Connectivity index of directed rough fuzzy graphs and its application in traffic flow network

The directed rough fuzzy graph (DRFG) is a fusion of rough and fuzzy theory, as it deals with incomplete and vague information simultaneously. Connection or the strength of connectivity (SC)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\mathcal{S}\mathcal{C})$$\end{document} is vital in the realm of circuits or networks that are linked to the real world. As a result, SC\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{S}\mathcal{C} $$\end{document} is one of the most essential aspects of a directed rough fuzzy network system. The neighborhood connectivity index (NCI)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\mathcal {NCI})$$\end{document} is one such parameter that has a variety of applications in network theory. In this study, our main objective is to present a new topological index NCI\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {NCI}$$\end{document} based on DRFGs to solve complicated problems. Motivated by the modeling of networks, the strength of vertices to their neighboring vertices, and the efficiency of DRFGs to solve complex problems, we aim to study the NCI of DRFGs. In this paper, we successfully introduce a notion NCI based on DRFGs to deal with the uncertainties that arise in real-world problems. Based on the strength of vertices to neighboring vertices, we provide several lower and upper bounds for the NCI\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {NCI}$$\end{document} of DRFGs with reference to other graph invariants such as the number of vertices, edges, and degree distance. When we study NCI\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {NCI}$$\end{document} in operations for DRFGs with a large number of vertices, the degree of vertices in a DRFG provides a confusing picture. Therefore, a mechanism to determine the NCI\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {NCI}$$\end{document} for DRFG operations is therefore required. Therefore, generalized formulas for the NCI\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {NCI}$$\end{document} of DRFGs obtained by operations such as union, composition, and Cartesian product are also developed. An algorithm for obtaining the NCI\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {NCI}$$\end{document} of DRFGs is also proposed. Further, an application of the NCI\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {NCI}$$\end{document} of DRFGs in traffic flow networks was discussed to identify the busiest intersection using the proposed algorithm. Finally, we illustrate a comparative analysis and analysis table of the established approach (NCI\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {NCI}$$\end{document}) with existing techniques (connectivity index (CI)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\mathcal{C}\mathcal{I})$$\end{document} and Wiener index (WI)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\mathcal{W}\mathcal{I})$$\end{document}) are shown to demonstrate the validity of the presented approach.


