Connectivity Indices of Intuitionistic Fuzzy Graphs and Their Applications in Internet Routing and Transport Network Flow

Connectivity index (CI) has a vital role in real-world problems especially in Internet routing and transport network flow. Intuitionistic fuzzy graphs (IFGs) allow to describe two aspects of information using membership and nonmembership degrees under uncertainties. Keeping in view the importance ofCIs in real life problems and comprehension of IFGs, we aim to develop someCIs in the environment of IFGs. We introduce two types ofCIs, namely,CI and averageCI, in the frame of IFGs. In spite of that, certain kinds of nodes called IF connectivity enhancing node (IFCEN), IF connectivity reducing node (IFCRN), and IF neutral node are introduced for IFGs. We have introduced strongest strong cycles, θ-evaluation of vertices, cycle connectivity, and CI of strong cycle. Applications of the CIs in two different types of networks are done, Internet routing and transport network flow, followed by examples to show the applicability of the proposed work.


Introduction
Zadeh [1] presented the idea of fuzzy set (FS) by giving membership grades to the objects of a set ranging from zero to one. Many concepts of crisp set theory like inclusion, union, intersection, complement, etc., were established for FSs. FS theory opened the way to fuzzy logic and fuzzy control systems. In the beginning, probability theory was the only tool to handle problems of uncertainty, facing science, technology, and real life problems. FS theory has many applications in different areas such as inventory control model [2], decision-making problems [3], and intelligence science [4]. More applications can be found in [5,6]. A recent research to treat COVID-19 disease with FS approach is given in [7].
In 1975, Rosenfeld [8] studied fuzzy graphs (FGs). After that, Yeh and Bang [9] presented the same concept independently during the same period. Rosenfeld defined some basic properties of fuzzy relations including fuzzy bridges and trees with their properties, while Yeh and Bang gave the concept of connectedness of FGs along with applications. Mordeson [10] proposed the work for fuzzy line graphs. Massa'deh [11,12] introduced complete FGs, regular FGs, complement of FGs, and some other properties. Mathew and Sunitha introduced types of arcs such as α-strong, β-strong, and δ-arcs in FGs [13]. e authors gave various concepts like strong arcs in [14], fuzzy end nodes in [15], and geodesic in [16]. Recently, Akram presented concepts like bipolar FGs in [17] and energy of bipolar FGs in [18]. Jan et al. studied the concept of cubic bipolar FGs with an application in a social network [19]. FGs are useful in representing relationships under uncertainty. FGs are used in various areas like human trafficking, disaster management system, decision-making method, etc. e extension of FS is intuitionistic fuzzy set (IFS) presented by Atanassov [20] in 1986. He added a new component in the definition of FS, which is known as the degree of nonmembership or falsity degree. IFS is the generalization of FS with the requirement that the sum of both degrees cannot exceed 1. Many researchers have applied IFSs in decision-making problems. Chen [21] proposed to measure the degree of similarity between vague sets. Similarity measures for discrete, as well as for continuous, sets are given in [22] and applied in pattern recognition problems. More work on IFSs can be found in [23][24][25]. Fields of applications of IFSs are Computer Science, Engineering, Medicine, Chemistry, Economics, etc.
e generalization of FG is intuitionistic fuzzy graph (IFGs) explained elaborately by Parvathi and Karunambigai [26]. ey also gave the concepts of path, bridge, and cut vertices in IFGs. Dhavudh and Srinivasan [27] defined IFG of second kind, and IFG of nth type was developed by Davvaz et al. [28]. Karunambigai and Buvaneswari [29] introduced arcs in IFGs like strong arcs, weakest arcs, strong path, α-strong, β-strong, and δ-weak arcs. Karunambigai and kalaivani [30] presented IFGs as the matrix representation. Mishra and Pal [31,32] discussed the product of two interval valued IFGs, their properties, and regular interval valued IFGs in 2013 and 2017. Fallatah et al. [33] and Alanser et al. [34] introduced new concepts as IF soft graphs and bipolar IFGs. Akram and Alshehri [35] introduced IF cycles and trees. Some misconceptions in the definitions of several generalizations of IFGs are corrected by Jan et al. [36]. IFGs are applied in different areas such as cellular network and decision support systems [37,38].
Connectivity is the most fundamental and normal parameter related to a network. e stability of a network depends on its connectivity. Binu et al. introduced two measures on connectivity, namely, cyclic CI and average cyclic CI of FGs [39]. Poulik and Ghorai brought the concept of CI, average CI, and types of connectivity nodes under bipolar fuzzy graph environment with applications [40]. Mathew and Mordeson [41] introduced CI and ACI, studied their properties, and investigated their applications. Binu et al. discussed the concept of Wiener index and relationship between Wiener index and connectivity index with an application to illegal immigration networks [42]. FGs describe only one type of opinion, that is, membership degree, while IFGs describe two types of opinions with the help of membership and nonmembership degrees.
In our paper, we have considered IFG and discussed certain concepts related to IFG. In Section 2, preliminary requirements are given for the work of this paper. In Section 3, we have extended concepts of CI and bounds of CI for IFG. Section 4 provides CI of vertex and edge deleted IF subgraphs. Section 5 presents concepts of the strongest strong cycles, θ-evaluation of vertices, cycle connectivity (CC), and CI of strong cycle for IFG. Section 6 deals with ACI along with its properties. Applications of CIs are discussed in Section 7. Finally Section 8 concludes this study. roughout this section, definitions and examples are  presented to recall concepts related to IFG, arcs in IFG, and  IF-cycles relevant to the present work. Most of the definitions in preliminaries are taken from [29,35,43]. e notion of IFG was proposed by Akram and Davvaz [43] and given as follows:

