Wiener index of a directed rough fuzzy graph and application to human trafficking

Abstract

In this paper, we introduced Wiener index $(WI)$ and average Wiener index $(AWI)$ of directed rough fuzzy graph (DRFG). $WI$ is the most extensively used index in graph theory. This index is based on the geodesic distance between two vertices. If there is no directed path from vertex x to vertex y in DRFG, we assume that the weight of geodesic from vertex x to vertex y is zero. In this paper, we investigate the connection between $WI$ and connectivity index ( $CI$ ), which is one of the most prominent index, by presenting several examples and results. We introduced the concept of complete directed rough fuzzy graph (CDRFG) along with some useful results like CDRFG have no weak edges. We also compute the $WI$ for CDRFG. Moreover, we discussed three types of vertices: Wiener enhancing vertex (WEV), Wiener reducing vertex (WRV), and Wiener neutral vertex (WNV). The proposed study of DRFG is suitable for modeling uncertainties and unclear data information in the real life circumstances. In the end, we proposed an application of the $WI$ in the human trafficking network. We also presented a detailed comparative analysis and comparison table by comparing our result for both $CI$ and $WI$ for the same human trafficking network.

Keywords

Directed rough fuzzy graph connectivity index wiener index human trafficking

1 Introduction and preliminaries

We have incomplete information about many real world problems. The vagueness in the description and the uncertainty have led to the development of fuzzy graph theory. Zadeh [36] was the first to propose a mathematical framework for describing uncertainty in real world situations. The theory of graphs seems to be important in handling the real life problems. The study of graphs is a simple way to describe data containing the relationships between different elements. By assigning the membership of [0,1] to vertices and edges, Kaufmann [16] introduced the concept of fuzziness to traditional graph theory in 1973. At the same time, Rosenfeld [31] proposed the fuzzy graph concepts, as well as various fuzzy analogues of graph-theoretic concepts such as path, cycles, and connectedness. Many researchers have contributed to this study of FGs. For example, Yeh and Bang [35] independently investigated the notion of FGs and described its applicability in cluster analysis. To describe a number of network ideas, Koczy [17] proposed fuzzy vertex graphs and fuzzy edge graphs. Bhattacharya and Suraweera [6] presented an approach for computing max-min powers and properties of FGs. Bhutani [7] studied FGs automorphism. Bhutani and Rosenfeld [8 –10] studied fuzzy endpoints, geodesics, and edge types based on connectivity in FGs. Samanta et al. [33] established the completeness and regularity of generalized FGs for the first time. Additional, fuzzy planar graphs were researched by Samanta and Pal [32]. The edges of a FG were defined by Mathew and Sunitha [20] as a α-strong, β-strong, and δ-edge. In FGs, Mathew and Sunitha [21, 23] discussed edge, vertex and cycle connectivity. Binu et al. [11] looked into the $CI$ of FG and how it may be used to combat human trafficking. For detail study about graphs and FGs, refers [15 , 26–28]. Rough refers to something that is approximate, imprecise, or inaccurate. Instead of using membership values, Pawlak’s rough set [29, 30] expresses uncertainty using the boundary region. If the boundary region is empty, the set under study is classical; otherwise, it is imprecise or rough. Equivalence classes are essential components of rough set theory for upper and lower approximations. Many researchers have addressed rough sets and their decision making or hybrid models in their research articles [18, 19]. Dubois and Prade [13] looked at both rough fuzzy sets (RFs) and fuzzy rough sets (FRs) and concluded that both hybrid models present unique possibilities for dealing with vagueness. Akram et al. [1, 2] proposed the concept of fuzzy rough digraphs and discussed a new approach based on fuzzy rough digraphs for decision making. Malik and Akram [14] presented a new technique for decision making based on intuitionistic fuzzy rough graphs.

A topological index is a mathematical technique that can be used to study the structural properties of a graph. Topological indices are used to handle many graph theoretical problems. In studying the boiling point of paraffin, Wiener [34] was the first to investigate the Wiener index ( $WI$ ). In the mid-1970s, following Harold Wiener’s seminal work on the subject, significant results related to $WI$ were described. $WI$ has been studied in graphs in a variety of domains, including chemistry, mathematics, and physics. Binu et al. [12] studied the $WI$ of FG as well as its application to unauthorized immigration networks. Recently, Islam and Pal [25] introduced the fuzzy hyper Wiener index with some bounds on the fuzzy hyper Wiener index for various fuzzy graphs like path, cycle and star.

In networks, directed fuzzy graphs (DFGs) are not able to deal with real life networking under incomplete data information or rough universe. Therefore, this drawback opens a way to introduce directed rough fuzzy graphs (DRFGs). The directed rough fuzzy model is a unique and innovative hybrid model for dealing with more complex problems of uncertainty. Topological indices based on graph connectivity analysis deal with decision-making problems by describing the characteristics of graphs. Sometimes in DRFG, the study of connectivity index is not sufficient to distinguish the properties of those graphs having same values of connectivity index, then in that case Wiener index can be used instead of connectivity index. Moreover, the concept of Wiener index exists both in crisp and fuzzy graph theory, but these models are not valid in their existing forms for graphical structures in all environments such as DRFGs. Therefore, our focus in this study is to generalize the concept of $WI$ from FGs to DRFGs.

Zafar and Akram [37] presented the idea of DRFG as a generalization of rough set and DFG. Recent work on properties and decision making of DRFGs was discussed in [3, 5]. The notion of strength of connection $(SC)$ between each pair of vertices in DRFGs was proposed by Akram and Zafar [4] and addressed various key concepts such as strongest DRF-path, strong DRF-path and DRF-bridge, α-strong DRF-edge, β-strong DRF-edge, δ-DRF-edge, and so on. Furthermore, they introduced the idea of connectivity index ( $CI),$ average connectivity index ( $ACI$ ) and related results in DRFG.

The motivation for our work is that $WI$ (a distance based index) and average Wiener index ( $AWI$ ) of fuzzy graphs are documented in the literature, while these indices are not known for DRFGs. These indices will allow for in-depth study of various DRFGs features. For this reason, we propose these notions for DRFGs. We also explore the correspondence between connectivity index ( $CI$ ) and $WI$ of DRFGs by presenting several examples and results. The aim of this paper is to deal with the complex problems that are not easily tackled by fuzzy graphs. Many scholars will be able to investigate DRFGs in more depth because to our approach.

The setting of our research article is given as follows. In Section 1, we provide basic DRFG definitions, outcomes, and expressions that are essential for the content development. In Section 2, we discuss $WI$ of DRFG and related results. Section 3 describes relationship between $CI$ and $WI$ of a DRFG along with complete directed rough fuzzy graph (CDRFG). In Section 4, we introduce average Wiener index ( $AWI),$ Wiener enhancing vertex (WEV), Wiener reducing vertex (WRV) and Wiener neutral vertex (WNV) of a DRFG. We define Wiener enhancing directed rough fuzzy graph (WEDRFG), Wiener reducing directed rough fuzzy graph (WRDRFG) and Wiener neutral directed rough fuzzy graph (WNDRFG). Section 5 explores how to use $WI$ of DRFG to determine much active countries on the paths of people trafficking from Pakistan. For this human trafficking network the detail comparative analysis between our research work and the existing method in [4] is presented in section 6. In section 7, we conclude our researchwork.

The following are the basic definitions and ideas of directed rough fuzzy graph (DRFG); the majority of them may be found in [4].

Definition 1.1. A DRFG on classical set $U^{*}$ is a pair $G = (\underline{G}, \bar{G}) = (RY, TZ),$ where $\underline{G} = (\underline{R} Y, \underline{T} Z)$ is a lower approximated directed fuzzy graph (DFG) and $\bar{G} = (\bar{R} Y, \bar{T} Z)$ is a upper approximated directed fuzzy graph (DFG) of $G = (\underline{G}, \bar{G})$ such that $\underline{T} Z (x_{0} x_{1}) \leq \min {(\underline{R} Y) (x_{0}), (\underline{R} Y) (x_{1})}$ and $\bar{T} Z (x_{0} x_{1}) \leq \min {(\bar{R} Y) (x_{0}), (\bar{R} Y) (x_{1})}, \forall x_{0}, x_{1} \in U^{*} .$ Here $R$ and $T$ are equivalence relations (ERs) and $Y$ and $Z$ are fuzzy subsets of $U^{*}$ and $U^{*} \times U^{*},$ respectively. Note that $SZ$ is called a rough fuzzy relation (RFR) on $RY$ a rough fuzzy set (RFs).

The underlying directed crisp graph of DRFG $G = (\underline{G}, \bar{G})$ having fuzzy membership value is represented by $G^{*} = (\underline{G^{*}}, \bar{G^{*}}),$ where $(\underline{R} Y)^{*} = {x_{0} \in U^{*} | (\underline{R} Y) (x_{0}) > 0},$ $(\bar{R} Y)^{*} = {x_{0} \in U^{*} | (\bar{R} Y) (x_{0}) > 0},$ $(\underline{T} Z)^{*} = {x_{0} x_{1} \in U^{*} \times U^{*} | (\underline{T} Z) (x_{0} x_{1}) > 0}$ and $(\bar{T} Z)^{*} = {x_{0} x_{1} \in U^{*} \times U^{*} | (\bar{T} Z) (x_{0} x_{1}) > 0} .$ Here ${\underline{G}}^{*} = ((\underline{R} Y)^{*}, (\underline{T} Z)^{*}))$ and ${\bar{G}}^{*} = ((\bar{R} Y)^{*}, (\bar{T} Z)^{*}) .$ A DRFG $\tilde{G} = (\underline{\tilde{G}}, \bar{\tilde{G}})$ is called partial directed rough fuzzy subgraph (partial DRF-subgraph) of DRFG $G = (\underline{G}, \bar{G})$ if $(\underline{R} \tilde{Y}) (x_{0}) \leq (\underline{R} Y) (x_{0}),$ $(\underline{T} \tilde{Z}) (x_{0}, x_{1}) \leq (\underline{T} Z) (x_{0}, x_{1}),$ $(\bar{R} \tilde{Y}) (x_{0}) \leq (\bar{R} Y) (x_{0})$ and $(\bar{T} \tilde{Z}) (x_{0}, x_{1}) \leq (\bar{T} Z) (x_{0}, x_{1})$ for all $x_{0} \in \tilde{Y}$ and $(x_{0}, x_{1}) \in \tilde{Z} .$ If $\tilde{G} = (\underline{\tilde{G}}, \bar{\tilde{G}})$ is partial DRF-subgraph of $G = (\underline{G}, \bar{G})$ such that $(\underline{R} \tilde{Y}) (x_{0}) = (\underline{R} Y) (x_{0}),$ $(\underline{T} \tilde{Z}) (x_{0}, x_{1}) = (\underline{T} Z) (x_{0}, x_{1}),$ $(\bar{R} \tilde{Y}) (x_{0}) = (\bar{R} Y) (x_{0})$ and $(\bar{T} \tilde{Z}) (x_{0}, x_{1}) = (\bar{T} Z) (x_{0}, x_{1})$ for all $x_{0} \in \tilde{Y}$ and $(x_{0}, x_{1}) \in \tilde{Z}$ , then $\tilde{G} = (\underline{\tilde{G}}, \bar{\tilde{G}})$ is called a directed rough fuzzy subgraph (DRF-subgraph) of DRFG $G = (\underline{G}, \bar{G}) .$

