2,z)) \le&I_{\underline{\mathbb {R}}(A_1)\ltimes \underline{\mathbb {R}}(A_2)}(x_1,z)\vee I_{\underline{\mathbb {R}}(A_1)\ltimes \underline{\mathbb {R}}(A_2)}(x_2,z)\\ F_{\underline{\mathbb {S}}(B_1)\ltimes \underline{\mathbb {S}}(B_2)}((x_1,z)(x_2,z)) \le&F_{\underline{\mathbb {R}}(A_1)\ltimes \underline{\mathbb {R}}(A_2)}(x_1,z)\vee F_{\underline{\mathbb {R}}(A_1)\ltimes \underline{\mathbb {R}}(A_2)}(x_2,z) \end{aligned}$$

Therefore, $\underline{\mathbb {S}}(B_1)\ltimes \underline{\mathbb {S}}(B_2)$ is a neutrosophic relation on $\underline{\mathbb {R}}(A_{1})\ltimes \underline{\mathbb {R}}(A_{2})$. Similarly, $\overline{\mathbb {S}}(B_1)\ltimes \overline{\mathbb {S}}(B_2)$ is a neutrosophic relation on $\overline{\mathbb {R}}(A_{1})\ltimes \overline{\mathbb {R}}(A_{2}).$ Hence, G is a neutrosophic rough graph.

Theorem 8.3

The cross product of two neutrosophic soft rough graphs is a neutrosophic soft rough graph.

Proof

Let ${G}_{1} =(\underline{{G}}_1,\overline{{G}}_1)$ and ${G}_{2} =(\underline{{G}}_2,\overline{{G}}_2)$ be two neutrosophic soft rough graphs. Let $G={G}_{1} \circledcirc {G}_{2} =(\underline{{G}}_1\circledcirc \underline{{G}}_2, \overline{{G}}_1\circledcirc \overline{{G}}_2)$ be the cross product of ${G}_{1}$ and ${G}_{2}$, where $\underline{{G}}_1\circledcirc \underline{{G}}_2 =(\underline{\mathbb {R}}(A_{1})\circledcirc \underline{\mathbb {R}}(A_{2}), \underline{\mathbb {S}}(B_1)\circledcirc \underline{\mathbb {S}}(B_2))$ and $\overline{{G}}_1\circledcirc \overline{{G}}_2 = (\overline{\mathbb {R}}(A_{1})\circledcirc \overline{\mathbb {R}}(A_{2}), \overline{\mathbb {S}}(B_1)\circledcirc \underline{\mathbb {S}}(B_2)).$ We claim that $G={G}_{1} \circledcirc {G}_{2}$ is a neutrosophic soft rough graph. It is enough to show that $\underline{\mathbb {S}}(B_1)\circledcirc \underline{\mathbb {S}}(B_2)$ and $\overline{\mathbb {S}}(B_1)\circledcirc \overline{\mathbb {S}}(B_2)$ are neutrosophic relations on $\underline{\mathbb {R}}(A_{1})\circledcirc \underline{\mathbb {R}}(A_{2})$ and $\overline{\mathbb {R}}(A_{1})\circledcirc \overline{\mathbb {R}}(A_{2}),$ respectively. First, we show that $\underline{\mathbb {S}}(B_1)\circledcirc \underline{\mathbb {S}}(B_2)$ is a neutrosophic relation on $\underline{\mathbb {R}}(A_{1})\circledcirc \underline{\mathbb {R}}(A_{2})$.

If $x_1x_2\in \underline{\mathbb {S}}(B_1)$, $y_1y_2\in \underline{\mathbb {S}}(B_2)$, then