Preliminaries
Definition 1 (see [43]). An IFG is a pair G � (N, M) such that representing the truth-membership degree and falsity-membership degree of the vertex and F M : E ⟶ [0, 1] being as follows: e next definition is related to complete IFG given by Parvathi et al. [29].
e role of path is extremely famous and important in IFGs. e following definition gives us the concept of path in IFGs.

Definition 3. [29]
A sequence u 1 , u 2 , u 3 , . . . , u n of distinct vertices is a path P in an IFG, provided it has one of the conditions given below for some i and j.
e strength of paths plays a significant role in IFG settings. e following definition gives us component-wise and whole strength of paths in IFGs.
Definition 4 (see [29]). Let P � u 1 , u 2 , u 3 , . . . , u n be a path in an IFG. en, (1) e T-strength of P is denoted and defined by S T � min T M (u i , u j ) ∀i, j (2) e F-strength of P is denoted and defined by S F � max F M (u i , u j ) ∀i, j (3) S P � (S T , S F ) is called the strength P if both S T and S F exist for the same edge 2 Mathematical Problems in Engineering e highly connected nodes have significant role to a network. e next definition is about the strength of connectedness between the nodes.
e T-strength of connectedness between two vertices u i and u j is defined by CONN T(G) (u i , u j ) � max S T and F-strength of connectedness between u i and u j is defined by CONN F(G) (u i , u j ) � min S F for all possible paths between u i and u j , the T-strength of connectedness and F-strength of connectedness between u i and u j attained by removing the edge (u i , u j ) from G, respectively. e following definition gives us the idea of a bridge whose deletion from an IFG increase its number of connected components.
Definition 6 (see [29]). An edge In other words, deletion of (u i , u j ) reduces the strength of connectedness between any pair of vertices. e concept of strong and weakest edges is of much importance in IFGs as well as in our study. e next definition is related to the notions of strong and weakest edges.
Definition 7 (see [29]). An edge (u i , u j ) in an IFG is e coming definition gives us the strongest paths between two vertices. is definition is relevant to our work.
Definition 8 (see [29]). A strongest path between two vertices in an IFG G is a path P having its strength equal to CONN T(G) (u i , u j ) and CONN F(G) (u i , u j ) lying in the same edge.
e following definition tells us the concept of strong path.
Definition 9 (see [29]). Let G � (N, M) be an IFG. A path P: u i − u j in G is called strong path if P consists of only strong edges. Example 1. In Figure 1, e next definition provides us types of strong arcs in IFG.
Definition 10 (see [29]). An arc (u i , u j ) in an IFG In Figure  2, the arcs e following definition gives us different types of strong paths in IFGs.
Definition 11 (see [29]). A path in an IFG containing only α-strong arcs is called α-strong and a path having only β-strong arcs is called β-strong. e concept of a cycle has a vital role in IFGs. e following definition gives us the concept of a cycle in IFGs environment.
Definition 12 (see [35]). ( Example 3. In this example, we take (T N (u),  Figure 3. e main goal of our study is to bring more accuracy and precision to the study of topological indices, especially in the context of connectivity indices. FGs have less information in comparison with IFGs. In particular situations like vagueness and uncertainty, FGs are described by only membership grades, but IFGs are characterized by the two grades known by membership and nonmembership. Due to the description of opinions using two membership grades, IFGs have less information loss as compared to FGs. So, that is why we aim to propose the concepts of several CIs for IFGs and study their applications.