A directed path $P : x_{0} \to x_{1} \to . . . \to x_{n}$ of length n is directed rough fuzzy path (DRF-path) in $G = (\underline{G}, \bar{G})$ if $P$ is directed fuzzy path (DF-path) of length n from w₀ to w_n in $\underline{G}$ likewise in $\bar{G} .$ The lowest membership value of edge is weakest edge in $G = (\underline{G}, \bar{G}) .$ The strength or intensity of the path $P$ is defined as the total of the degree of membership of weakest edge in $\underline{G}$ as well as in $\bar{G} .$ CONN_G(x₀, x₁) is used to represent the strength of connectedness ( $SC$ ) from x₀ to x₁ in $G = (\underline{G}, \bar{G})$ and is defined as the sum of highest of strengths of all directed paths from x₀ to x₁ in $\underline{G}$ and $\bar{G} .$ If the strength of a DRF-path $P$ in $G = (\underline{G}, \bar{G})$ is equal to CONN_G(x₀, x₁) , then $P$ is said to be a strongest x₀ - x₁ DRF-path. An directed rough fuzzy edge (DRF-edge) x₀x₁ of $G = (\underline{G}, \bar{G})$ is said to be a α- strong DRF-edge if x₀x₁ is α- strong fuzzy edge in $\underline{G}$ as well as in $\bar{G},$ i.e., $\underline{T} Z (x_{0}, x_{1}) > {CONN}_{\underline{G} - x_{0} x_{1}} (x_{0}, x_{1})$ and $\bar{T} Z (x_{0}, x_{1}) > {CONN}_{\bar{G} - x_{0} x_{1}} (x_{0}, x_{1}),$ respectively. An edge x₀x₁ of a DRFG $G = (\underline{G}, \bar{G})$ is said to be a β- strong DRF-edge if x₀x₁ is β- strong fuzzy edge in $\underline{G}$ as well as in $\bar{G},$ i.e., $\underline{T} Z (x_{0}, x_{1}) = {CONN}_{\underline{G} - x_{0} x_{1}} (x_{0}, x_{1})$ and $\bar{T} Z (x_{0}, x_{1}) = {CONN}_{\bar{G} - x_{0} x_{1}} (x_{0}, x_{1}),$ respectively. An edge x₀x₁ of a DRFG $G = (\underline{G}, \bar{G})$ is called δ- DRF-edge if x₀x₁ is δ- fuzzy edge in $\underline{G}$ as well as in $\bar{G},$ i.e., $\underline{T} Z (x_{0}, x_{1}) < {CONN}_{\underline{G} - x_{0} x_{1}} (x_{0}, x_{1})$ and $\bar{T} Z (x_{0}, x_{1}) < {CONN}_{\bar{G} - x_{0} x_{1}} (x_{0}, x_{1}),$ respectively. An edge is strong DRF-edge in $G = (\underline{G}, \bar{G})$ if it is either α-strong DRF-edge or β-strong DRF-edge. A directed path $P$ is said to be a strong DRF-path if its all edges are strong DRF-edges. If the elimination of an DRF-edge $x_{0} x_{1} \in U^{*} \times U^{*}$ decrease the $SC$ between several pair of vertices in $G = (\underline{G}, \bar{G}),$ so edge x₀x₁ is said to be a directed rough fuzzy bridge (DRF-bridge) of $G = (\underline{G}, \bar{G}) .$ In a similar fashion if the elimination of directed rough fuzzy vertex (DRF-vertex) x₀ decrease the $SC$ between several pair of vertices in $G = (\underline{G}, \bar{G}),$ therefore x₀ is called directed rough fuzzy cutvertex (DRF-cutvertex) of $G = (\underline{G}, \bar{G}) .$ A DRFG without DRF-cutvertices is called a directed rough fuzzy block (DRF-block) or just a block.

The definitions and considerations applied in this work are provided below.

Definition 1.2. Consider two DRFGs $G_{1} = ({\underline{G}}_{1}, {\bar{G}}_{1})$ and $G_{2} = ({\underline{G}}_{2}, {\bar{G}}_{2})$ , then there is a isomorphism b : G₁ → G₂ if there exists two isomorphisms $\underline{b} : \underline{G_{1}} \to \underline{G_{2}}$ and $\bar{b} : \bar{G_{1}} \to \bar{G_{2}},$ i.e., there is a pair of bijective mappings $(\underline{(} b), \bar{b}) : U^{*} \to U^{*}$ such that

(i) $(\underline{R} Y_{1}) (x_{1}) = (\underline{R} Y_{2}) (\underline{b} (x_{1})),$

$(\underline{T} Z_{1}) (x_{1} x_{2}) = (\underline{T} Z_{2}) (\underline{b} (x_{1}) \underline{b} (x_{2})), \forall x_{1} \in U^{*}, x_{1} x_{2} \in U^{*} \times U^{*} .$

(ii) $(\bar{R} Y_{1}) (x_{1}) = (\bar{R} Y_{2}) (\bar{b} (x_{1})),$

$(\bar{T} Z_{1}) (x_{1} x_{2}) = (\bar{T} Z_{2}) (\bar{b} (x_{1}) \bar{b} (x_{2})), \forall x_{1} \in U^{*}, x_{1} x_{2} \in U^{*} \times U^{*} .$

Definition 1.3. Consider a DRFG $G = (\underline{G}, \bar{G})$ and x₀, x_n∈ $G = (\underline{G}, \bar{G}) .$ A strong DRF-path $P$ from x₀ to x_n is said to be a directed rough fuzzy geodesic (DRF-geodesic) in $G = (\underline{G}, \bar{G})$ if it is directed fuzzy geodesic (DF-geodesic) in $\underline{G}$ as well as in $\bar{G} .$ That is, there is no shorter strong path $P$ from vertex x₀ to vertex x_n in both $\underline{G}$ and $\bar{G} .$ A geodesic’s value is the total of the degrees of membership of all of DRF-edges in the geodesic.

Example 1.4. Consider a DRFG $G = (\underline{G}, \bar{G})$ given in Fig. 1. on $U^{*} = {x_{1}, x_{2}, x_{3}, x_{4}} .$ Here $\underline{G} = (\underline{R} Y, \underline{T} Z)$ and $\bar{G} = (\bar{R} Y, \bar{T} Z)$ depict directed fuzzy graphs (DFGs). The $SC$ between vertices of DRFG for edges are as follows: $\begin{matrix} {CONN}_{\underline{G} - (x_{1}, x_{2})} (x_{1}, x_{2}) = 0.1, {CONN}_{\bar{G} - (x_{1}, x_{2})} (x_{1}, x_{2}) = 0.3, {CONN}_{\underline{G} - (x_{2}, x_{1})} (x_{2}, x_{1}) = 0.2, {CONN}_{\bar{G} - (x_{2}, x_{1})} (x_{2}, x_{1}) = 0.2, \\ {CONN}_{\underline{G} - (x_{1}, x_{3})} (x_{1}, x_{3}) = 0.1, {CONN}_{\bar{G} - (x_{1}, x_{3})} (x_{1}, x_{3}) = 0.3, {CONN}_{\underline{G} - (x_{3}, x_{1})} (x_{3}, x_{1}) = 0.2, {CONN}_{\bar{G} - (x_{3}, x_{1})} (x_{3}, x_{1}) = 0.3, \\ {CONN}_{\underline{G} - (x_{1}, x_{4})} (x_{1}, x_{4}) = 0.1, {CONN}_{\bar{G} - (x_{1}, x_{4})} (x_{1}, x_{4}) = 0.3, {CONN}_{\underline{G} - (x_{4}, x_{1})} (x_{4}, x_{1}) = 0.1, {CONN}_{\bar{G} - (x_{4}, x_{1})} (x_{4}, x_{1}) = 0.3, \\ {CONN}_{\underline{G} - (x_{2}, x_{3})} (x_{2}, x_{3}) = 0.1, {CONN}_{\bar{G} - (x_{2}, x_{3})} (x_{2}, x_{3}) = 0.3, {CONN}_{\underline{G} - (x_{3}, x_{2})} (x_{3}, x_{2}) = 0.0, {CONN}_{\bar{G} - (x_{3}, x_{2})} (x_{3}, x_{2}) = 0.0, \\ {CONN}_{\underline{G} - (x_{2}, x_{4})} (x_{2}, x_{4}) = 0.3, {CONN}_{\bar{G} - (x_{2}, x_{4})} (x_{2}, x_{4}) = 0.5, {CONN}_{\underline{G} - (x_{4}, x_{2})} (x_{4}, x_{2}) = 0.1, {CONN}_{\bar{G} - (x_{4}, x_{2})} (x_{4}, x_{2}) = 0.3, \\ {CONN}_{\underline{G} - (x_{3}, x_{4})} (x_{3}, x_{4}) = 0.2, {CONN}_{\bar{G} - (x_{3}, x_{4})} (x_{3}, x_{4}) = 0.3, {CONN}_{\underline{G} - (x_{4}, x_{3})} (x_{4}, x_{3}) = 0.1, {CONN}_{\bar{G} - (x_{4}, x_{3})} (x_{4}, x_{3}) = 0.3 . \end{matrix}$ It is easy to verify that the edges x₁x₄, x₂x₃, x₂x₁, x₄x₁, x₃x₂ are α-strong fuzzy edges in $\underline{G}$ and x₁x₄, x₂x₁, x₄x₁, x₃x₂ are α-strong fuzzy edges in $\bar{G},$ respectively. Therefore, the edges x₁x₄, x₂x₁, x₄x₁, x₃x₂ are α-strong DRF-edges in $G = (\underline{G}, \bar{G})$ the DRFG. Similarly, x₁x₂, x₄x₃ are β-strong DRF-edges in $G = (\underline{G}, \bar{G})$ the DRFG and x₂x₄ is δ DRF-edge in $G = (\underline{G}, \bar{G})$ the DRFG. So x₁x₄, x₂x₃, x₂x₁, x₄x₁, x₃x₂, x₁x₂, x₄x₃ are strong DRF-edges of $G = (\underline{G}, \bar{G}) .$

In this example all DRF-edges excluding x₂x₄ are strong. Therefore, x₂ → x₁ → x₄ is the only shorter directed strong path connecting x₂ and x₄ in both $\underline{G}$ and $\bar{G} .$ Therefore this directed path is a geodesic which connecting the vertex x₂ and vertex x₄ in $\underline{G}$ as well as in $\bar{G}$ and values of geodesic’s in $\underline{G}$ and $\bar{G}$ are 0.7 and 1.1 respectively.

Fig. 1

$G = (\underline{G}, \bar{G})$ the DRFG.