Introduction
Graph theory plays a significant role in numerous innovative approaches and measurement model for strategic planning in range of disciplines such as geometry, algebra, computational mathematics, topology, decision theory, optimization, computer programming and social systems.Network approach tends to be beneficial in tackling real issues.Graph analysis is a straightforward technique to analyze data that includes relationships between diverse components.
In 1965, the innovation of Zadeh's fuzzy set (FS) theory (Zadeh 1965) provided mathematics with a global idea with which to assess uncertainty.As ambiguity and uncertainty came to dominate membership criteria, where each point in the underlying set is linked with a value in the interval [0,1].Kauffman (1973), started the idea of FGs by assigning membership [0,1] to vertices and edges.Yeh and Bang (1975) explained fuzzy graph (FG) theory.Several graph-theoretic concepts like paths, connectedness, bridges, cycles and trees were obtained by (Rosenfeld 1975).Many researchers contributed to this FG study.Later, Bhattacharya (1987) discovered that a fuzzy group is naturally connected with a FG.Geodesics based on FGs connectivity were introduced by (Bhutani and Rosenfeld 2003).Mathew andSunitha (2010, 2013) explore edge connectivity, vertex connectivity, and cycle connectivity in FGs.Binu et al. (2019) conducted research on the connectivity index ðCI Þ of FGs and application in human trafficking.For further knowledge of graphs and FGs see (Jicy 2004, Mathew and Sunitha 2017, Morddesonet al. 2018).Mordeson and Nair (1998) developed the objective of complement of FGs and investigated several FGs operations.Wu and Kao (1985) explore about the construction of fuzzy digraphs.For the first time, Samanta et al. (2016) set up the completeness and regularity in FGs.For additional study on multi-attribute decision making, the readers are referred to (Feng et al. 2021, Akram et al. 2023).
Data which is approximate, imprecise, or inaccurate leads to rough set.Pawlak's rough set (RS) (Pawlak 1982;Pawlak and Skowron 2007) revealed ambiguity using the boundaries region instead of membership functions.Dubois and Prade (1990) investigated rough fuzzy sets (RFs) and fuzzy rough sets (FRs).Akram et al. (2018a, b) presented the notion of fuzzy rough digraphs (FRDGs) and debated a modern decision making strategy based on FRDGs.El-Atik et al. (2022) discussed the topological visualization of rough sets using neighbors and graphs based on a cardiac application.Akram and Ali (2020a) worked on hybrid models for decision making based on rough Pythagorean fuzzy bipolar soft information.
A topological index is a numerical approach that associate to graph representing a network to characterize the underline network of the graph.In the beginning Wiener 1947 discover the Wiener index (WI ) when examining the heat capacity of paraffin.Binu et al. (2020) investigated the distance-based index which is WI in FGs and application to illegal migration routes.Islam and Pal (2021) recently stated the fuzzy hyper Wiener index (FHWI), along with certain limitations on the FHWI for various FGs such as path, cycle, and star.Tong and Zheng (1996) devised a method for calculating the power of connection matrix of a FG.Ahmad et al. (2023) presented fuzzy topological indices with application to cybercrime problem.
Directed fuzzy graphs (DFGs) in networks are incapable of dealing with real circuits in the condition of insufficient data information or a congested world.As a result, this disadvantage provides an opportunity to add directed rough fuzzy graphs (DRFGs).The model of DRFGs is a novel and inventive compound model for addressing more sophisticated uncertainty challenges.Topological indices depending on graph strength of connection ðSCÞ analysis address decision-making issues by defining graph features.When the study of CI refuses to discriminate the features of pair of graphs in DRFG, we use other connectivity-based topological index such as the neighborhood connectivity index ðNCI Þ instead.Josy et al. (2022) investigated the NCI in FGs (which is based on strength of neighbor vertices of graph) and its application to human trafficking.The notion of NCI present in crisp graph theory and FG theory, however, these models are absolutely not applicable to all graphical structures, such as DRFGs.As a result, the focus of this study is on applying the concept of NCI to DRFGs.Zafar and Akram (2018) proposed DRFG as a extension of RS and DFG.Akram andZafar (2019a, 2020b) describe latest investigation on the characteristics and decision making of DRFGs.DRFG also introduced basic operations and outcomes such as union, intersection, Cartesian product and composition, and so on.Akram and Zafar (2019b) introduced the approach of SC among the set of vertices in DRFGs and handled numerous notions like strongest path, strong path and bridge, strong edges, weak edges, and so on.Furthermore, they presented the concept of CI , average connectivity index (ACI ), and associated findings in DRFGs.Ahmad and Nawaz (2022) recently researched connectivity in DRFGs and generalized Menger's theorem of vertices and edges in DRFGs.Ahmad and Nawaz (2023) introduced the Wiener index (distance-based index) of a DRFG and its application to organized crime.Ahmad and Batool (2023) introduced the concept of domination in rough fuzzy digraphs with application.Connectivity is the most fundamental and common network property.The connection of a network determines its stability.There are various topological indices concepts in crisp graph theory and FG theory, but these models are neither applicable to all graphical forms.We focus on the generalization of the concept of NCI of DRFGs in this paper.
-The DRFG is a hybrid model of fuzzy and rough theory and help to overcome the limitations of fuzzy graph theory and rough theory.The disadvantage of fuzzy and rough theory is that it cannot handle incomplete and vague information at the same time.As a result, we may lose some information.However, the DRFGs model deals with vague, incomplete information through a pair of fuzzy sets as it specifies the lower and upper approximations of the given knowledge set.
Therefore, the DRFGs model deals with more complex problems and provides more accurate results.The aforementioned properties of DRFGs motivate us to present our work in a DRFGs environment.
-When modeling networks, such as transportation and water supply networks, the potential traffic strength between vertices and neighboring vertices should be known.The strength of edges between vertices can be adjusted in the modeling phase itself.The notion of NCI is the topological index based on the strength of links between vertices and their neighbors, as it provides more practical results in real-life problems.-The NCI of DRFGs has some outstanding features, such as less time consuming, more stable and accurate results, since it depends on the neighboring vertices.
The authors believe that this concept is important in modern networks and can also be used in assignment problems, traffic control problems, and routing problems.
From the above discussion, there is no work on the NCI of DRFGs in the existing literature.Motivated by the salient features of NCI , we extended the concept of NCI for DRFGs.The main contributions and innovations of this paper can be summarized as follows: • In this study, our goal is to establish an NCI approach for DRFGs based on the strength of neighboring vertices.We explore the notion of NCI based on the strength of vertices to deal with real life problems.We establish some important results of NCI of DRFGs with the help of a suitable illustration.We also present some bounds of the NCI of DRFGs.It is difficult to compute NCI for some operations of DRFGs with a large number of vertices.Therefore, we also introduce an NCI approach for some operations of DRFGs such as union, composition, and Cartesian product under certain conditions.• To show the applicability of our proposed approach, we apply it to a real-world problem: We implemented the proposed NCI concept to identify the busiest vertex in a network flow problem.• To show the applicability and robustness of our work, we perform a comparative analysis of the established strategy with the existing approaches in the literature such as connectivity index and Wiener index of DRFGs.
The following is the structure of our study work.Section 2 contains fundamental DRFG definitions and results that are required for content production.In Sect.3, we discuss the NCI of DRFGs and related results, along with its bounds.In Sect.4, we establish how the NCI acts on some operations such as union, composition, and Cartesian product under some conditions, along with several examples on DRFGs.In Sect.5, we present an algorithm (lay out) to evaluate the NCI to explore how to use the NCI of DRFG to identify the busiest vertex or junction of the traffic flow network.Section 6 discusses a comprehensive analysis among our research work and the existing methods in (Akram and Zafar 2019b) and (Ahmad and Nawaz 2023) in terms of human trafficking.Section 7 brings our investigation to a conclusion.