Connectivity Index for Intuitionistic Fuzzy Graphs
When we talk about the network like Internet or transport network, naturally, we think about the connectivity of this network. e connectivity means how stable and dynamic this network is! So, we can say that this measure of connectivity is the most fundamental and natural. e measure of connectivity is already available in FGs. But IFG is the generalization of FG, and it gives better results in situations where FGs are not preferable. So, because of this reason, we have proposed this concept of connectivity from FGs to IFGs. We have made some results of connectivity of FGs to IFGs. We define CI formally as follows. Example 4. Refer to Figure 1, It may be observed that TCI(G) > FCI(G), which shows that the level of FCI(G) is lower than the level of TCI(G) in this problem. is comparison is interesting and useful in applications of connectivity index. en, Proof. Let v 1 be the vertex having least truth-membership value t 1 . For a complete IFG, Similarly, for vertex v 2 , we obtain and so on, for vertex v n− 1 By adding all the above equations, we get Now, let v 1 be the vertex with the largest falsity-membership value s 1 . For a complete IFG, Similarly, for vertex v 2 , we have and for vertex v 3 and so on, for vertex v n− 1

Mathematical Problems in Engineering
By adding all the above equations, we get Hence, by the definition of connectivity, we see □ Example 5. In Figure 4, it can be easily seen that k 3 is a complete IFG. So, erefore, CI(G) � TCI(G) + FCI(G) � 0.285 + 0.223 � 0.508. Now, we use above theorem Adding these two summations, we get Hence, it is verified that

Edge Deleted and Vertex Deleted IFGs with Connectivity Index
e CI is affected or not by deleting a vertex or an edge. It is based on the nature of vertex and edge to be removed.    (v 1 , v 4 )) < CI(G), which means that CI of G has been reduced by deleting α-strong edge (v 1 , v 4 ). e IFG, Figure 6 and Figure 6(b). Similarly, 6 Mathematical Problems in Engineering Mathematical Problems in Engineering when we delete the δ-arc (v 1 , v 2 ), then the strength of connectedness between every pair of vertices does not change, and so is the CI. e graph of G − (v 1 , v 2 ) is shown in Figure 6(c).  Proof. Take uv as a bridge. According to the definition, there exit u and v such that their strength of connectedness will be decreased. So, we conclude that CI(G) > CI(H) or CI(G) < CI(H).
Conversely, suppose that CI(G) > CI(H) or CI(G) < CI(H) and consider the possibilities given below.
Case 1: suppose that uv is a δ-arc.
is implies that there is another u − v path different from uv edge. erefore, the removal of the arc uv will have no effect on the strength of connectedness between u and v. So, CI(G) � CI(H). Case 3: now, take uv as α-strong edge. en,  Proof. Suppose that G 1 � (N 1 , M 1 ) and G 2 � (N 2 , M 2 ) are isomorphic IFGs. en, ∃ is a mapping h: As G 1 and G 2 are isomorphic, then the strength of any strongest path between u i and u j is equal to that between h(u i ) and h(u j ) in G 2 . us, for u, v ∈ N * So, we have 8 Mathematical Problems in Engineering us, is implies that CI(G 1 ) � CI(G 2 ).