Definition 1.5. Consider a DRFG $G = (\underline{G}, \bar{G}) .$ The connectivity index ( $CI$ ) of $G = (\underline{G}, \bar{G})$ is $CI (G) = CI (\underline{G}) + CI (\bar{G}),$ where $\begin{matrix} CI (\underline{G}) = \sum_{x_{0}, x_{1} \in (\underline{R} Y)^{*}} (\underline{R} Y) (x_{0}) (\underline{R} Y) (x_{1}) {CONN}_{\underline{G}} (x_{0}, x_{1}), \\ CI (\bar{G}) = \sum_{x_{0}, x_{1} \in (\bar{R} Y)^{*}} (\bar{R} Y) (x_{0}) (\bar{R} Y) (x_{1}) {CONN}_{\bar{G}} (x_{0}, x_{1}), \end{matrix}$

Note that ${CONN}_{\underline{G}} (x_{0}, x_{1})$ and ${CONN}_{\bar{G}} (x_{0}, x_{1})$ is the $SC$ between x₀ and x₁ in $\underline{G}$ and $\bar{G}$ respectively.

Definition 1.6. Consider a DRFG $G = (\underline{G}, \bar{G}) .$ The average connectivity index ( $ACI$ ) of $G = (\underline{G}, \bar{G})$ is $\bar{G}) .$ $ACI (G) = ACI (\underline{G}) + ACI (\bar{G}),$ where

$ACI (\underline{G}) = \frac{1}{^{n} C_{2}} \sum_{x_{0}, x_{1} \in (\underline{R} Y)^{*}} (\underline{R} Y) (x_{0}) (\underline{R} Y) (x_{1}) {CONN}_{\underline{G}} (x_{0}, x_{1}),$

$ACI (\bar{G}) = \frac{1}{^{n} C_{2}} \sum_{x_{0}, x_{1} \in (\bar{R} Y)^{*}} (\bar{R} Y) (x_{0}) (\bar{R} Y) (x_{1}) {CONN}_{\bar{G}} (x_{0}, x_{1}) .$ Here n represent the total number of vertices in a DRFG $G = (\underline{G}, \bar{G})$ .

Example 1.7. Suppose a DRFG $G = (\underline{G}, \bar{G})$ on $U^{*} = {x_{1}, x_{2}, x_{3}, x_{4}} .$ Therefore, $\underline{G} = (\underline{R} Y, \underline{T} Z)$ and $\bar{G} = (\bar{R} Y, \bar{T} Z)$ in Fig. 2. depict DFGs. By direct computations, the $SC$ between vertices of DRFG are as follows:

$\begin{matrix} {CONN}_{\underline{G}} (x_{1}, x_{2}) = 0.7, {CONN}_{\bar{G}} (x_{1}, x_{2}) = 0.7, {CONN}_{\underline{G}} (x_{2}, x_{1}) = 0.2, {CONN}_{\bar{G}} (x_{2}, x_{1}) = 0.4, \\ {CONN}_{\underline{G}} (x_{1}, x_{3}) = 0.5, {CONN}_{\bar{G}} (x_{1}, x_{3}) = 0.6, {CONN}_{\underline{G}} (x_{3}, x_{1}) = 0.2, {CONN}_{\bar{G}} (x_{3}, x_{1}) = 0.4, \\ {CONN}_{\underline{G}} (x_{1}, x_{4}) = 0.5, {CONN}_{\bar{G}} (x_{1}, x_{4}) = 0.6, {CONN}_{\underline{G}} (x_{4}, x_{1}) = 0.2, {CONN}_{\bar{G}} (x_{4}, x_{1}) = 0.4, \\ {CONN}_{\underline{G}} (x_{2}, x_{3}) = 0.3, {CONN}_{\bar{G}} (x_{2}, x_{3}) = 0.4, {CONN}_{\underline{G}} (x_{3}, x_{2}) = 0.2, {CONN}_{\bar{G}} (x_{3}, x_{2}) = 0.4, \\ {CONN}_{\underline{G}} (x_{2}, x_{4}) = 0.5, {CONN}_{\bar{G}} (x_{2}, x_{4}) = 0.6, {CONN}_{\underline{G}} (x_{4}, x_{2}) = 0.2, {CONN}_{\bar{G}} (x_{4}, x_{2}) = 0.4, \\ {CONN}_{\underline{G}} (x_{3}, x_{4}) = 0.3, {CONN}_{\bar{G}} (x_{3}, x_{4}) = 0.4, {CONN}_{\underline{G}} (x_{4}, x_{3}) = 0.3, {CONN}_{\bar{G}} (x_{4}, x_{3}) = 0.4 . \end{matrix}$ Therefore, the $CI$ and $ACI$ of $G = (\underline{G}, \bar{G})$ are asfollows:

$CI (\underline{G}) = 1.713$ , $CI (\bar{G}) = 4.139$ , and $CI (G) = 5.852$ , $ACI (\underline{G}) = 0.285$ , $ACI (\bar{G}) = 0.689$ , and $ACI (G) = 0.974$ .

Fig. 2

$G = (\underline{G}, \bar{G})$ the DRFG.

Definition 1.8. Let $G = (Y, Z)$ be a FG on $U^{*}$ . The Wiener index ( $WI$ ) of $G = (Y, Z)$ is

$WI (G) = \sum_{x, y \in U^{*}} (Y) (x) (Z) (y) d_{s} (x, y) .$ Here d_s(x, y) depict value of those geodesics from vertex x to vertex y whose total is minimum in $G = (Y, Z)$ .

Definition 1.9. Let $G = (Y, Z)$ be a DFG on $U^{*}$ . The DFG is complete directed fuzzy graph (CDFG) if for every pair of directed adjacent vertices $Z (x, y) = \min {Y (x), Z (y)} .$

2 Wiener index of a directed rough fuzzy graph

In this segment, we introduced the notion of the Wiener index ( $WI$ ) for DRFG. Practically topological indices are graph invariants that have been used for a long time to create molecules with required characteristics, such as in drug design. $WI$ is also a topological index which is used in a variety of domains, including medical, facility location, communication, and cryptology. There are numerous scenarios which impressively modeled by a DRFG. The conventional explanation to estimate $WI$ of DRFG is provided next.

Definition 2.1. Let a DRFG $G = (\underline{G}, \bar{G}) .$ The Wiener index ( $WI$ ) of $G = (\underline{G}, \bar{G})$ is

$WI (G) = WI (\underline{G}) + WI (\bar{G}),$ where

$WI (\underline{G}) = \sum_{x, y \in (\underline{R} Y)^{*}} (\underline{R} Y) (x) (\underline{R} Y) (y) d_{s_{\underline{G}}} (x, y),$ $WI (\bar{G}) = \sum_{x, y \in (\bar{R} Y)^{*}} (\bar{R} Y) (x) (\bar{R} Y) (y) d_{s_{\bar{G}}} (x, y) .$ Here $d_{s_{\underline{G}}} (x, y)$ and $d_{s_{\bar{G}}} (x, y)$ depict values of those geodesics from vertex x to vertex y whose total is minimum in $\underline{G}$ as well as in $\bar{G},$ respectively.

Example 2.2. Suppose a DRFG $G = (\underline{G}, \bar{G})$ on $U^{*} = {x_{1}, x_{2}, x_{3}, x_{4}} .$ Therefore, $\underline{G} = (\underline{R} Y, \underline{T} Z)$ and $\bar{G} = (\bar{R} Y, \bar{T} Z)$ in Fig. 3. depict DFGs. By direct computations, the weight of geodesic between vertices of DRFG whose sum is minimum are as follows: $\begin{matrix} d_{s_{\underline{G}}} (x_{1}, x_{2}) = 0.7, d_{s_{\bar{G}}} (x_{1}, x_{2}) = 0.7, d_{s_{\underline{G}}} (x_{2}, x_{1}) = 0.7, d_{s_{\bar{G}}} (x_{2}, x_{1}) = 1.0, \\ d_{s_{\underline{G}}} (x_{1}, x_{3}) = 0.5, d_{s_{\bar{G}}} (x_{1}, x_{3}) = 0.6, d_{s_{\underline{G}}} (x_{3}, x_{1}) = 0.9, d_{s_{\bar{G}}} (x_{3}, x_{1}) = 1.4, \\ d_{s_{\underline{G}}} (x_{1}, x_{4}) = 1.2, d_{s_{\bar{G}}} (x_{1}, x_{4}) = 1.3, d_{s_{\underline{G}}} (x_{4}, x_{1}) = 0.2, d_{s_{\bar{G}}} (x_{4}, x_{1}) = 0.4, \\ d_{s_{\underline{G}}} (x_{2}, x_{3}) = 1.2, d_{s_{\bar{G}}} (x_{2}, x_{3}) = 1.6, d_{s_{\underline{G}}} (x_{3}, x_{2}) = 0.2, d_{s_{\bar{G}}} (x_{3}, x_{2}) = 0.4, \\ d_{s_{\underline{G}}} (x_{2}, x_{4}) = 0.5, d_{s_{\bar{G}}} (x_{2}, x_{4}) = 0.6, d_{s_{\underline{G}}} (x_{4}, x_{2}) = 0.9, d_{s_{\bar{G}}} (x_{4}, x_{2}) = 1.1, \\ d_{s_{\underline{G}}} (x_{3}, x_{4}) = 0.7, d_{s_{\bar{G}}} (x_{3}, x_{4}) = 1.0, d_{s_{\underline{G}}} (x_{4}, x_{3}) = 0.7, d_{s_{\bar{G}}} (x_{4}, x_{3}) = 1.0 . \end{matrix}$ By above computations and from Fig. 3, we have $WI (\underline{G}) = 3.542,$ $WI (\bar{G}) = 7.985,$ and $WI (G) = 11.527 .$

Fig. 3

$G = (\underline{G}, \bar{G})$ the DRFG.

Note: Suppose $G = (\underline{G}, \bar{G})$ a DRFG and $H = (\underline{H}, \bar{H})$ be the partial DRF-subgraph of $G = (\underline{G}, \bar{G}),$ then it is not compulsory that $WI (H)$ be lower than or equal to $WI (G) .$ This is demonstrated in the succeeding example.

