Abstract
Connectivity index has a vital role in real-world problems especially in Internet routing and transport network flow. Intuitionistic fuzzy graphs allow to describe two aspects of information using membership and nonmembership degrees under uncertainties. Keeping in view the importance of in real life problems and comprehension of , we aim to develop some in the environment of . We introduce two types of , namely, and average , in the frame of . In spite of that, certain kinds of nodes called connectivity enhancing node , connectivity reducing node , and neutral node are introduced for . We have introduced strongest strong cycles, -evaluation of vertices, cycle connectivity, and of strong cycle. Applications of the 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.
1. Introduction
Zadeh [1] presented the idea of fuzzy set 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 . 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 approach is given in [7].
In 1975, Rosenfeld [8] studied fuzzy graphs . 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 along with applications. Mordeson [10] proposed the work for fuzzy line graphs. Massa’deh [11, 12] introduced complete , regular , complement of , and some other properties. Mathew and Sunitha introduced types of arcs such as -strong, -strong, and -arcs in [13]. The authors gave various concepts like strong arcs in [14], fuzzy end nodes in [15], and geodesic in [16]. Recently, Akram presented concepts like bipolar in [17] and energy of bipolar in [18]. Jan et al. studied the concept of cubic bipolar with an application in a social network [19]. are useful in representing relationships under uncertainty. are used in various areas like human trafficking, disaster management system, decision-making method, etc.
The extension of is intuitionistic fuzzy set 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. is the generalization of with the requirement that the sum of both degrees cannot exceed 1. Many researchers have applied 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 can be found in [23–25]. Fields of applications of are Computer Science, Engineering, Medicine, Chemistry, Economics, etc.
The generalization of is intuitionistic fuzzy graph explained elaborately by Parvathi and Karunambigai [26]. They also gave the concepts of path, bridge, and cut vertices in . Dhavudh and Srinivasan [27] defined of second kind, and of nth type was developed by Davvaz et al. [28]. Karunambigai and Buvaneswari [29] introduced arcs in like strong arcs, weakest arcs, strong path, -strong, -strong, and -weak arcs. Karunambigai and kalaivani [30] presented as the matrix representation. Mishra and Pal [31, 32] discussed the product of two interval valued , their properties, and regular interval valued in 2013 and 2017. Fallatah et al. [33] and Alanser et al. [34] introduced new concepts as soft graphs and bipolar . Akram and Alshehri [35] introduced cycles and trees. Some misconceptions in the definitions of several generalizations of are corrected by Jan et al. [36]. 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. The stability of a network depends on its connectivity. Binu et al. introduced two measures on connectivity, namely, cyclic and average cyclic of [39]. Poulik and Ghorai brought the concept of , average , and types of connectivity nodes under bipolar fuzzy graph environment with applications [40]. Mathew and Mordeson [41] introduced and , 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]. describe only one type of opinion, that is, membership degree, while describe two types of opinions with the help of membership and nonmembership degrees.
In our paper, we have considered and discussed certain concepts related to . In Section 2, preliminary requirements are given for the work of this paper. In Section 3, we have extended concepts of and bounds of for . Section 4 provides of vertex and edge deleted subgraphs. Section 5 presents concepts of the strongest strong cycles, -evaluation of vertices, cycle connectivity , and of strong cycle for . Section 6 deals with along with its properties. Applications of are discussed in Section 7. Finally Section 8 concludes this study.
2. Preliminaries
Throughout 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].
The notion of was proposed by Akram and Davvaz [43] and given as follows:
Definition 1 (see [43]). An is a pair such that(1) with and representing the truth-membership degree and falsity-membership degree of the vertex and for each .(2) with and being as follows:and for every edge .The next definition is related to complete given by Parvathi et al. [29].
Definition 2 (see [29]). An is complete if and for each.
The role of path is extremely famous and important in . The following definition gives us the concept of path in .
Definition 3. [29] A sequence of distinct vertices is a path in an , provided it has one of the conditions given below for someand.(1) and (2) and (3) and The strength of paths plays a significant role in settings. The following definition gives us component-wise and whole strength of paths in .
Definition 4 (see [29]). Let be a path in an . Then, (1)The T-strength of is denoted and defined by (2)The F-strength of is denoted and defined by (3) is called the strength if both and exist for the same edgeThe highly connected nodes have significant role to a network. The next definition is about the strength of connectedness between the nodes.
Definition 5 (see [29]). The T-strength of connectedness between two vertices and is defined by and F-strength of connectedness between and is defined by for all possible paths between and ,
where , denote the T-strength of connectedness and F-strength of connectedness between and attained by removing the edge from , respectively.
The following definition gives us the idea of a bridge whose deletion from an increase its number of connected components.
Definition 6 (see [29]). An edge is called a bridge in if eitherIn other words, deletion of reduces the strength of connectedness between any pair of vertices.
The concept of strong and weakest edges is of much importance in as well as in our study. The next definition is related to the notions of strong and weakest edges.
Definition 7 (see [29]). An edge in an is(1)Strong if and for each (2)Weakest if and for each The coming definition gives us the strongest paths between two vertices. This definition is relevant to our work.
Definition 8 (see [29]). A strongest path between two vertices in an is a path having its strength equal to and lying in the same edge.
The following definition tells us the concept of strong path.
Definition 9 (see [29]). Let be an . A path in is called strong path if consists of only strong edges.
Example 1. In Figure 1, and , which implies that is a strong arc. Similarly, , , are strong arcs and , are weakest arcs. Here, is a strong path. In fact, it is the strongest path.
The next definition provides us types of strong arcs in .