Preliminaries
The preceding are the primary interpretations and notions of directed rough fuzzy graph (DRFG); greater part of above notions possibly discovered in (Akram and Zafar 2019b).
A path P : r 0 !r 1 !::: !r n from r 0 to r n with n length is directed path in G ¼ ðG; GÞ a DRFG provided that P is directed path from r 0 to r n of length n in G and additionally in G; respectively.An edge with smallest degree is weakest edge in DRFG.The potency or strength of the path P is stated as the sum of the degree of weakest edge in G likewise in G. CONN G ðr 0 ; r 1 Þ is put to use to symbolize the power of connection or strength of connectedness (SC) from r 0 to r 1 in G ¼ ðG; GÞ and is explained as the sum of maximum of powers from r 0 to r 1 for all paths in G as well as in G. Provided that the intensity of a path P in G ¼ ðG; GÞ a DRFG is equal to CONN G ðr 0 ; r 1 Þ; then path P is considered to be a strongest r 0 À r 1 path.An edge r 0 r 1 is considered to be alphaÀ strong edge in i.e., N Eðr 0 ; r 1 Þ\CONN GÀr 0 r 1 ðr 0 ; r 1 Þ and N Eðr 0 ; r 1 Þ\CONN GÀr 0 r 1 ðr 0 ; r 1 Þ; respectively.An edge is strong in G ¼ ðG; GÞ a DRFG if its alpha or beta edge of G ¼ ðG; GÞ: A path P in G ¼ ðG; GÞ a DRFG is strong path provided that its every edges or links are strong edges.
Providing that the elimination of an edge r 0 r 1 2 Z Â Z reduced the SC among the set of vertices in a DRFG G ¼ ðG; GÞ; then edge r 0 r 1 is a bridge in a DRFG G ¼ ðG; GÞ: In a parallel way supposing that the elimination of an vertex r 0 reduced the SC among the set of vertices in a DRFG G ¼ ðG; GÞ; that is why r 0 is cutvertex of a DRFG G ¼ ðG; GÞ: A DRFG G ¼ ðG; GÞ which not contain the cutvertices is named as block.A G ¼ ðG; GÞ the DRFG is considered to be a complete directed rough fuzzy graph (CDRFG) if its complete directed fuzzy graph (CDFG) in G as well as in G, respectively.Mathematically we can write: The terminologies and essential results employed in this research study are given below.
Definition 6 (Akram and Zafar 2019b) Suppose that G ¼ ðG; GÞ a DRFG, then connectivity index (CI ) of G ¼ ðG; GÞ is where where Here, d G ðr 1 ; r 2 Þ and d G ðr 1 ; r 2 Þ indicated values of such geodesics from vertices r 1 to r 2 whose sum is least in G as well as in G; respectively.
3 Neighborhood connectivity index of a directed rough fuzzy graph From this section, we state a new parameter related to the potential of vertices and its neighborhood namely, Neighborhood connectivity index ðNCI Þ for a DRFG.Furthermore a graph index, recognized as a structural descriptor, is a arithmetical formula that may be employed to any graph that represents some type of structural features.This index may be used to analyze various physical aspects of a graph.NCI is a topological index that is employed in a number of fields like medicine, capacity planning, networking, and so on.The standard explanation for estimating NCI of a DRFG as follows.
Definition 9 Suppose that G ¼ ðG; GÞ a DRFG.The Neighborhood connectivity index (NCI ) of G ¼ ðG; GÞ a DRFG is where are maximum of strengths of each and every paths from r 1 to r 2 in G and additionally in G, respectively.
Remark 1 The potential of a vertex in a DRFG G ¼ ðG; GÞ is denoted by Rðr 1 Þ in G as well as in G; respectively.
Remark 2 The cardinality of edge set of G ¼ ðG; GÞ a DRFG is zero if and only if its NCI is zero.
For any vertex r 1 ; and maxfN Eðr 1 r 2 Þ : r 2 2 Rðr 1 Þg ¼ y 2 then strengths of P and P are less than or equal to y 1 and y 2 , respectively.In particular, if N Eðr 1 r 3 Þ ¼ y 1 and N Eðr 1 r 3 Þ ¼ y 2 then r 1 r 3 is a strongest path with strengths nðr 1 Þ and nðr 1 Þ in G and in G, respectively.This implies both the definitions of nðr 1 Þ are equivalent.Similar case is for nðr 1 Þ.
Mainly we consider (i), from concept of isomorphism.Since the values of the edges and vertices are preserved by an isomorphism, therefore   2 and 3 as follows: From Tables 2 and 3 In the following results, the paper establishes some bounds for the index.
Theorem 6 Let G be a DFG and 'a' represent the number of vertices in G: Then, 0\NCI ðGÞ aða À 1Þ: Proof Let G be a DFG.In case 1, assuming ðN EÞ Proof Suppose that G ¼ ðG; GÞ be a DRFG with j Z j¼ a: By Theorem 6 and Corollary 2, we have following      4 NCI and some operations of DRFG There are several DRFG operations in DRFG theory.This section deals with the NCI of DRFGs obtained by some of these operations.As defined earlier, in this section (1) Where are the edge sets of G 1 and G 2 , respectively.And j is the number of edges arising from the vertex ''r'' which lies in G 1 as well as in G 2 : (2) Where ðN E 1 Þ 00 and ðN E 2 Þ 00 are the edge sets of G 1 and G 2 respectively.And is the number of edges arising from the vertex ''r'' which lies in G 1 as well as in G 2 : Here, G 1 ; G 2 ; G 1 and G 2 are DFGs.We prove this result by supposing three cases.
Case (i): there is no new neighbor by construction so m G 2 ðrÞ ¼j : Since in this case there is no new edge originating from 0 r 0 and there is no change in weight for the existing edges.Therefore, n G 2 ðrÞ ¼ 0: Similar case arises when r 2 G 2 : Case (ii): In second case, we take r Case (iii): In third and last case, we take r Since in last case, the edges are taking the maximum weight and the maximum will be any of the n G 1 ðrÞ and n G 2 ðrÞ: Clearly, from all the above three cases, result (1) hold.
(2).Proof is similar to the proof of (1).h 4 and 5, respectively depicts DFGs.Also, here 6 depicts DFGs.By direct computations from Fig. 6, the 4 as follows: From Table 4, we have Now by direct computations from Figs. 4 and 5, the are given in Tables 5 and 6, respectively, as follows: Now using Theorem 10, we can find this without actually finding 5 and 6.
Here, G 1 ; G 2 ; G 1 and G 2 are DFGs.First, we consider G 1 and G 2 .By definition, Next, we find potential of the vertex ðr; r 1 Þ: and and n G1½G2 ðr; r 1 Þ ¼ max ðr;r1Þ2ðMV1Þ 00 ÂðMV2Þ 00 Now, we can analyze the all parts of the theorem. ( Therefore, (2): Proof is similar to the proof of (1).h Here G 1 ; G 2 ; G 1 and G 2 are DFGs.Firstly we consider G 1 and G 2 and prove above results.By definition, Similarly, We can find m G 1 nG 2 ðr; r 1 Þ and n G 1 nG 2 ðr; r 1 Þ separately.
Since by construction, the Cartesian product of DFGs differ from composition only by the set of edges fðf ; gÞ : Similarly, Now, we can analyze the all parts of the theorem. (1): (2): Proof is similar to the proof of ( 1).h 7 and 8, respectively depicts DFGs.Also, here Additionally, here 10 depicts DFGs.By direct computations from Figs. 9 and 10, the 7 and 8, respectively, as follows: From Tables 7 and 8, we have NCI ðG 1 nG 2 Þ ¼ 6:3; and NCI ðG 1 nG 2 Þ ¼ 9:1: Now by direct computations from Figs. 7 and 8, the are given in Tables 9 and 10, respectively, as follows: Since MV 1 !N E 2 and MV 2 !N E 1 ; now using (1) and ( 2) from Theorem 11, we can find this without actually  9 and Table 10.
Clearly, both gives same value of NCI ðG 1 nG 2 Þ which is 9.1.