Example 2.3. Consider $G = (\underline{G}, \bar{G})$ the DRFG on $X = {x_{1}, x_{2}, x_{3}, x_{4}} .$ And $H = (\underline{H}, \bar{H})$ a partial DRF-subgraph of $G = (\underline{G}, \bar{G})$ as shown in Figs. 4 and 5, respectively. Here $\underline{G} = (\underline{R} Y, \underline{T} Z),$ $\bar{G} = (\bar{R} Y, \bar{T} Z),$ $\underline{H} = (\underline{R} \tilde{Y}, \underline{T} \tilde{Z})$ and $\bar{H} = (\bar{R} \tilde{Y}, \bar{T} \tilde{Z})$ in Figs. 4 and 5, depict DFGs. By direct computations, the weight of geodesic between vertices of $G = (\underline{G}, \bar{G})$ and $H = (\underline{H}, \bar{H}),$ respectively, whose sum is minimum are as follows: $\begin{matrix} d_{s_{\underline{G}}} (x_{1}, x_{2}) = 0.5, d_{s_{\bar{G}}} (x_{1}, x_{2}) = 0.5, d_{s_{\underline{G}}} (x_{2}, x_{1}) = 0.7, d_{s_{\bar{G}}} (x_{2}, x_{1}) = 0.7, \\ d_{s_{\underline{G}}} (x_{1}, x_{3}) = 0.9, d_{s_{\bar{G}}} (x_{1}, x_{3}) = 0.9, d_{s_{\underline{G}}} (x_{3}, x_{1}) = 0.3, d_{s_{\bar{G}}} (x_{3}, x_{1}) = 0.3, \\ d_{s_{\underline{G}}} (x_{1}, x_{4}) = 1.1, d_{s_{\bar{G}}} (x_{1}, x_{4}) = 1.1, d_{s_{\underline{G}}} (x_{4}, x_{1}) = 0.2, d_{s_{\bar{G}}} (x_{4}, x_{1}) = 0.4, \\ d_{s_{\underline{G}}} (x_{2}, x_{3}) = 0.4, d_{s_{\bar{G}}} (x_{2}, x_{3}) = 0.4, d_{s_{\underline{G}}} (x_{3}, x_{2}) = 0.8, d_{s_{\bar{G}}} (x_{3}, x_{2}) = 0.8, \\ d_{s_{\underline{G}}} (x_{2}, x_{4}) = 0.6, d_{s_{\bar{G}}} (x_{2}, x_{4}) = 0.6, d_{s_{\underline{G}}} (x_{4}, x_{2}) = 0.7, d_{s_{\bar{G}}} (x_{4}, x_{2}) = 0.9, \\ d_{s_{\underline{G}}} (x_{3}, x_{4}) = 1.4, d_{s_{\bar{G}}} (x_{3}, x_{4}) = 1.4, d_{s_{\underline{G}}} (x_{4}, x_{3}) = 0.2, d_{s_{\bar{G}}} (x_{4}, x_{3}) = 0.4, \end{matrix}$ and $\begin{matrix} d_{s_{\underline{H}}} (x_{1}, x_{2}) = 0.5, d_{s_{\bar{H}}} (x_{1}, x_{2}) = 0.5, d_{s_{\underline{H}}} (x_{2}, x_{1}) = 0.7, d_{s_{\bar{H}}} (x_{2}, x_{1}) = 0.7, \\ d_{s_{\underline{H}}} (x_{1}, x_{3}) = 0.9, d_{s_{\bar{H}}} (x_{1}, x_{3}) = 0.9, d_{s_{\underline{H}}} (x_{3}, x_{1}) = 0.3, d_{s_{\bar{H}}} (x_{3}, x_{1}) = 0.3, \\ d_{s_{\underline{H}}} (x_{1}, x_{4}) = 1.1, d_{s_{\bar{H}}} (x_{1}, x_{4}) = 1.1, d_{s_{\underline{H}}} (x_{4}, x_{1}) = 0.5, d_{s_{\bar{H}}} (x_{4}, x_{1}) = 0.7, \\ d_{s_{\underline{H}}} (x_{2}, x_{3}) = 0.4, d_{s_{\bar{H}}} (x_{2}, x_{3}) = 0.4, d_{s_{\underline{H}}} (x_{3}, x_{2}) = 0.8, d_{s_{\bar{H}}} (x_{3}, x_{2}) = 0.8, \\ d_{s_{\underline{H}}} (x_{2}, x_{4}) = 0.6, d_{s_{\bar{H}}} (x_{2}, x_{4}) = 0.6, d_{s_{\underline{H}}} (x_{4}, x_{2}) = 1.0, d_{s_{\bar{H}}} (x_{4}, x_{2}) = 1.2, \\ d_{s_{\underline{H}}} (x_{3}, x_{4}) = 1.4, d_{s_{\bar{H}}} (x_{3}, x_{4}) = 1.4, d_{s_{\underline{H}}} (x_{4}, x_{3}) = 0.2, d_{s_{\bar{H}}} (x_{4}, x_{3}) = 0.4 . \end{matrix}$ By above computations and from Figs. 4 and 5, we have $WI (\underline{G}) = 2.556$ , $WI (\bar{G}) = 4.326$ , and $WI (G) = 6.882$ . $WI (\underline{H}) = 2.772$ , $WI (\bar{H}) = 4.596$ , and $WI (H) = 7.368$ .

Clearly, from above calculations $WI (G) < WI (H) .$

Fig. 4

DRFG $G = (\underline{G}, \bar{G})$ with $WI (G) < WI (H)$ .

Fig. 5

$H = (\underline{H}, \bar{H})$ the partial DRF-subgraph of $G = (\underline{G}, \bar{G})$ .

Theorem 2.4. Consider a DRFG $G_{1} = ({\underline{G}}_{1}, {\bar{G}}_{1})$ and a DRFG $G_{2} = ({\underline{G}}_{2}, {\bar{G}}_{2})$ and G₁ ≅ G₂ then $WI (G_{1}) = WI (G_{2}) .$

Proof. Let $G_{1} = ({\underline{G}}_{1}, {\bar{G}}_{1})$ and $G_{2} = ({\underline{G}}_{2}, {\bar{G}}_{2})$ be two DRFGs. Also $G_{1} = ({\underline{G}}_{1}, {\bar{G}}_{1})$ and $G_{2} = ({\underline{G}}_{2}, {\bar{G}}_{2})$ are isomorphic graphs such that G₁ ≅ G₂ . Then there is a pair of bijective mappings $(\underline{b}, \bar{b}) : U^{*} \to U^{*}$ such that

(i) $(\underline{R} Y_{1}) (x_{1}) = (\underline{R} Y_{2}) (\underline{b} (x_{1})),$

$(\underline{T} Z_{1}) (x_{1} x_{2}) = (\underline{T} Z_{2}) (\underline{b} (x_{1}) \underline{b} (x_{2})), \forall x_{1} \in U^{*}, x_{1} x_{2} \in U^{*} \times U^{*} .$

(ii) $(\bar{R} Y_{1}) (x_{1}) = (\bar{R} Y_{2}) (\bar{b} (x_{1})),$

$(\bar{T} Z_{1}) (x_{1} x_{2}) = (\bar{T} Z_{2}) (\bar{b} (x_{1}) \bar{b} (x_{2})), \forall x_{1} \in U^{*}, x_{1} x_{2} \in U^{*} \times U^{*} .$

First we consider (i), from above definition of isomorphism. For $x_{1}, x_{2} \in (\underline{R} Y_{1})^{*},$ let ${\underline{P}}_{1}$ be the directed x₁ - x₂ path in ${\underline{G}}_{1}$ which gives $d_{s_{{\underline{G}}_{1}}} (x_{1}, x_{2}) .$ Similarly for every directed edge $w_{1} w_{2} \in {\underline{P}}_{1},$ there corresponds an directed edge $\underline{b} (w_{1}) \underline{b} (w_{2})$ in ${\underline{G}}_{2}$ such that $(\underline{T} Z_{1}) (w_{1} w_{2}) = (\underline{T} Z_{2}) (\underline{b} (w_{1}) \underline{b} (w_{2})) .$ Thus straightforwardly we can conclude that corresponding to the x₁ - x₂ path ${\underline{P}}_{1}$ in ${\underline{G}}_{1},$ there is a directed $\underline{b} (x_{1}) - \underline{b} (x_{2})$ path ${\underline{P}}_{2}$ in ${\underline{G}}_{2}$ such that total of the degrees of membership of edges of ${\underline{P}}_{2}$ is smallest among every single shortest strong directed paths from $\underline{b} (w_{1})$ to $\underline{b} (w_{2}) .$ Therefore, $d_{s_{{\underline{G}}_{1}}} (x_{1}, x_{2}) = d_{s_{{\underline{G}}_{2}}} (\underline{b} (x_{1}), \underline{b} (x_{2})) .$

Hence

$WI ({\underline{G}}_{1}) = \sum_{x_{1}, x_{2} \in (\underline{R} Y_{1})^{*}} (\underline{R} Y_{1}) (x_{1}) (\underline{R} Y_{1}) (x_{2}) d_{s_{{\underline{G}}_{1}}} (x_{1}, x_{2})$ $WI ({\underline{G}}_{1}) = \sum_{\underline{b} (x_{1}), \underline{b} (x_{2}) \in (\underline{R} Y_{2})^{*}} (\underline{R} Y_{2}) \underline{b} (x_{1}) (\underline{R} Y_{2}) \underline{b} (x_{2}) d_{s_{{\underline{G}}_{2}}} (\underline{b} (x_{1}), \underline{b} (x_{2})) = WI ({\underline{G}}_{2})$ Thus, $WI ({\underline{G}}_{1}) ≅ WI ({\underline{G}}_{2}) .$ Similarly, $WI ({\bar{G}}_{1}) ≅ WI ({\bar{G}}_{2}) .$

Hence, $WI (G_{1}) ≅ WI (G_{2}) .$

3 Relationship between

WI

and

CI

of a DRFG

In this section, we discuss the link between $CI$ and $WI$ of $G = (\underline{G}, \bar{G})$ a DRFG. To understand the connection between $CI (G)$ and $WI (G)$ some related results and important examples are presented below.

Example 3.1. Suppose $G = (\underline{G}, \bar{G})$ a DRFG on $U^{*} = {x_{1}, x_{2}, x_{3}} .$ Therefore, $\underline{G} = (\underline{R} Y, \underline{T} Z)$ and $\bar{G} = (\bar{R} Y, \bar{T} Z)$ in Fig. 6. depict DFGs. By direct computations, the $SC$ and the weight of geodesic between vertices of DRFG whose sum is minimum are as follows: $\begin{matrix} {CONN}_{\underline{G}} (x_{1}, x_{2}) = 0.3, {CONN}_{\bar{G}} (x_{1}, x_{2}) = 0.3, {CONN}_{\underline{G}} (x_{2}, x_{1}) = 0.0, {CONN}_{\bar{G}} (x_{2}, x_{1}) = 0.0, \\ {CONN}_{\underline{G}} (x_{1}, x_{3}) = 0.0, {CONN}_{\bar{G}} (x_{1}, x_{3}) = 0.0, {CONN}_{\underline{G}} (x_{3}, x_{1}) = 0.0, {CONN}_{\bar{G}} (x_{3}, x_{1}) = 0.0, \\ {CONN}_{\underline{G}} (x_{2}, x_{3}) = 0.0, {CONN}_{\bar{G}} (x_{2}, x_{3}) = 0.0, {CONN}_{\underline{G}} (x_{3}, x_{2}) = 0.1, {CONN}_{\bar{G}} (x_{3}, x_{2}) = 0.3, \end{matrix}$ and $\begin{matrix} d_{s_{\underline{G}}} (x_{1}, x_{2}) = 0.3, d_{s_{\bar{G}}} (x_{1}, x_{2}) = 0.3, d_{s_{\underline{G}}} (x_{2}, x_{1}) = 0.0, d_{s_{\bar{G}}} (x_{2}, x_{1}) = 0.0, \\ d_{s_{\underline{G}}} (x_{1}, x_{3}) = 0.0, d_{s_{\bar{G}}} (x_{1}, x_{3}) = 0.0, d_{s_{\underline{G}}} (x_{3}, x_{1}) = 0.0, d_{s_{\bar{G}}} (x_{3}, x_{1}) = 0.0, \\ d_{s_{\underline{G}}} (x_{2}, x_{3}) = 0.0, d_{s_{\bar{G}}} (x_{2}, x_{3}) = 0.0, d_{s_{\underline{G}}} (x_{3}, x_{2}) = 0.1, d_{s_{\bar{G}}} (x_{3}, x_{2}) = 0.3 . \end{matrix}$ By above computations and from Fig. 6, we get

$CI (\underline{G}) = 0.092,$ $CI (\bar{G}) = 0.132,$ and $CI (G) = 0.224,$

$WI (\underline{G}) = 0.092,$ $WI (\bar{G}) = 0.132,$ and $WI (G) = 0.224 .$

Clearly, from above calculations, $CI (G) = WI (G) .$

Fig. 6

DRFG $G = (\underline{G}, \bar{G})$ with $CI (G) = WI (G)$ .