Definition 10 (see [29]). An arc in an is(1)Called -strong if and (2)Called -strong if and (3)Called -weak if and
Example 2. In Figure 2, the arcs are -strong, is -strong, and are -weak.
The following definition gives us different types of strong paths in .

Definition 11 (see [29]). A path in an containing only -strong arcs is called -strong and a path having only -strong arcs is called -strong.
The concept of a cycle has a vital role in . The following definition gives us the concept of a cycle in environment.
Definition 12 (see [35]). (1) is called a cycle if is cycle(2) is called an cycle if is cycle and unique such that
Example 3. In this example, we take for all . Here, and . Clearly, is an IF cycle. The graph is shown in Figure 3.
The 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. have less information in comparison with . In particular situations like vagueness and uncertainty, are described by only membership grades, but are characterized by the two grades known by membership and nonmembership. Due to the description of opinions using two membership grades, have less information loss as compared to . So, that is why we aim to propose the concepts of several for and study their applications.

3. 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. The connectivity means how stable and dynamic this network is! So, we can say that this measure of connectivity is the most fundamental and natural. The measure of connectivity is already available in . But is the generalization of , and it gives better results in situations where are not preferable. So, because of this reason, we have proposed this concept of connectivity from to . We have made some results of connectivity of to . We define formally as follows.
Definition 13. The of an is defined aswhere is -connectivity index of , is -connectivity index of , and and are -strength of connectedness and -strength of connectedness between and
Example 4. Refer to Figure 1,So, .
It may be observed that , which shows that the level of is lower than the level of in this problem. This comparison is interesting and useful in applications of connectivity index.
Theorem 1. Let be a complete IFG with such that and , where and . Then,
Proof. Let be the vertex having least truth-membership value . For a complete IFG, for each . So, ; and hence, ; . Taking summation over , we haveSimilarly, for vertex , we obtainand for vertex and so on, for vertex By adding all the above equations, we getNow, let be the vertex with the largest falsity-membership value . For a complete IFG, for each . Thus, ; and hence, ; . Summing over , we haveSimilarly, for vertex , we haveand for vertex and so on, for vertex By adding all the above equations, we getHence, by the definition of connectivity, we see
Example 5. In Figure 4, it can be easily seen that is a complete IFG. So,Therefore, .
Now, we use above theoremAdding these two summations, we getHence, it is verified that

4. Edge Deleted and Vertex Deleted IFGs with Connectivity Index
The is affected or not by deleting a vertex or an edge. It is based on the nature of vertex and edge to be removed.
Example 6. Taking the shown in Figure 5. Here, are -strong arcs, are -strong arcs, and are -arcs. Then,So, we have .
So, we have . Thus, , which means that of has been reduced by deleting -strong edge . The , is shown in Figure 6(a). If we delete the -strong edge , then the strength of connectedness between every pair of vertices is invariant, i.e., and , so . The graph of is shown in Figure 6(b). Similarly, when we delete the -arc, then the strength of connectedness between every pair of vertices does not change, and so is the . The graph of is shown in Figure 6(c).