$$\begin{aligned} T_{\underline{\mathbb {S}}(B_1)\circledcirc \underline{\mathbb {S}}(B_2)}((x_1,y_1)(x_2,y_2)) =&T_{\underline{\mathbb {S}}(B_1))}(x_1x_2)\wedge T_{\underline{\mathbb {S}}(B_2)}(y_1y_2) \\ \le&(T_{\underline{\mathbb {R}}(A_1))}(x_1)\wedge T_{\underline{\mathbb {R}}(A_1))}(x_2)\wedge (T_{\underline{\mathbb {R}}(A_2)}(y_1)\wedge T_{\underline{\mathbb {R}}(A_2)}(y_2)) \\ =&(T_{\underline{\mathbb {R}}(A_1))}(x_1)\wedge T_{\underline{\mathbb {R}}(A_2)}(x_2))\wedge (T_{\underline{\mathbb {R}}(A_1))}(y_1)\wedge T_{\underline{\mathbb {R}}(A_2)}(y_2)) \\ =&T_{\underline{\mathbb {R}}(A_1)\circledcirc \underline{\mathbb {R}}(A_2)}(x_1,x_2)\wedge T_{\underline{\mathbb {R}}(A_1)\circledcirc \underline{\mathbb {R}}(A_2)}(y_1,y_2) \\ T_{\underline{\mathbb {S}}(B_1)\circledcirc \underline{\mathbb {S}}(B_2)}((x_1,x_2)(y_1,y_2)) \le&T_{\underline{\mathbb {R}}(A_1)\circledcirc \underline{\mathbb {R}}(A_2)}(x_1,y_1)\wedge T_{\underline{\mathbb {R}}(A_1)\circledcirc \underline{\mathbb {R}}(A_2)}(x_2,y_2)\\ \text {Similarly, } I_{\underline{\mathbb {S}}(B_1)\circledcirc \underline{\mathbb {S}}(B_2)}((x_1,y_1)(x_2,y_2)) \le&I_{\underline{\mathbb {R}}(A_1)\circledcirc \underline{\mathbb {R}}(A_2)}(x_1,y_1)\vee I_{\underline{\mathbb {R}}(A_1)\circledcirc \underline{\mathbb {R}}(A_2)}(x_2,y_2)\\ F_{\underline{\mathbb {S}}(B_1)\circledcirc \underline{\mathbb {S}}(B_2)}((x_1,y_1)(x_2,y_2)) \le&F_{\underline{\mathbb {R}}(A_1)\circledcirc \underline{\mathbb {R}}(A_2)}(x_1,y_1)\vee F_{\underline{\mathbb {R}}(A_1)\circledcirc \underline{\mathbb {R}}(A_2)}(x_2,y_2) \end{aligned}$$

Thus, $\underline{\mathbb {S}}(B_1)\circledcirc \underline{\mathbb {S}}(B_2)$ is a neutrosophic relation on $\underline{\mathbb {R}}(A_{1})\circledcirc \underline{\mathbb {R}}(A_{2})$. Similarly, we can show that $\overline{\mathbb {S}}(B_1)\circledcirc \overline{\mathbb {S}}(B_2)$ is a neutrosophic relation on $\overline{\mathbb {R}}(A_{1})\circledcirc \overline{\mathbb {R}}(A_{2})$. Hence, G is a neutrosophic soft rough graph.

8.3 Application of Neutrosophic Soft Rough Graphs

In this section, we apply the concept of neutrosophic soft rough sets to a decision-making problem. In recent times, the object recognition problem has gained considerable importance. The object recognition problem can be considered as a decision-making problem, in which final identification of object is founded on given set of information. A detailed description of the algorithm for the selection of most suitable object based on available set of alternatives is given, and purposed decision-making method can be used to calculate lower and upper approximation operators to progress deep concerns of the problem. The presented algorithms can be applied to avoid lengthy calculations when dealing with a large number of objects. This method can be applied in various domains for multicriteria selection of objects.

Selection of Most Suitable Generic Version of Brand Name Medicine

In pharmaceutical industry, different pharmaceutical companies develop, produce and discover pharmaceutical medicines (drugs) for use as medication. These pharmaceutical companies deals with “brand name medicine” and “generic medicine”. Brand name medicine and generic medicine are bioequivalent, generic medicine rate and element of absorption. Brand name medicine and generic medicine have the same active ingredients, and the inactive ingredients may differ. The most important difference is cost. Generic medicine is less expensive as compared to brand name comparators. Usually generic drug manufacturers have competition to produce cost less products. The product may possibly be slightly dissimilar in colour, shape or markings. The major difference is cost. We consider a brand name drug “$u=\text {Loratadine}$” used for seasonal allergies medication. Consider