Remark: In $G = (\underline{G}, \bar{G})$ a DRFG, if $CI (\underline{G}) = WI (\underline{G})$ and $CI (\bar{G}) = WI (\bar{G})$ then $CI (G) = WI (G) .$

Note: For DRFG in Example 3.1, it could be noted that $CI (G) = WI (G) .$ This equality, however, does not have to occur all of the time. Consider the succeeding example to demonstrate this.

Example 3.2. Suppose a DRFG $G = (\underline{G}, \bar{G})$ on $U^{*} = {x_{1}, x_{2}, x_{3}} .$ Therefore, $\underline{G} = (\underline{R} Y, \underline{T} Z)$ and $\bar{G} = (\bar{R} Y, \bar{T} Z)$ in Fig. 7. depict DFGs. By direct computations, the $SC$ and the weight of geodesic between vertices of DRFG whose sum is minimum are as follows: $\begin{matrix} {CONN}_{\underline{G}} (x_{1}, x_{2}) = 0.5, {CONN}_{\bar{G}} (x_{1}, x_{2}) = 0.6, {CONN}_{\underline{G}} (x_{2}, x_{1}) = 0.0, {CONN}_{\bar{G}} (x_{2}, x_{1}) = 0.0, \\ {CONN}_{\underline{G}} (x_{1}, x_{3}) = 0.0, {CONN}_{\bar{G}} (x_{1}, x_{3}) = 0.0, {CONN}_{\underline{G}} (x_{3}, x_{1}) = 0.2, {CONN}_{\bar{G}} (x_{3}, x_{1}) = 0.4, \\ {CONN}_{\underline{G}} (x_{2}, x_{3}) = 0.0, {CONN}_{\bar{G}} (x_{2}, x_{3}) = 0.0, {CONN}_{\underline{G}} (x_{3}, x_{2}) = 0.3, {CONN}_{\bar{G}} (x_{3}, x_{2}) = 0.4, \end{matrix}$ and $\begin{matrix} d_{s_{\underline{G}}} (x_{1}, x_{2}) = 0.5, d_{s_{\bar{G}}} (x_{1}, x_{2}) = 0.6, d_{s_{\underline{G}}} (x_{2}, x_{1}) = 0.0, d_{s_{\bar{G}}} (x_{2}, x_{1}) = 0.0, \\ d_{s_{\underline{G}}} (x_{1}, x_{3}) = 0.0, d_{s_{\bar{G}}} (x_{1}, x_{3}) = 0.0, d_{s_{\underline{G}}} (x_{3}, x_{1}) = 0.2, d_{s_{\bar{G}}} (x_{3}, x_{1}) = 0.4, \\ d_{s_{\underline{G}}} (x_{2}, x_{3}) = 0.0, d_{s_{\bar{G}}} (x_{2}, x_{3}) = 0.0, d_{s_{\underline{G}}} (x_{3}, x_{2}) = 0.7, d_{s_{\bar{G}}} (x_{3}, x_{2}) = 1.0, \end{matrix}$ By above computations and from Fig. 7, we obtain

$CI (\underline{G}) = 0.402,$ $CI (\bar{G}) = 0.976,$ and $CI (G) = 1.378 .$

$WI (\underline{G}) = 0.581,$ $WI (\bar{G}) = 1.414,$ and $WI (G) = 1.995 .$

Fig. 7

$G = (\underline{G}, \bar{G})$ the DRFG.

Note: For DRFG in Example 3.2, we found that $CI (G) \neq WI (G) .$ So we can say that a DRFG with 3 vertices may have different values for $CI (G)$ and $WI (G) .$

Example 3.3. Suppose $G = (\underline{G}, \bar{G})$ a DRFG on $U^{*} = {x_{1}, x_{2}, x_{3}, x_{4}} .$ Therefore, $\underline{G} = (\underline{R} Y, \underline{T} Z)$ and $\bar{G} = (\bar{R} Y, \bar{T} Z)$ in Fig. 8. depict DFGs. By direct computations, the $SC$ and the weight of geodesic between vertices of DRFG whose sum is minimum are as follows: $\begin{matrix} {CONN}_{\underline{G}} (x_{1}, x_{2}) = 0.7, {CONN}_{\bar{G}} (x_{1}, x_{2}) = 0.7, {CONN}_{\underline{G}} (x_{2}, x_{1}) = 0.2, {CONN}_{\bar{G}} (x_{2}, x_{1}) = 0.4, \\ {CONN}_{\underline{G}} (x_{1}, x_{3}) = 0.5, {CONN}_{\bar{G}} (x_{1}, x_{3}) = 0.6, {CONN}_{\underline{G}} (x_{3}, x_{1}) = 0.2, {CONN}_{\bar{G}} (x_{3}, x_{1}) = 0.4, \\ {CONN}_{\underline{G}} (x_{1}, x_{4}) = 0.2, {CONN}_{\bar{G}} (x_{1}, x_{4}) = 0.6, {CONN}_{\underline{G}} (x_{4}, x_{1}) = 0.2, {CONN}_{\bar{G}} (x_{4}, x_{1}) = 0.4, \\ {CONN}_{\underline{G}} (x_{2}, x_{3}) = 0.2, {CONN}_{\bar{G}} (x_{2}, x_{3}) = 0.4, {CONN}_{\underline{G}} (x_{3}, x_{2}) = 0.2, {CONN}_{\bar{G}} (x_{3}, x_{2}) = 0.4, \\ {CONN}_{\underline{G}} (x_{2}, x_{4}) = 0.5, {CONN}_{\bar{G}} (x_{2}, x_{4}) = 0.6, {CONN}_{\underline{G}} (x_{4}, x_{2}) = 0.2, {CONN}_{\bar{G}} (x_{4}, x_{2}) = 0.4, \\ {CONN}_{\underline{G}} (x_{3}, x_{4}) = 0.2, {CONN}_{\bar{G}} (x_{3}, x_{4}) = 0.4, {CONN}_{\underline{G}} (x_{4}, x_{3}) = 0.3, {CONN}_{\bar{G}} (x_{4}, x_{3}) = 0.4, \end{matrix}$ and $\begin{matrix} d_{s_{\underline{G}}} (x_{1}, x_{2}) = 0.7, d_{s_{\bar{G}}} (x_{1}, x_{2}) = 0.7, d_{s_{\underline{G}}} (x_{2}, x_{1}) = 0.7, d_{s_{\bar{G}}} (x_{2}, x_{1}) = 1.0, \\ d_{s_{\underline{G}}} (x_{1}, x_{3}) = 0.5, d_{s_{\bar{G}}} (x_{1}, x_{3}) = 0.6, d_{s_{\underline{G}}} (x_{3}, x_{1}) = 0.9, d_{s_{\bar{G}}} (x_{3}, x_{1}) = 1.4, \\ d_{s_{\underline{G}}} (x_{1}, x_{4}) = 1.2, d_{s_{\bar{G}}} (x_{1}, x_{4}) = 1.3, d_{s_{\underline{G}}} (x_{4}, x_{1}) = 0.2, d_{s_{\bar{G}}} (x_{4}, x_{1}) = 0.4, \\ d_{s_{\underline{G}}} (x_{2}, x_{3}) = 1.2, d_{s_{\bar{G}}} (x_{2}, x_{3}) = 1.6, d_{s_{\underline{G}}} (x_{3}, x_{2}) = 0.2, d_{s_{\bar{G}}} (x_{3}, x_{2}) = 0.4, \\ d_{s_{\underline{G}}} (x_{2}, x_{4}) = 0.5, d_{s_{\bar{G}}} (x_{2}, x_{4}) = 0.6, d_{s_{\underline{G}}} (x_{4}, x_{2}) = 0.9, d_{s_{\bar{G}}} (x_{4}, x_{2}) = 1.1, \\ d_{s_{\underline{G}}} (x_{3}, x_{4}) = 0.7, d_{s_{\bar{G}}} (x_{3}, x_{4}) = 1.0, d_{s_{\underline{G}}} (x_{4}, x_{3}) = 0.7, d_{s_{\bar{G}}} (x_{4}, x_{3}) = 1.0 . \end{matrix}$ By above computations and from Fig. 8, we get

$CI (\underline{G}) = 1.713,$ $CI (\bar{G}) = 4.139,$ and $CI (G) = 5.852,$

$WI (\underline{G}) = 3.542,$ $WI (\bar{G}) = 7.985,$ and $WI (G) = 11.527 .$

Clearly, from above calculations, $CI (G) < WI (G) .$

Fig. 8

DRFG $G = (\underline{G}, \bar{G})$ with $CI (G) < WI (G)$ .

Note: For DRFG with 4 vertices as shown in Example 3.3, it could be noted that $CI (G) < WI (G) .$ This inequality, however, does not have to occur all of the time. Consider the following example to demonstrate this.