Application to transport network
In this section, we study the application of NCI of DRFG to a traffic flow networking problem.Traffic flow networking focuses on the movement of cars on the road and their interactions with other vehicles, pedestrians, and road signals.Traffic flow is affected by many factors, including design speed, the percentage of heavy vehicles, and the number of lanes and intersections on the road.The number of vehicles using a particular roadway per unit time during an hour is referred to as traffic flow on a roadway and is quantified by traffic counts at a specific location on the roadway over a period of time.The counts are conducted at different times of the day.Rush hour traffic is considered in any traffic flow analysis.Measuring traffic flow is necessary to determine the point on the route where congestion occurs, the need for traffic signals at an intersection, the capacity of the roadway for the current traffic flow, and similar tasks.In addition, traffic flow is affected by traffic speed and vehicle density on the route.The problem is to identify the busiest junction from a junctions network to get from one place to another in a city.
First, we develop an algorithm to rank the alternatives from high to low preference and make an optimal decision.The steps of our proposed method as an algorithm are shown in Table 11.
After using algorithm the direct computation from Tables 16 and 17, we get NCI ðG À J 4 Þ and NCI ðG À J 4 Þ of G À J 4 and G À J 4 as follows: For the given traffic network G ¼ ðG; GÞ of Fig. 11, we have We can arrange the list of or intersections or junctions according to their NCI values of given traffic flow model from less busy junction to most busy junction of traffic network as follows: The NCI of the junction deleted DRF-subgraph G À J 2 is lowest and G À J 3 is highest.So the junction J 2 i.e., the junction in traffic network is the busiest junction and the vertex J 3 i.e., the junction in traffic network is the lowest busiest junction according to the NCI .Hence, the traffic load in this network at J 2 is highest and the traffic load in this network at J 3 is lowest.