$$\begin{aligned} X = \{&x_1=\text {Triamcinolone}, x_2= \text {Cetirizine/Pseudoephedrine},&\\&x_3= \text {Pseudoephedrine}, x_4= \text {loratadine/pseudoephedrine},&\\ {}&x_5= \text {Fluticasone} \}&\end{aligned}$$

is a set of generic versions of “Loratadine”. We want to select the most suitable generic version of Loratadine on the basis of parameters $e_1 =$ highly soluble, $e_2 =$ highly permeable, $e_3 =$ rapidly dissolving. $M=\{e_1, e_2, e_3\}$ be a set of paraments. Let $\mathbb {R}$ be a neutrosophic soft relation from X to parameter set M, describes truth-membership, indeterminacy-membership and false-membership degrees of generic version medicines corresponding to the parameters as shown in Table 8.12.

Table 8.12 Neutrosophic soft set $\mathbb {R}=(F, M)$

Suppose $A=\{(e_{1}, 0.2, 0.4, 0.5), (e_{2}, 0.5, 0.6, 0.4), (e_{3}, 0.7, 0.5, 0.4)\}$ is most favourable object which is a neutrosophic set on the parameter set M under consideration. Then $(\underline{\mathbb {R}}(A), \overline{\mathbb {R}}(A))$ is a neutrosophic soft rough set in neutrosophic soft approximation space $(X, M, \mathbb {R}),$ where

$$\begin{aligned} \overline{\mathbb {R}}(A)= & {} \{\!(x_1, 0.6, 0.5,0.4), (x_2, 0.7, 0.4, 0.4), (x_3, 0.7,0.4,0.4),(x_4, 0.7, 0.6,0.5), (x_5, 0.7,0.5,0.5)\!\}, \\ \underline{\mathbb {R}}(A)= & {} \{\!(x_1, 0.5,0.6,0.4), (x_2, 0.5, 0.6, 0.5), (x_3, 0.3, 0.3, 0.5),(x_4, 0.5, 0.6, 0.5), (x_5, 0.4, 0.5, 0.5)\!\}. \end{aligned}$$

Let $E=\{x_{1}x_{2}, x_{1}x_{3}, x_{4}x_{1}, x_{2}x_{3}, x_{5}x_{3}, x_{2}x_{4}, x_{2}x_{5}\} \subseteq \acute{X}$ and $L=\{e_{1}e_{3}, e_{2}e_{1}, e_{3}e_{2}\} \subseteq \acute{M}$.

Then a neutrosophic soft relation S on E (from L to E) can be defined in Table 8.13 as follows:

Table 8.13 Neutrosophic soft relation S

Let $B=\{(e_1e_2, 0.2, 0.4, 0.5),(e_2e_3, 0.5, 0.4, 0.4),(e_1e_3, 0.5, 0.2, 0.5)\}$ be a neutrosophic set on L which describes some relationship between the parameters under consideration, then $SB=(\underline{S}(B), \overline{S}(B))$ is a neutrosophic soft rough relation, where

$$\begin{aligned} \overline{\mathbb {S}}(B)=\{(x_1x_2, 0.5,0.4,0.4),(x_1x_3, 0.4,0.2,0.4),(x_ 4x_1, 0.4,0.2,0.4), (x_2x_3, 0.5, 0.3,0.4),&\\ (x_5x_3, 0.5, 0.2, 0.4),(x_2x_4, 0.4, 0.3, 0.4), (x_2x_5, 0.5, 0.3, 0.4)\},&\\ \underline{\mathbb {S}}(B)=\{(x_1x_2, 0.2,0.4,0.4)(x_1x_3, 0.5,0.4,0.4),(x_4x_1, 0.5,0.4,0.4), (x_2x_3, 0.4, 0.4,0.5),&\\ (x_5x_3, 0.2, 0.4, 0.4),(x_2x_4, 0.2, 0.4, 0.4), (x_2x_5, 0.4, 0.4, 0.5)\}.&\end{aligned}$$

Thus, $G=(\underline{G}, \overline{G})$ is a neutrosophic soft rough graph as shown in Fig. 8.12.

Fig. 8.12

Neutrosophic soft rough graph $G=( \underline{G}, \overline{G})$

The sum of two neutrosophic numbers is defined as follows.