Example 3.4. Suppose $G = (\underline{G}, \bar{G})$ a DRFG on $U^{*} = {x_{1}, x_{2}, x_{3}, x_{4}} .$ Therefore, $\underline{G} = (\underline{R} Y, \underline{T} Z)$ and $\bar{G} = (\bar{R} Y, \bar{T} Z)$ in Fig. 9. depict DFGs. By direct computations, the $SC$ and the weight of geodesic between vertices of DRFG whose sum is minimum are as follows: $\begin{matrix} {CONN}_{\underline{G}} (x_{1}, x_{2}) = 0.3, {CONN}_{\bar{G}} (x_{1}, x_{2}) = 0.7, {CONN}_{\underline{G}} (x_{2}, x_{1}) = 0.3, {CONN}_{\bar{G}} (x_{2}, x_{1}) = 0.7, \\ {CONN}_{\underline{G}} (x_{1}, x_{3}) = 0.3, {CONN}_{\bar{G}} (x_{1}, x_{3}) = 0.7, {CONN}_{\underline{G}} (x_{3}, x_{1}) = 0.3, {CONN}_{\bar{G}} (x_{3}, x_{1}) = 0.7, \\ {CONN}_{\underline{G}} (x_{1}, x_{4}) = 0.5, {CONN}_{\bar{G}} (x_{1}, x_{4}) = 0.8, {CONN}_{\underline{G}} (x_{4}, x_{1}) = 0.5, {CONN}_{\bar{G}} (x_{4}, x_{1}) = 0.8, \\ {CONN}_{\underline{G}} (x_{2}, x_{3}) = 0.3, {CONN}_{\bar{G}} (x_{2}, x_{3}) = 0.7, {CONN}_{\underline{G}} (x_{3}, x_{2}) = 0.3, {CONN}_{\bar{G}} (x_{3}, x_{2}) = 0.7, \\ {CONN}_{\underline{G}} (x_{2}, x_{4}) = 0.3, {CONN}_{\bar{G}} (x_{2}, x_{4}) = 0.7, {CONN}_{\underline{G}} (x_{4}, x_{2}) = 0.3, {CONN}_{\bar{G}} (x_{4}, x_{2}) = 0.7, \\ {CONN}_{\underline{G}} (x_{3}, x_{4}) = 0.3, {CONN}_{\bar{G}} (x_{3}, x_{4}) = 0.7, {CONN}_{\underline{G}} (x_{4}, x_{3}) = 0.3, {CONN}_{\bar{G}} (x_{4}, x_{3}) = 0.7, \end{matrix}$ and $\begin{matrix} d_{s_{\underline{G}}} (x_{1}, x_{2}) = 0.3, d_{s_{\bar{G}}} (x_{1}, x_{2}) = 0.7, d_{s_{\underline{G}}} (x_{2}, x_{1}) = 0.3, d_{s_{\bar{G}}} (x_{2}, x_{1}) = 0.7, \\ d_{s_{\underline{G}}} (x_{1}, x_{3}) = 0.3, d_{s_{\bar{G}}} (x_{1}, x_{3}) = 0.7, d_{s_{\underline{G}}} (x_{3}, x_{1}) = 0.3, d_{s_{\bar{G}}} (x_{3}, x_{1}) = 0.7, \\ d_{s_{\underline{G}}} (x_{1}, x_{4}) = 0.5, d_{s_{\bar{G}}} (x_{1}, x_{4}) = 0.8, d_{s_{\underline{G}}} (x_{4}, x_{1}) = 0.5, d_{s_{\bar{G}}} (x_{4}, x_{1}) = 0.8 \\ d_{s_{\underline{G}}} (x_{2}, x_{3}) = 0.3, d_{s_{\bar{G}}} (x_{2}, x_{3}) = 0.7, d_{s_{\underline{G}}} (x_{3}, x_{2}) = 0.3, d_{s_{\bar{G}}} (x_{3}, x_{2}) = 0.7, \\ d_{s_{\underline{G}}} (x_{2}, x_{4}) = 0.3, d_{s_{\bar{G}}} (x_{2}, x_{4}) = 0.7, d_{s_{\underline{G}}} (x_{4}, x_{2}) = 0.3, d_{s_{\bar{G}}} (x_{4}, x_{2}) = 0.7, \\ d_{s_{\underline{G}}} (x_{3}, x_{4}) = 0.3, d_{s_{\bar{G}}} (x_{3}, x_{4}) = 0.7, d_{s_{\underline{G}}} (x_{4}, x_{3}) = 0.3, d_{s_{\bar{G}}} (x_{4}, x_{3}) = 0.7 . \end{matrix}$

By above computations and from Fig. 9, we get $CI (\underline{G}) = 0.664$ , $CI (\bar{G}) = 4.846$ , and $CI (G) = 5.51$ , $WI (\underline{G}) = 0.664$ , $WI (\bar{G}) = 4.846$ , and $WI (G) = 5.51$ .

Clearly, from above calculations, $CI (G) = WI (G) .$

Fig. 9

$G = (\underline{G}, \bar{G})$ the DRFG.

Note: For DRFG in Example 3.4, we found that $CI (G) = WI (G) .$ So we can say that a DRFG with 4 vertices may have same values for $CI (G)$ and $WI (G) .$

Definition 3.5. A DRFG $G = (\underline{G}, \bar{G})$ is said to be a complete directed rough fuzzy graph (CDRFG) if it is complete directed fuzzy graph (CDFG) in $\underline{G}$ and $\bar{G},$ respectively. Mathematically we can write: $(\underline{T} Z) (x_{1} x_{2}) = \min {(\underline{R} Y) (x_{1}), (\underline{R} Y) (x_{2})}$ $(\bar{T} Z) (x_{1} x_{2}) = \min {(\bar{R} Y) (x_{1}), (\bar{R} Y) (x_{2})}, \forall x_{1}, x_{2} \in U^{*} .$

Lemma 3.6. Consider a CDFG $\underline{G} .$ Then $\underline{G}$ have no δ DF-edges.

Proof. Consider a CDFG $\underline{G} .$ Suppose on contrary that G have a δ DF-edge, x₁x₂ . Therefore, $(\underline{T} Z) (x_{1} x_{2}) < {CONN}_{\underline{G} - (x_{1} x_{2}) (x_{1}, x_{2})} .$ That is, there is a stronger directed path $P$ other than the edge x₁x₂ from x₁ to x₂ in $\underline{G} .$ Take $(\underline{T} Z) (x_{1} x_{2}) = t$ and the strength of directed path $P$ be s . This implies t < s . Now choose u to be the first vertex in the directed path $P$ after x₁. Then, $(\underline{T} Z) (x_{1} u) > t (1)$ In the similar manner, let v be the last vertex in $P$ before x₂. Then, $(\underline{T} Z) ({vx}_{2}) > t (2)$ Since $(\underline{T} Z) (x_{1} x_{2}) = t,$ at least one of $(\underline{R} Y) (x_{1})$ or $(\underline{R} Y) (x_{2})$ must be t. Therefore, $\underline{G}$ being a CDFG, (1) provides a contradiction if $(\underline{R} Y) (x_{1}) = t$ and (2) provides a contradiction if $(\underline{R} Y) = t$ ; which completes the proof.

Example 3.7. Suppose $G = (\underline{G}, \bar{G})$ a CDRFG on $U^{*} = {x_{1}, x_{2}, x_{3}} .$ Therefore, $\underline{G} = (\underline{R} Y, \underline{T} Z)$ and $\bar{G} = (\bar{R} Y, \bar{T} Z)$ in Fig. 10. depict CDFGs. By direct calculation, all DRF-edges are strong DRF-edges or no δ DRF-edge exist in CDRFG.

Fig. 10

$G = (\underline{G}, \bar{G})$ the CDRFG.

Lemma 3.8. There exists exactly one α-strong DF-edge in a CDFG.

Using 3.6, we can derive the following theorem.

Theorem 3.9. Consider a CDRFG $G = (\underline{G}, \bar{G}) .$ Then $G = (\underline{G}, \bar{G})$ have no δ DRF-edges.

Proof. Suppose a CDRFG $G = (\underline{G}, \bar{G}) .$ Since $\underline{G}$ and $\bar{G}$ are CDFGs. Also from 2, $\underline{G}$ and $\bar{G}$ being CDFGs does not include any δ DF-edges, which completes the proof.

Lemma 3.10. There exists exactly one α strong DRF-edge in a CDRFG.

Theorem 3.11. Consider a DRFG $G = (\underline{G}, \bar{G})$ and fulfills the following requirements.

(i) $G = (\underline{G}, \bar{G})$ with no δ DRF-edge.

(ii) For every two vertices $x_{1}, x_{2} \in U^{*}, x_{1} x_{2} \in U^{*} \times U^{*} .$

Then $CI (G) = WI (G) .$

Proof. Suppose a DRFG $G = (\underline{G}, \bar{G})$ with no δ DRF-edge and for some two vertices $x_{1}, x_{2} \in U^{*}, x_{1} x_{2} \in U^{*} \times U^{*}$ in $G = (\underline{G}, \bar{G}) .$ As $G = (\underline{G}, \bar{G})$ have no δ DRF-edge by hypothesis, all DRF-edges are α strong or β strong DRF-edges. In either case, $(T Z) (x_{1} x_{2}) = {CONN}_{G} (x_{1}, x_{2}) = d_{s_{G}} (x_{1}, x_{2}) .$

That is, $(\underline{T} Z) (x_{1} x_{2}) = {CONN}_{\underline{G}} (x_{1}, x_{2}) = d_{s_{\underline{G}}} (x_{1}, x_{2})$ in $\underline{G}$ and $(\bar{T} Z) (x_{1} x_{2}) = {CONN}_{\bar{G}} (x_{1}, x_{2}) = d_{s_{\bar{G}}} (x_{1}, x_{2})$ in $\bar{G} .$ Clearly from above argument $CI (\underline{G}) = WI (\underline{G})$ and $CI (\bar{G}) = WI (\bar{G})$ holds. Hence $CI (G) = WI (G) .$

Example 3.12. Consider a DRFG $G = (\underline{G}, \bar{G})$ given in Example 3.7 on $U^{*} = {x_{1}, x_{2}, x_{3}} .$ Consequently, $\underline{G} = (\underline{R} Y, \underline{T} Z)$ and $\bar{G} = (\bar{R} Y, \bar{T} Z)$ in Fig. 10. depict DFGs. For every pair $x_{1}, x_{2} \in U^{*}$ there exist $x_{1} x_{2} \in U^{*} \times U^{*}$ and $G = (\underline{G}, \bar{G})$ does not include any δ DRF-edge. Thus by direct calculation and Fig. 10, $CI (G) = WI (G) = 2.186 .$

Corollary 3.13 Consider a CDRFG $G = (\underline{G}, \bar{G}) .$ Then $CI (G) = WI (G) .$

4 Average Wiener index of directed rough fuzzy graph

In this section, we can determine how much flow in the directed network is steady by measuring the average flow. Therefore, a new DRFG parameter entitled average Wiener index ( $AWI$ ) will be introduced.

Definition 4.1. Consider a DRFG $G = (\underline{G}, \bar{G}) .$ The average Wiener index ( $AWI$ ) of $G = (\underline{G}, \bar{G})$ is $AWI (G) = AWI (\underline{G}) + AWI (\bar{G}),$ where $AWI (\underline{G}) = \frac{1}{^{n} P_{2}} \sum_{x, y \in (\underline{R} Y)^{*}} (\underline{R} Y) (x) (\underline{R} Y) (y) d_{s_{\underline{G}}} (x, y),$ $AWI (\bar{G}) = \frac{1}{^{n} P_{2}} \sum_{x, y \in (\bar{R} Y)^{*}} (\bar{R} Y) (x) (\bar{R} Y) (y) d_{s_{\bar{G}}} (x, y) .$

Fig. 11

$G = (\underline{G}, \bar{G})$ the DRFG.