Strongest Strong Cycles, θ-Evaluation of Vertices, Cycle Connectivity, and CI of Strong Cycle
is section contains some concepts about IF cycles. IF strongest strong cycles, θ-evaluation of IF vertices, IF cycle connectivity, and CI of IF strong cycles are defined in the current section. Also, some properties related to these concepts are studied.

Definition 14.
e truth and falsity values of the weakest edge in a cycle C are defined to be the strength of C in an IFG G.
Definition 15. Let C denote a cycle in an IFG G. en, C is called IF strongest strong cycle (IFSSC) if it is the union of two strongest strong u − v paths for each of u and v in C with the exception when uv in C is an IF bridge of G. Remark 1. We observe that when uv is an IF bridge of G and it lies in C, the condition for C to be the union of two strongest strong u to v paths can be omitted for u and v. Also, Definition 16. If C is a cycle in an IFG G, then C is said to be strong, provided each of its edges is strong. Figure 7, we take (T N (u), F N (u)) � (0.6, 0.3) for all u ∈ N * . e edges (v 1 , v 2 ), (v 3 , v 5 ) are bridges in G.

Example 7. In
is not the union of two strongest strong v 3 − v 5 paths. So, it is not a strong cycle. Also, Definition 17. Let G be an IFG. en, Tθ-evaluation of two vertices u i and u j in G is the set θ T (u i , u j ) defined by where α represents T-strength of a SC passing through both u i and u j . Similarly, Fθ-evaluation of u i and u j is the set where β represents F-strength of the same SC passing through both u i and u j .

Note 1.
If cycles through u and v do not exist, then θ T (u i , u j ) � ϕ and θ F (u i , v j ) � ϕ. With this θ-evaluation, we define another connectivity measure in IFGs called cyclic connectivity (CC).
Definition 18. Let G be an IFG. en, cycle T-connectivity between u i and u j in G is denoted and defined by Similarly, cycle F-connectivity between u i and u j in G is denoted and defined by Note 2. If θ T (u i , u j ) � ϕ and θ F (u i , u j ) � ϕ for any two vertices u i and u j , then we define cycle T-connectivity and cycle F-connectivity to be zero, i.e., TC G u,v � 0 and FC G u,v � 0.

Theorem 4.
Let G be an IFG and for any u i , u j ∈ N * , both u i and u j lie on a common SC. en, Mathematical Problems in Engineering 9 Proof. Suppose that u i , u j ∈ N * such that both of them lie on a common IFSSC. en, TC G u i ,u j � max α: erefore, CONN T(G) (u i , u j ) � TC G u i ,u j and hence, Similarly, FC G u i ,u j � min β: β ∈ θ F (u i , u j ); u i , u j ∈ N * , where θ F (u i , u j ) � β ∈ (0, 1] : β is F − strength of a strong cycle through cycle through u and v}. us, we obtain CONN F(G) (u i , u j ) � FC G u i ,u j and hence, So, we obtain and the average F-connectivity index of G is denoted and defined as e average connectivity index of G is defined to be the sum of average Tconnectivity index and average Fconnectivity index of G, i.e., where CONN T(G) (u, v) is the T-strength of connectedness, and CONN F(G) (u, v) is the F-strength of connectedness between the nodes u and v.
Note 3. Obviously, CI(G) will not be enhanced by the removal of an edge, and therefore, ACI(G) also. For an IFG, 0 ≤ ACI(G) ≤ 1.

Definition 21.
Let G � (N, M) be an IFG and u ∈ N * . en, u is said to be an IF connectivity reducing node (IFCRN) of G if ACI(G − u) < ACI(G). u is said to be an IF connectivity enhancing node Example 10. Consider the IFG G as shown in Figure 10. We have taken here (T N (u), F N (u)) � (0.5, 0.5) for all u ∈ N * .
us v 1 , v 3 are IFCRNs, v 2 is an IF neutral node, and v 4 is a IFCEN. We characterize these nodes using CI in the following theorem.