Comparative analysis
In this section, a comparative analysis of NCI of the DRFG model with existing approaches such as CI of the DRFG model and WI of the DRFG model is conducted.The advantages of NCI are discussed to show its significance and importance compared to other CI and WI.
Rough fuzzy network (RFN) is based on indistinguishable relationships between objects and can be used in all cases where the data is based on crisp coding, i.e., whether there is indistinguishability between objects or not.Topological indices are numerical numbers that mathematically characterize the topology of chemical composition.

Input
Step 1. Suppose a directed rough fuzzy model (DRFM) of traffic flow network.
Step 2. In the selected model, arrow indicate the direction from junction to junction.
Step 3. Input the vertex or junction set Z and the equivalence relation (ER) on Z is M.
Step 4. Input the association set E and ER on E is N : Step 5. On Z, Input V the vertex FS and on E, input E the edge FS.
Step 6. Evaluate the degree of vertex MV ¼ ðMV; MVÞ in DRFGs through the formula ðMVÞðrÞ ¼ Step 6 and step 7 should be repeated for all pair of junctions in traffic flow.
Step 9. Construct the fuzzy matrices D G ¼ ½r ij and D G ¼ ½r ij with i, j are rows and columns respectively.
Step 10.Find the largest membership value in each row of the matrix D G and D G .Let it be u i and u i respectively.
Step 11.Find the number of non-zero entries in each row of the matrix D G and D G .Let it be v i and v i respectively.