Example 4.2. Suppose a DRFG $G = (\underline{G}, \bar{G})$ given in Example 3.2 on $U^{*} = {x_{1}, x_{2}, x_{3}} .$ Therefore, $\underline{G} = (\underline{R} Y, \underline{T} Z)$ and $\bar{G} = (\bar{R} Y, \bar{T} Z)$ in Fig. 7. depict DFGs. Thus by direct computations and from Fig. 7, we obtain

$WI (\underline{G}) = 0.581$ , $WI (\bar{G}) = 1.414$ , and $WI (G) = 1.995$ .

$AWI (\underline{G}) = 0.096$ , $AWI (\bar{G}) = 0.235$ , and $AWI (G) = 0.331$ .

Definition 4.3. Suppose a DRFG $G = (\underline{G}, \bar{G})$ and some vertex $x_{1} \in U^{*} .$ Then

•x₁ is said to be a Wiener enhancing vertex (WEV) of $G = (\underline{G}, \bar{G})$ if x₁ is WEV in $\underline{G}$ as well as in $\bar{G} .$ That is, $AWI (\underline{G} - x_{1}) > AWI (\underline{G})$ and $AWI (\bar{G} - x_{1}) > AWI (\bar{G}) .$

•x₁ is said to be a Wiener reducing vertex (WRV) of $G = (\underline{G}, \bar{G})$ if

(i) x₁ is WRV in $\underline{G}$ as well as in $\bar{G} .$ That is, $AWI (\underline{G} - x_{1}) < AWI (\underline{G})$ and $AWI (\bar{G} - x_{1}) < AWI (\bar{G}) .$

$OR$

(ii) x₁ is WRV in $\underline{G}$ and WEV in $\bar{G},$ respectively. That is, $AWI (\underline{G} - x_{1}) < AWI (\underline{G})$ and $AWI (\bar{G} - x_{1}) > AWI (\bar{G}) .$ $OR$

(iii) x₁ is WEV in $\underline{G}$ and WRV in $\bar{G},$ respectively. That is, $AWI (\underline{G} - x_{1}) > AWI (\underline{G})$ and $AWI (\bar{G} - x_{1}) < AWI (\bar{G}) .$

•x₁ is said to be a Wiener neutral vertex (WNV) of $G = (\underline{G}, \bar{G})$ if x₁ is WNV in $\underline{G}$ as well as in $\bar{G} .$ That is, $AWI (\underline{G} - x_{1}) = AWI (\underline{G})$ and $AWI (\bar{G} - x_{1}) = AWI (\bar{G}) .$

Definition 4.4. Consider a DRFG $G = (\underline{G}, \bar{G}) .$ Then

• $G = (\underline{G}, \bar{G})$ is said to be a Wiener enhancing directed rough fuzzy graph (WEDRFG) if $G = (\underline{G}, \bar{G})$ have at least one WEV.

• $G = (\underline{G}, \bar{G})$ is said to be a Wiener reducing directed rough fuzzy graph (WRDRFG) if $G = (\underline{G}, \bar{G})$ contains no WEV and minimum one WRV.

• $G = (\underline{G}, \bar{G})$ is said to be a Wiener neutral directed rough fuzzy graph (WNDRFG) if every vertex of $G = (\underline{G}, \bar{G})$ is WNV.

Example 4.5. Consider a DRFG $G = (\underline{G}, \bar{G})$ on $U^{*} = {x_{1}, x_{2}, x_{3}}$ as shown in Fig. 11. By direct computations, the weight of geodesic between vertices of $G = (\underline{G}, \bar{G})$ whose sum is minimum are as follows: $\begin{matrix} d_{s_{\underline{G}}} (x_{1}, x_{2}) = 0.0, d_{s_{\bar{G}}} (x_{1}, x_{2}) = 0.0, d_{s_{\underline{G}}} (x_{2}, x_{1}) = 0.2, d_{s_{\bar{G}}} (x_{2}, x_{1}) = 0.3, \\ d_{s_{\underline{G}}} (x_{1}, x_{3}) = 0.4, d_{s_{\bar{G}}} (x_{1}, x_{3}) = 0.5, d_{s_{\underline{G}}} (x_{3}, x_{1}) = 0.4, d_{s_{\bar{G}}} (x_{3}, x_{1}) = 0.5, \\ d_{s_{\underline{G}}} (x_{2}, x_{3}) = 0.2, d_{s_{\bar{G}}} (x_{2}, x_{3}) = 0.3, d_{s_{\underline{G}}} (x_{3}, x_{2}) = 0.0, d_{s_{\bar{G}}} (x_{3}, x_{2}) = 0.0 . \end{matrix}$ By above computations and from Fig. 11, we have $AWI (\underline{G}) = 0.056,$ $AWI (\bar{G}) = 0.1626,$ and $AWI (G) = 0.2186 .$

If we eliminate the vertex “x₂” so the DRF-subgraph $G - x_{2} = (\underline{G} - x_{2}, \bar{G} - x_{2})$ is represented in Fig. 12. By direct computations, the weight of geodesic between vertices of $G - x_{2} = (\underline{G} - x_{2}, \bar{G} - x_{2}$ whose sum is minimum are as follows: $\begin{matrix} d_{s_{\underline{G}}} (x_{1}, x_{3}) = 0.4, d_{s_{\bar{G}}} (x_{1}, x_{3}) = 0.5, \\ d_{s_{\underline{G}}} (x_{3}, x_{1}) = 0.4, d_{s_{\bar{G}}} (x_{3}, x_{1}) = 0.5 . \end{matrix}$

By above computations and from Fig. 12, we obtain $AWI (\underline{G} - x_{2}) = 0.1 > 0.056 = AWI (\underline{G})$ $AWI (\bar{G} - x_{2}) = 0.32 > 0.1626 = AWI (\bar{G})$ Thus, x₂ is a WEV in $G = (\underline{G}, \bar{G}) .$ In similar manner, we can show that “x₁” and “x₃” both are WRV. Therefore, $G = (\underline{G}, \bar{G})$ is a WEDRFG.

Proposition 4.6. Consider a DRFG $G = (\underline{G}, \bar{G})$ and for any vertex $x_{1} \in U^{*}$ with $n = | U^{*} | \geq 3 .$ Suppose $k = \frac{WI (G)}{WI (G - x_{1})} .$ Then

(i) x₁ is a WHV if $k < \frac{n}{n - 2}$ and vice versa.

(ii) x₁ is a WRV if $k > \frac{n}{n - 2}$ and vice versa.

(iii) x₁ is a WNV if $k = \frac{n}{n - 2}$ and vice versa.

Proof. (i) x₁ is a WHV. $\Leftrightarrow AWI (\underline{G} - x_{1}) > AWI (\underline{G})$ and $AWI (\bar{G} - x_{1}) > AWI (\bar{G}) .$ $\Leftrightarrow \frac{1}{^{(n - 1)} P_{2}} WI (\underline{G} - x_{1}) > \frac{1}{^{n} P_{2}} WI (\underline{G}),$ and $\frac{1}{^{(n - 1)} P_{2}} WI (\bar{G} - x_{1}) > \frac{1}{^{n} P_{2}} WI (\bar{G}) .$ $\Leftrightarrow \frac{1}{^{(n - 1)} P_{2}} WI (G - x_{1}) > \frac{1}{^{n} P_{2}} WI (G) .$ $\Leftrightarrow \frac{WI (G)}{WI (G - x_{1})} < \frac{n}{n - 2} \Leftrightarrow k < \frac{n}{n - 2} .$

Similarly other two cases can be proved.

Fig. 12

$G - x_{2} = (\underline{G} - x_{2}, \bar{G} - x_{2})$ the DRFG.

5 Application

This section builds on the author’s work on human trafficking, which can be found in [4]. A dealing or trading in a commodity or service, which is frequently illegal is called trafficking. According to international laws trafficking of all kind like: human trafficking, illegal weapons, expensive animals and drug trafficking is serious crime. Trafficking in people, also known as human trafficking, is a kind of modern-day slavery that includes the deceiving movement of people for the sake of labour, sex slavery, or economic benefit for others. A growing number of migrants from Asia are attempting to enter Europe’s various countries. However, a large number of people from Pakistan are involved in trafficking of human.

In this application we apply the concept of $WI$ to find the much effective countries paths of human trafficking (HT) from Pakistan for DRFG model. Akram and zafar [4] in 2019 use the concept of $CI$ in human trafficking model of DRFG. In this paper, we will use the same human trafficking model of DRFG.

Consider $G = (\underline{G}, \bar{G})$ the DRFG on $U^{*} = {x_{1} = Greece, x_{2} = Iran, x_{3} = Middle East, x_{4} = Oman, x_{5} = Pakistan, x_{6} = Spain, x_{7} = Turkey, x_{8} = UAE, x_{9} = West Africa}$ as shown in Fig. 13. We create an ER $R$ on $U^{*}$ which depicts that all destination countries (terminal countries) go to identical equivalence class (EC). Similarly, all passing and all source or originated countries go to two identical ECs, respectively. Let $Y$ be a Fs on $U^{*}$ which shows the vulnerability of every country and $RY = (\underline{R} Y, \bar{R} Y)$ a RFs. Let $Z^{*}$ be a subset of $U^{*} \times U^{*}$ and $T$ be ER on $Z^{*},$ where $T$ describes the ECs of “relationships among various countries”. Let $Z$ be a Fs on $Z^{*}$ which depict the degree of membership of unlawful migration from one country to another. Let $TZ = (\underline{T} Z, \bar{T} Z)$ be a RFR, where $\underline{T} Z$ is lower approximation and $\bar{T} Z$ is upper approximation of $Z$ . Here, $\underline{G}$ and $\bar{G}$ in Fig. 13 depict DFGs. From Figs. 13 and 14 with conventional computations, we obtain $CI (\underline{G}) = 0.3372,$ $CI (\bar{G}) = 1.3248,$ and $CI (G) = 1.662$ $CI (\underline{G} - x_{2}) = 0.2592,$ $CI (\bar{G} - x_{2}) = 0.7778,$ and $CI (G - x_{2}) = 1.037$ By direct computations, the weight of geodesic between vertices of $G = (\underline{G}, \bar{G})$ whose sum is minimum are as follows: $\begin{matrix} d_{s_{\underline{G}}} (x_{2}, x_{1}) = 0.31, d_{s_{\bar{G}}} (x_{2}, x_{1}) = 0.44, \\ d_{s_{\underline{G}}} (x_{2}, x_{4}) = 0.15, d_{s_{\bar{G}}} (x_{2}, x_{4}) = 0.20, \\ d_{s_{\underline{G}}} (x_{2}, x_{7}) = 0.15, d_{s_{\bar{G}}} (x_{2}, x_{7}) = 0.20, \\ d_{s_{\underline{G}}} (x_{2}, x_{8}) = 0.31, d_{s_{\bar{G}}} (x_{2}, x_{8}) = 0.44, \\ d_{s_{\underline{G}}} (x_{3}, x_{6}) = 0.60, d_{s_{\bar{G}}} (x_{3}, x_{6}) = 0.60, \\ d_{s_{\underline{G}}} (x_{3}, x_{9}) = 0.30, d_{s_{\bar{G}}} (x_{3}, x_{9}) = 0.30, \\ d_{s_{\underline{G}}} (x_{4}, x_{8}) = 0.16, d_{s_{\bar{G}}} (x_{4}, x_{8}) = 0.24, \\ d_{s_{\underline{G}}} (x_{5}, x_{1}) = 0.41, d_{s_{\bar{G}}} (x_{5}, x_{1}) = 0.54, \\ d_{s_{\underline{G}}} (x_{5}, x_{2}) = 0.10, d_{s_{\bar{G}}} (x_{5}, x_{2}) = 0.10, \\ d_{s_{\underline{G}}} (x_{5}, x_{3}) = 0.15, d_{s_{\bar{G}}} (x_{5}, x_{3}) = 0.15, \\ d_{s_{\underline{G}}} (x_{5}, x_{4}) = 0.25, d_{s_{\bar{G}}} (x_{5}, x_{4}) = 0.30, \\ d_{s_{\underline{G}}} (x_{5}, x_{8}) = 0.41, d_{s_{\bar{G}}} (x_{5}, x_{8}) = 0.54, \\ d_{s_{\underline{G}}} (x_{5}, x_{7}) = 0.25, d_{s_{\bar{G}}} (x_{5}, x_{7}) = 0.30, \\ d_{s_{\underline{G}}} (x_{5}, x_{9}) = 0.45, d_{s_{\bar{G}}} (x_{5}, x_{9}) = 0.45, \\ d_{s_{\underline{G}}} (x_{5}, x_{6}) = 0.75, d_{s_{\bar{G}}} (x_{5}, x_{6}) = 0.75, \\ d_{s_{\underline{G}}} (x_{7}, x_{1}) = 0.16, d_{s_{\bar{G}}} (x_{7}, x_{1}) = 0.24, \\ d_{s_{\underline{G}}} (x_{9}, x_{6}) = 0.30, d_{s_{\bar{G}}} (x_{9}, x_{6}) = 0.30 . \end{matrix}$ By above computations and from Fig. 13, we otain $WI (\underline{G}) = 0.729,$ $WI (\bar{G}) = 2.5602,$ and $WI (G) = 3.2892 .$ $AWI (\underline{G}) = 0.010125,$ $AWI (\bar{G}) = 0.03555,$ and $AWI (G) = 0.045675 .$