Definition 8.19

Let C and D be two single-valued neutrosophic numbers, and the sum of two single-valued neutrosophic numbers is defined as follows:

$$\begin{aligned} C\oplus D = <T_C+T_D-T_C\times T_D, I_C\times I_D, F_C\times F_D >. \end{aligned}$$

(8.1)

The sum of upper neutrosophic soft rough set $\overline{\mathbb {R}}A$ and the lower neutrosophic soft rough set $\underline{\mathbb {R}}A$ and sum of lower neutrosophic soft rough relation $\underline{\mathbb {S}}B$ and the upper neutrosophic soft rough relation $\overline{\mathbb {S}}B$ are neutrosophic sets $\overline{\mathbb {R}}A\oplus \underline{\mathbb {R}}A$ and $\overline{\mathbb {S}}B\oplus \underline{\mathbb {S}}B$, respectively, defined by

The score function $\gamma (x_{k})$ defines for each generic version medicine ${x}_{i}\in X$,

$$\begin{aligned} \gamma (x_{i})=\sum \limits _{x_ix_j \in E}\frac{ T_{\alpha }(x_j)+I_{\alpha }(x_j)-F_{\alpha }(x_j)}{3-(T_{\beta }(x_ix_j)+I_{\beta }(x_ix_j)-F_{\beta }(x_ix_j))} \end{aligned}$$

(8.2)

and $x_k$ with the larger score value $x_k=\max \limits _{i}\gamma (x_{i})$ is the most suitable generic version medicine. By calculations, we have

$$\begin{aligned} \gamma (x_1)=0.88, \gamma (x_2)=0.69,~\gamma (x_3)=0.26~\gamma (x_4)=0.57,~~\text {and}~~\gamma (x_5)=0.33. \end{aligned}$$

(8.3)

Here, $x_1$ is the optimal decision, and the most suitable generic version of “Loratadine” is “Triamcinolone”. We have used software MATLAB for calculating the required results in the application. The algorithm is given in Algorithm 8.3.1.

Algorithm 8.3.1

Algorithm for selection of most suitable objects

• 1.   Input the number of elements in vertex set $X=\{x_1,x_2,\ldots , x_n\}$.

• 2.   Input the number of elements in parameter set $M=\{e_1,e_2,\ldots , e_m\}$.

• 3. Input a neutrosophic soft relation $\mathbb {R}$ from X to M.

• 4. Input a neutrosophic set A on M.

• 5. Compute neutrosophic soft rough vertex set $\mathbb {R}A=(\underline{\mathbb {R}}(A),\overline{\mathbb {R}}(A))$.

• 6. Input the number of elements in edge set $E=\{x_1x_1,x_1x_2,\ldots , x_kx_1\}$.

• 7. Input the number of elements in parameter set $\acute{M}=\{e_1e_1,e_1e_2,\ldots , e_le_1\}$.

• 8. Input a neutrosophic soft relation $\mathbb {S}$ from $\acute{X}$ to ${\acute{M}}$.

• 9. Input a neutrosophic set B on ${\acute{M}}$.

• 10. Compute neutrosophic soft rough edge set $\mathbb {S}B=(\underline{\mathbb {S}}(B),\overline{\mathbb {S}}(B))$.

• 11. Compute neutrosophic set $\alpha =(T_{\alpha }(x_i), I_{\alpha }(x_i), F_{\alpha }(x_i))$, where

  

• 12. Compute neutrosophic set $\beta =(T_{\beta }(x_ix_i), I_{\beta }(x_ix_j), F_{\beta }(x_ix_j))$, where

  

• 13. Calculate the score values of each object $x_i$, and score function is defined as follows:

  $$\begin{aligned} \gamma (x_{i})=\sum \limits _{x_ix_j \in E}\dfrac{ T_{\alpha }(x_j)+I_{\alpha }(x_j)-F_{\alpha }(x_j)}{3-(T_{\beta }(x_ix_j)+I_{\beta }(x_ix_j)-F_{\beta }(x_ix_j))}; \end{aligned}$$

• 14. The decision is ${x}_{i}$ if ${\gamma }_{i} = \max \limits \limits _{i=1}^{n}\gamma _{i}$.

• 15. If i has more than one value, then any one of ${x}_{i}$ may be chosen.