Output
Step 12. (i): Then NCI ðGÞ ¼ An important area of applied mathematics is the study of network theory, which is based on the structure of networks.Because of its global applications, this theory can be used to model real world situations.Different topological indices have been proposed to deal with uncertainty, which help in strategic planning and are compared in different contexts, sometimes arguing that one is more flexible or helpful than the other, depending on the nature of the problem.In this Part, we propose that the ideas of the Connectivity Index CI, the Wiener Index WI , and the Neighborhood Connectivity Index (NCI ) of DRFGs proposed here serve different purposes.The WI (distancebased index) of a DRFGs (Ahmad and Nawaz 2023) explains the shortest strong path and the intensity or    12 with standard calculations, we get CI ðGÞ ¼ 0:3372; CIðGÞ ¼ 1:3248; and Similarly, from Fig. 12 with standard computations, we obtain WI ðGÞ ¼ 0:729; WIðGÞ ¼ 2:5602; and From Fig. 12 with standard calculations, we get NCI ðGÞ ¼ 1:52; NCI ðGÞ ¼ 1:78; and NCI ðGÞ ¼ 3:3 For the given DRFG G ¼ ðG; GÞ of Fig. 12, we have NCI ðG À IranÞ ¼ 2:3, NCI ðG À OmanÞ ¼ 2:55, NCI ðG À UAEÞ ¼ 2:9, NCI ðG À TurkeyÞ ¼ 2:55, NCI ðG À GreeceÞ ¼ 2:9, NCI ðG À Middle EastÞ ¼ 2:3, NCI ðG À West AfricaÞ ¼ 2:1, NCI ðG À SpainÞ ¼ 2:78.
We can arrange the list of countries according to their NCI values of given HT model from less active country to most active country on the routes of HT from Pakistan as: Therefore, we can see that using the idea of NCI , the most active country on the routes of HT from Pakistan is West Africa with lowest NCI : Since by deleting the vertices, we will get subgraph of G ¼ ðG; GÞ a DRFG.According to CI (Akram and Zafar 2019b), the most active country on the routes of HT from Pakistan is Iran.Similarly according to WI (Ahmad and Nawaz 2023), the most active country on the routes of HT from Pakistan is Iran.But with the help of NCI ; the most active country on the routes of HT from Pakistan is West Africa.But the above method and the technique of NCI ; presented in this paper says that ''West Africa'' is most active country among others countries on the HT routes from Pakistan.
Our proposed approach is more effective and practical in the way that it provide best decision about the most active country among others countries on the HT routes from -Rough Fuzzy Sets (Dubois and Prade 1990)    They will be useful in studying various properties of union, Cartesian product and composition of DRFGs.In addition, the concept of NCI of DRFGs to a real-world problem involving the identification of the busiest junction.Since neighborhood traffic is relevant in most modern networks, the concepts of this work can be used in a variety of problems.This can also be used in analyzing the effectiveness of planning and routing in different domains.Finally, using the identical human trafficking network (Akram and Zafar 2019b), we conducted a comprehensive comparative analysis and comparison table by briefly presenting the results for the connectivity index (CI), the Wiener index (WI ), and the NCI of a DRFG.We concluded that the presented approach is an increasingly precise and flexible approach, as it allows more accurate decisions based on the strength of the neighboring vertices.This study can be extended to (1) directed rough fuzzy graphs, (2) soft directed rough fuzzy graphs, and (3) soft directed rough fuzzy graphs.
Author contributions Investigation: UA, IN; writing original draft: IN and SB; writing review and editing.

;n
GÞ is a partial subgraph of a DRFG G ¼ ðG; GÞ, then n GðrÞ n G ðrÞ; r 2 ðMVÞ 00 GðrÞ n G ðrÞ; r 2 ðMVÞ 00 : Proof Suppose that G ¼ ð G; GÞ is a partial subgraph of G ¼ ðG; GÞ a DRFG.Let r and r 1 be an arbitrary vertices in ðMVÞ 00 : By Theorem 2, we have

Fig. 11 G
Fig. 11 G ¼ ðG; GÞ the DRFG representing traffic flow among different junctions NCI ðG À UAEÞ ¼ NCI ðG À GreeceÞ [ NCI ðG À SpainÞ [ NCI ðG À TurkeyÞ ¼ NCI ðG À OmanÞ [ NCI ðG À IranÞ NCI ðG À Middle EastÞ [ NCI ðG À West AfricaÞ: is a powerful tool for discussing objective and subjective uncertainty simultaneously.This hybrid model of rough set and fuzzy model deals with uncertainty and unclear information in terms of upper and lower approximations.The rough fuzzy model deals with more complex problems than rough and fuzzy theory.The combination of rough fuzzy information with graphs facilitates the study of network problems with uncertainty without additional information.-Topologicalindices, which depend on the analysis of the connectivity strength of graphs ððSCÞ, deal with decision making problems by defining graph features.In the proposed study, this mathematical approach NCI of DRFGs is based on the strength of connections between nodes and their neighboring vertices.-The study of WI (distance-based index) does not provide information about the possible shortest strong path of the DRFGs along all strong paths from vertices to their neighboring vertices.-Similarly, the connectivity-based index CI refuses to distinguish the features of graph pairs in DRFG.Therefore, we use other connectivity-based topological indices such as NCI , which is not only strength-based but also depends on the strength of neighboring vertices.NCI is a more effective graph structure