Fig. 13

$G = (\underline{G}, \bar{G})$ the DRFG.

If we eliminate the country “x₂ = Iran” from $G = (\underline{G}, \bar{G})$ then the DRF-subgraph $G - x_{2} = (\underline{G} - x_{2}, \bar{G} - x_{2})$ is represented in Fig. 14. By direct computations, the weight of geodesic between vertices of $G - x_{2} = (\underline{G} - x_{2}, \bar{G} - x_{2})$ whose sum is minimum are as follows: $\begin{matrix} d_{s_{\underline{G} - x_{2}}} (x_{3}, x_{6}) = 0.60, d_{s_{\bar{G} - x_{2}}} (x_{3}, x_{6}) = 0.60, \\ d_{s_{\underline{G} - x_{2}}} (x_{3}, x_{9}) = 0.30, d_{s_{\bar{G} - x_{2}}} (x_{3}, x_{9}) = 0.30, \\ d_{s_{\underline{G} - x_{2}}} (x_{4}, x_{8}) = 0.16, d_{s_{\bar{G} - x_{2}}} (x_{4}, x_{8}) = 0.24, \\ d_{s_{\underline{G} - x_{2}}} (x_{5}, x_{1}) = 0.41, d_{s_{\bar{G} - x_{2}}} (x_{5}, x_{1}) = 0.54, \\ d_{s_{\underline{G} - x_{2}}} (x_{5}, x_{3}) = 0.15, d_{s_{\bar{G} - x_{2}}} (x_{5}, x_{3}) = 0.15, \\ d_{s_{\underline{G} - x_{2}}} (x_{5}, x_{4}) = 0.25, d_{s_{\bar{G} - x_{2}}} (x_{5}, x_{4}) = 0.30, \\ d_{s_{\underline{G} - x_{2}}} (x_{5}, x_{8}) = 0.41, d_{s_{\bar{G} - x_{2}}} (x_{5}, x_{8}) = 0.54, \\ d_{s_{\underline{G} - x_{2}}} (x_{5}, x_{7}) = 0.25, d_{s_{\bar{G} - x_{2}}} (x_{5}, x_{7}) = 0.30, \\ d_{s_{\underline{G} - x_{2}}} (x_{5}, x_{9}) = 0.45, d_{s_{\bar{G} - x_{2}}} (x_{5}, x_{9}) = 0.45, \\ d_{s_{\underline{G} - x_{2}}} (x_{5}, x_{6}) = 0.75, d_{s_{\bar{G} - x_{2}}} (x_{5}, x_{6}) = 0.75, \\ d_{s_{\underline{G} - x_{2}}} (x_{7}, x_{1}) = 0.16, d_{s_{\bar{G} - x_{2}}} (x_{7}, x_{1}) = 0.24, \\ d_{s_{\underline{G} - x_{2}}} (x_{9}, x_{6}) = 0.30, d_{s_{\bar{G} - x_{2}}} (x_{9}, x_{6}) = 0.30 . \end{matrix}$ By above computations and from Fig. 14, we obtain $WI (\underline{G} - x_{2}) = 0.4152,$ $WI (\bar{G} - x_{2}) = 0.9168,$ and $WI (G - x_{2}) = 1.332 .$

Fig. 14

$G - Iran = (\underline{G} - Iran, \bar{G} - Iran)$ the DRFG.

Here, $k = \frac{WI (G)}{WI (G - x_{2})} = \frac{3.2892}{1.332} = 2.469 and \overset{´}{k} = \frac{n}{n - 2} = 1.285 .$

Since $k < \overset{´}{k} .$ As a result, “x₂ = Iran” has a WRV of $G = (\underline{G}, \bar{G}) .$ In similar manner, we can see that all countries in this model are WRVs. Therefore, we can say that using the concept of $WI$ , the country with high significance on the trafficking pathways from x₅ = Pakistan are x₂ = Iran .

6 Comparative Analysis

In classical graph theory, the $CI$ between the vertex x and vertex y is equal to the product of the degrees of membership of vertices x, y, and SC between the vertex x and the vertex y, which is either 0 or 1. On the other hand, in fuzzy graphs (FGs), the $CI$ between the vertex x and the vertex y is equal to the product of the degrees of membership of vertices x, y and SC between the vertex x and the vertex y, which is in the interval [0,1]. A directed fuzzy graph (DFG) is an extension of a directed crisp graph whose values lie in the interval [0,1], and in the case of classical theory, the degree of membership of vertices x and y is 1. Also in classical graph theory, the Wiener index $WI$ between every pair of vertices can be computed by finding the minimum of the total distance between every pair of vertices, which is any integer value. But $WI$ in the case of DFG explains itself differently. In DFG, the $WI$ between the vertex x to vertex y is equal to the product of the degrees of membership of vertices x, y and d_s(x, y), a distance between the vertex x to the vertex y is the value of those strong geodesics whose total is minimum. Rough and fuzzy sets are alternative ways to deal with ambiguity. The DRFG is a combination of rough fuzzy sets and DFGs. The $WI$ for DFNs cannot handle real world problems under the roughness of DFNs. Therefore, roughness of DFN gives us a reason to extend the concept of $WI$ into DRFG. In DRFG, the concept of $CI$ is already define see [4].

The results and considerations are employed to determine more active countries in the people trafficking pathways from Pakistan. The concept of a DRFG’s $WI$ is offered as one such measure. In [4], the concept of $CI$ is used to determine the more important countries on the trafficking pathways from Pakistan. This method uses the idea of $SC$ between every pair of vertices (countries). The above method and the technique presented in our work both agree that “x₂ = Iran” is more important country among others efficent countries on the trafficking routes from Pakistan. Table 1 contains the comparative analysis of the optimal results proposed by $CI$ Akram et al. (2019) and Uzma et al. (2022) $WI$ of a DRFG.

Table 1
Comparison analysis

Graph Model Topological Index $G = (\underline{G}, \bar{G})$ DRFG Result

DRFG Connectivity Index Connectivity Enhancing x₂ = Iran

[Akram and Zafar [4]] DRFG

DRFG Wiener Index Wiener Reducing x₂ = Iran

[Proposed] [DRFG]

Graph Model	Topological Index	$G = (\underline{G}, \bar{G})$ DRFG	Result
DRFG	Connectivity Index	Connectivity Enhancing	x₂ = Iran
	[Akram and Zafar [4]]	DRFG
DRFG	Wiener Index	Wiener Reducing	x₂ = Iran
	[Proposed]	[DRFG]

7 Conclusion

A topological index is a mathematical technique that can be used to study the structural properties of a graph. Topological indices are used to handle many graph theoretical problems. There are several topological indices in crisp and fuzzy graph theory, but these models are not valid for graphical structures in all environments like DRFG. The aim of this study is to generalize the concept of $WI$ from fuzzy graphs to DRFGs. In many DRFG problems, sometimes the study of connectivity index is not able to distinguish the properties of two graphs. Therefore to overcome this difficulty, we use the concept of $WI$ . In this paper, we proposed the notion of $WI,$ a novel concept for DRFG based on the distance between vertices of a DRFG. Through several examples, this paper explores the interrelationship between $CI$ and $WI$ of a DRFG. We also discussed the idea of a CDRFG and many interesting results related to the $WI$ of a CDRFG. We have discussed the idea of $AWI$ which is based on the average of the distances between the vertices of the DRFG. We have studied three types of vertices, namely WEV, WRV and WNV. The term $WI$ was also employed in this study to discover the more effective countries for people trafficking. The result is in agreement with that found in [4]. That is, Iran is more effective country among the high effective countries for people trafficking. Finally, we presented a detailed comparative analysis and table comparing our results for both $CI$ and $WI$ for the same trafficking network. In the coming years, we plan to extend our research to Soft Directed Rough Fuzzy Graphs (SDRFGs) and the connectivity index of SDRFGs. More similar outcomes and methods will be discussed in new research articles.

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Wiener index of a directed rough fuzzy graph and application to human trafficking

Abstract

Keywords

1 Introduction and preliminaries

Table 1 Comparison analysis Graph Model Topological Index G = ( G _ , G ¯ ) DRFG Result DRFG Connectivity Index Connectivity Enhancing x2 = Iran [Akram and Zafar [4]] DRFG DRFG Wiener Index Wiener Reducing x2 = Iran [Proposed] [DRFG]

References

Table 1
Comparison analysis

Graph Model Topological Index $G = (\underline{G}, \bar{G})$ DRFG Result

DRFG Connectivity Index Connectivity Enhancing x₂ = Iran

[Akram and Zafar [4]] DRFG

DRFG Wiener Index Wiener Reducing x₂ = Iran

[Proposed] [DRFG]