(a)

(b)

(c)
Theorem 2. Let be the IF subgraph of an IFG formed by removing an edge from . Then, or iff is a bridge.
Proof. Take as a bridge. According to the definition, there exit and such that their strength of connectedness will be decreased. So, we conclude that or .
Conversely, suppose that or and consider the possibilities given below. Case 1: suppose that is a -arc. Then, and . So, we have and and therefore, . Case 2: take as -strong edge. Then, , . This implies that there is another path different from edge. Therefore, the removal of the arc will have no effect on the strength of connectedness between and . So, . Case 3: now, take as -strong edge. Then, , . So, the only strongest path is edge having strength equal to . Then, clearly , or or since -strong edges are bridges. This implies that is a bridge.
Corollary 1. Let be the IF subgraph of an IFG formed by removing an edge from . Then, iff is either edge or -strong.
Corollary 2. Suppose that an edge of a complete IFG . Then, iff is a unique IF bridge of .
Proof. Let be a complete . Suppose that . Then, is the only bridge of .
Conversely, let be a unique bridge of . Hence, by Theorem 2, it follows that .
Theorem 3. Let and be the two isomorphic IFGs. Then, .
Proof. Suppose that and are isomorphic . Then, is a mapping such that is bijective and and for all as well as and for . As and are isomorphic, then the strength of any strongest path between and is equal to that between and in . Thus, for So, we haveThus, . This implies that .
5. Strongest Strong Cycles, -Evaluation of Vertices, Cycle Connectivity, and CI of Strong Cycle
This section contains some concepts about cycles. strongest strong cycles, -evaluation of vertices, cycle connectivity, and of strong cycles are defined in the current section. Also, some properties related to these concepts are studied.
Definition 14. The truth and falsity values of the weakest edge in a cycle are defined to be the strength of in an .
Definition 15. Let denote a cycle in an . Then, is called strongest strong cycle (IFSSC) if it is the union of two strongest strong paths for each of and in with the exception when in is an bridge of .
Remark 1. We observe that when is an bridge of and it lies in , the condition for to be the union of two strongest strong to paths can be omitted for and . Also, and .
Definition 16. If is a cycle in an , then is said to be strong, provided each of its edges is strong.
Example 7. In Figure 7, we take for all . The edges are bridges in . and are strongest strong cycles as is not the union of two strongest strong paths. So, it is not a strong cycle. Also, and . But has no strongest path. Moreover, we see that is a fuzzy bridge of that is outside of .

Definition 17. Let be an . Then, -evaluation of two vertices and in is the set defined bywhere represents T-strength of a passing through both and . Similarly, -evaluation of and is the set defined bywhere represents F-strength of the same passing through both and .
Note 1. If cycles through and do not exist, then and . With this -evaluation, we define another connectivity measure in called cyclic connectivity (CC).
Definition 18. Let be an . Then, cycle T-connectivity between and in is denoted and defined bySimilarly, cycle F-connectivity between and in is denoted and defined by
Note 2. If and for any two vertices and , then we define cycle T-connectivity and cycle F-connectivity to be zero, i.e., and .
Example 8. From Figure 8, we have and hence, . Similarly, and therefore, .

Theorem 4. Let be an IFG and for any , both and lie on a common SC. Then,
Proof. Suppose that such that both of them lie on a common IFSSC. Then, , where . Therefore, and hence,Similarly, , where . Thus, we obtain and hence,So, we obtain
6. Average Connectivity Index of an IFG
The concept of average connectivity index is present in the literature of . So, we introduced this concept for . The stability of a network is guaranteed by its average flow.
Example 9. Let be the in Figure 9, with for all . Then, by routine calculations, we have and .
From Example 10, and the number of pairs in is . By averaging the and , we get , and . Now, consider , and we have and . On averaging them, we have and . The overall connectivity of is increased by deleting vertex from . The following definitions and results are led by this example.

(a)

(b)
Definition 19. Let be an . Then, the average -connectivity index of is denoted and defined asand the average -connectivity index of is denoted and defined aswhere is the - strength of connectedness, and is the - strength of connectedness between the nodes and .
Definition 20. Let be an . The average connectivity index of is defined to be the sum of average - connectivity index and average - connectivity index of , i.e.,where is the -strength of connectedness, and is the -strength of connectedness between the nodes and .
Note 3. Obviously, will not be enhanced by the removal of an edge, and therefore, also. For an , .
Definition 21. Let be an and . Then, is said to be an connectivity reducing node of if . is said to be an connectivity enhancing node of if . is said to be an neutral node of if .
Example 10. Consider the as shown in Figure 10. We have taken here for all . , and .Thus are s, is an neutral node, and is a . We characterize these nodes using in the following theorem.