Table 11 Algorithm
Algorithm: Identifying the busiest junction of city in traffic model

Table 13
Equivalence relation N on E N ðJ 1

Table 14
(Akram and Zafar 2019b)ble15Adjacency matrix of G DRFGs over the entire strong path, while the CI (degree-based index) of a DRFG(Akram and Zafar 2019b)gives the overall strength between each pair of vertices for the specified model.aDRFGS'sCIprovidesonly the overall strength of connectivity between each set of vertices.But unlike CI, the proposed term NCI (neighbor degree based index) of a DRFG gives not only the overall strength of the connection between each pair of vertices, but also knowledge about the possible strength or power of the connection of vertices to their neighboring vertices.Therefore, in this part, we have described a method for comparing CI, WI and NCI of DRFGs.Here, we present the numerical comparison of CI, WI and NCI of DRFGs.Akram and Zafar (2019a)discussed this topic for the DRFG model.The following discussion evolves from the author's intention to identify the most successful human trafficking (HT) country pathways from Pakistan for the DRFG model, which can be found in(Akram and Zafar 2019b).The DRFG G ¼ ðG; GÞ on Z ¼ fr 1 ¼ Greece; r 2 ¼ Iran; r 3 ¼ Middle East; r 4 ¼ Oman; r 5 ¼ Pakistan; r 6 ¼ Spain; r 7 ¼ Turkey; r 8 ¼ UAE; r 9 ¼ West Africag as shown in Fig.12.The equivalence relation (ER) M on Z which indicate that every destination countries lie to same equivalence class (EC).Similar manner, every going and every initiated countries lie to same ECs, respectively.Let V be a fuzzy set (Fs) on Z which indicate the vulnerability of each country and MV ¼ ðMV; MVÞ a rough fuzzy set (RFs).Let E be a subset of Z Â Z and N be ER on E, where N characterizes the ECs of ''correlation among several countries''.Let E be a Fs on Z which indicate the membership value of unlawful migration from country to country.Let N E ¼ ðN E; N EÞ be a RFR, where N E is lower approximation (LA) and N E is upper approximation (UA) of E. Where, G and G in Fig.12indicate DFGs.Provided we erase }r 2

Table 16
Adjacency Matrix of G À J 4 Table 17 Adjacency Matrix of G À J 4Pakistan to their neighbors countries.For example, ''West Africa'' is most active country among others countries on the HT routes from Pakistan corresponding countries.Table18speak about the comparative study of the best possible results proposed by CI (Akram and Zafar 2019b), WI (Ahmad and Nawaz 2023) and(Ahmad et al. 2023) NCI of a DRFG.The main points of comparison are summarized as follows:

Table 18
Comparison analysisDirected rough fuzzy graph (DRFG) is an essential hybrid graph model for dealing with vagueness and ambiguity in the presence of incomplete information in terms of the lower and upper approximation spaces.This model provides the system with more compatibility, precision and flexibility than the fuzzy model and rough model.The DRFGs are widely applied in today's science and technology, especially in social circuits, machine intelligence, transportation networks, deliberation, etc. Connectivity or the strength of connectivity ðSCÞ is always considered as a cornerstone in network theory.In the past, topological indices have been studied and used to solve many different graph problems, and numerous extensions have been presented.A new parameter, the neighborhood connectivity index ðNCI Þ, which is related to SC and has many applications in network theory, is studied in this paper.The proposed topological index NCI is able to make a more appropriate choice based on the strength of the neighboring vertices.Moreover, various upper and lower bounds for the NCI of DRFGs are established.The NCI in a DRFG gives a complicated picture when we study operations for DRFGs with a large number of vertices.Therefore, a technique for determining the NCI for DRFG operations is needed.For this reason, we also develop a generalized formula for the NCI of DRFGs obtained by operations such as union, composition and Cartesian product obtained under certain conditions, and illustrate it with examples.