Theorem 5. Let G � (N, M) be an IFG and
Proof. Suppose that u is an IF neutral node. en, by definition, ACI(G − u) � ACI(G). By definition of average connectivity index, we obtain From here, we get ( 0 . 6 , 0 . 1 ) ( 0 . 9 , 0 . 1 ) e converse part can be proved by reversing the arguments. Similarly, we can prove other cases.
Proof. Let w be an IF neutral node. en, by definition, ACI(G − w) � ACI(G). We see that e converse part can be proved by reversing these steps. Similarly, we can prove other parts.
At this stage, we can classify an IFG depending on the nature of vertices in it. □ Definition 22. An IFG G containing at least one IFCEN is called an IF connectivity enhancing graph. If there is no IFCEN node in G and at least one IFCRN, then G is said to be an IF connectivity reducing graph. If G has all vertices as IF neutral nodes, then it is said to be an IF neutral graph.

Applications
In this section, two applications are given. One is on Internet routing, and the other is on transport network flow.

Internet Routing.
e strength of connectedness between points in a network has much importance in various areas, for example, shortest path problem, routing problem, network flow problem, and maximum band width problem. Consider a network G that connects routers in a part of a network. For convenience in calculations, we have taken (T N (v), F N (v)) � (0.9, 0.1) for all v. Here, the edge values represent maximum bandwidth between the corresponding routers. Membership value of the edge represents correct information and nonmembership value for incorrect information. Also, if P is a path connecting two routers in the network, then S T (P) denotes the truth bandwidth of P and similarly, S F (P) is for falsity bandwidth of P. Hence, CONN T(G) (u, v) and CONN F(G) (u, v) denote the maximum possible truth bandwidth and the minimum possible falsity bandwidth between u and v. Consider the fuzzified network as shown in Figure 11.
is implies that v 2 is IFCEN, and v 11 is a IFCRN.
is problem has less incorrect information because TCI(G) > FCI(G). Also, the average bandwidth of the network is increased by the removal of router v 2 and removal of v 11 causes reduction in average bandwidth.

Transport Network Flow.
Consider a directed network G of traffic flow as shown in Figure 12, which is fuzzified. Conveniently, we have taken (T N (v), F N (v)) � (0.8, 02) for all v ∈ V(G). e connectivity of directed IFG and undirected IFG is similar. So, we can extend these concepts for directed IFG. e vertices are junctions containing correct and incorrect values for vehicles. e edges represent roads connecting two junctions, and their weights indicate number of vehicles consisting of correct and incorrect information. Now, we discuss some connectivity properties of the network flow. Firstly, we find the associated T-connectivity matrix TM(G As the graph is directed, the above matrix is not symmetric. So, we need to sum up all the elements of the matrix.
We have after calculations, As   Table 1 shows that there is a small difference between ACI(G ← ) and ACI(G ← − v 4 ). So, the removal of v 4 has no too much effect on the network.
We also see that the difference between ACI(G ← ) and ACI(G ← − v 3 ) is highest than other differences, so the removal of v 3 has maximum negative effects on the connectivity.

Conclusion
We have developed some CIs in the IFGs framework due to the reason that IFGs cover uncertainty and vagueness with the help of two membership grades. Some key results of our study are as follows: (i) We introduced the notion of CI for IFGs and developed results on CI. Examples are also given to support results of CI. (ii) CI of edge and vertex deleted IFGs with an example is also given. (iii) We developed SSC, θ-evaluation of vertices, CC, and CI of SC and related results. (iv) ACI of IFG is defined.
(v) Types of connectivity nodes, namely, CEN, CRN, and neutral node, and results on them are introduced. (vi) Applications in two types of networks, namely, Internet routing and transport flow network.
In the future, we aim to extend our work to the environment of picture fuzzy graphs [44] and T-spherical fuzzy graphs [45]. We also aim to introduce some other connectivity indices in IFGs and investigate their applications.

Advantages
e main advantages and characteristics of our study are as follows: (i) e main feature of our study is to develop the concept of CIs under IFGs environment due to the fact that IFGs handle uncertain information with two membership grades. (ii) IFGs are described by two types of components that is membership and nonmembership, while FGs are characterized by only one component. (iv) Comparatively, our study proves that IFGs would have less loss of information as compared to FGs.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare no conflicts of interest about the publication of the research article.