Theorem 5. Let be an IFG and with . Let . is a iff . is a iff . is an neutral node iff .
Proof. Suppose that is an neutral node. Then, by definition, . By definition of average connectivity index, we obtainFrom here, we getThe converse part can be proved by reversing the arguments. Similarly, we can prove other cases.
Note 4. An isolated vertex in an , and is an . Further, if denote complement of , then and therefore,
Theorem 6. Let be an IFG with . If is an end vertex of , let . Then,(1) is an if (2) is an if (3) is an neutral node if
Proof. Let be an neutral node. Then, by definition, . We see thatThe converse part can be proved by reversing these steps. Similarly, we can prove other parts.
At this stage, we can classify an depending on the nature of vertices in it.
Definition 22. An containing at least one is called an connectivity enhancing graph. If there is no node in and at least one , then is said to be an connectivity reducing graph. If has all vertices as neutral nodes, then it is said to be an neutral graph.
7. Applications
In this section, two applications are given. One is on Internet routing, and the other is on transport network flow.
7.1. Internet Routing
The 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 that connects routers in a part of a network. For convenience in calculations, we have taken for all . 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 is a path connecting two routers in the network, then denotes the truth bandwidth of and similarly, is for falsity bandwidth of . Hence, and denote the maximum possible truth bandwidth and the minimum possible falsity bandwidth between and . Consider the fuzzified network as shown in Figure 11.

After computations, we have , . By removal routers and , we obtain , , . Hence, , . We see that and . This implies that is , and is a .
This problem has less incorrect information because . Also, the average bandwidth of the network is increased by the removal of router and removal of causes reduction in average bandwidth.
7.2. Transport Network Flow
Consider a directed network of traffic flow as shown in Figure 12, which is fuzzified. Conveniently, we have taken for all . The connectivity of directed and undirected is similar. So, we can extend these concepts for directed . The vertices are junctions containing correct and incorrect values for vehicles. The 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 -connectivity matrix of the directed IFG.

As the graph is directed, the above matrix is not symmetric. So, we need to sum up all the elements of the matrix. Thus, . Now, the associated - connectivity matrix is given as follows.
By adding all the entries of , we obtain . Thus, .
Consider . It is also directed . The matrices and are given by
By calculations, we have and . Thus, . As , which implies that is .
Next, we consider . The matrices and are given by
We have after calculations, and . Thus, . As , which implies that is also .
Consider . The matrices and are given by
We calculate using above matrices and . Thus, . As , which implies that is also . Now, we consider . The matrices and are given by
From the above matrices, we have and . Thus, . As , which implies that is also . Finally, we consider . The matrices and are given by
From the above matrices, we obtain and . Thus, . As , which implies that is . So, the removal of junction increases the average connectivity amongst the other junctions. Table 1 shows that there is a small difference between and . So, the removal of has no too much effect on the network.
We also see that the difference between and is highest than other differences, so the removal of has maximum negative effects on the connectivity.
8. Conclusion
We have developed some in the framework due to the reason that 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 for and developed results on . Examples are also given to support results of .(ii) of edge and vertex deleted with an example is also given.(iii)We developed , -evaluation of vertices, , and of and related results.(iv) of is defined.(v)Types of connectivity nodes, namely, , 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 and investigate their applications.
9. Advantages
The main advantages and characteristics of our study are as follows:(i)The main feature of our study is to develop the concept of under environment due to the fact that handle uncertain information with two membership grades.(ii) are described by two types of components that is membership and nonmembership, while are characterized by only one component.(iii)Our findings are the generalization of the results of in . For example, if we neglect the second component, then our results are converted into the results of . Thus, the results of in become the special case of our results in .(iv)Comparatively, our study proves that would have less loss of information as compared to .
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare no conflicts of interest about the publication of the research article.
Acknowledgments
The authors are grateful to the Deanship of Scientific Research, King Saud University for supporting through Vice Deanship of Scientific Research Chairs.