Abstract
The Wiener index is one of the connectivity parameters used to know the biochemical and physicochemical properties of compounds depending upon their molecular structures. Intuitionistic fuzzy graphs are a convenient tool to represent the objects and relations between them with two types of information using truth membership degree and falsity membership degree. This research work presents the concept of under the structure , trees, and cycles. Some bounds on are investigated. The relationship between and connectivity index is also studied. In the end of this study, an application of the in transport network flow is proposed.
1. Introduction
Zadeh [1] was the first person who presented his first publication on fuzzy sets (FSs) in 1965. He generalized crisp set by giving membership grades to every object of the set in the interval . Various concepts of ordinary sets were established for FSs. Further explorations in the work of Zadeh were made by Goguen [2] in 1967. Zadeh [3] discussed models including constraints and goals for the betterment and development of different sections of a country’s society in 1970. Interval-valued FSs were presented bySambuc [4] in 1975 under the name of H-fuzzy set, which is proposed by Zadeh [5] taken as a special case of type 2 FSs. In 1999, the authors in [6] studied a method for the calculation of correlation coefficient under fuzzy data. The value obtained by this formula provides the strength of correlation between the sets and positivity or negativity of this correlation. Ashraf et al. introduced spherical FSs and proposed their applications [7]. Mahmood et al. [8] introduced spherical and T-spherical FSs along with some operations on them. They discussed some problems using these sets. There have been a lot of publications on FSs and more to come. There are various areas in which FSs are applied, for example, pattern recognition [9], psychology [10], and decision-making [11].
The graph is a convenient source by which objects and their relationships are presented by vertices and edges. When these objects and relationships contain vague information, it is necessary to present them by a FG. The first step in the field of FGs was taken by Rosenfeld [12] in 1975. In 1994, J.N. Mordeson [13] defined some operations on FGs. Bhattacharya [14] showed an association between FG and fuzzy group in 1987. The researchers completed a lot of work on FGs in the duration of 1981 to 1990. After that, the concept of domination in the environment of FGs was introduced by Somasundaram [15] in 1998. Akram [16] developed the concept of bipolar FGs in 2011, discussed different ways of their construction, and investigated some properties on them. By the same author [17], irregular bipolar FGs of different types were introduced in 2013. They also developed the concept of regular bipolar FGs [18]. FGs have applications in various fields such as controversial problems, decision-making, and social networks [19–21].
FSs contain only membership information. Atanassov [22] was the first one who made the generalization of FSs by considering nonmembership information and invented a new set called intuitionistic fuzzy set (IFS). After this consideration, researchers started to generalize various concepts of FSs into the frame of IFSs. In 1991, Gerstenkorn and Manko [23] defined a function to measure the correlation of IFSs and introduced a coefficient of this correlation to analyze some of its properties. Angelov [24] extended the concept of fuzzy optimization and introduced IF optimization in 1997. The authors in [25] proposed the concept of distance between IFSs in 2000. Atanassov [26] introduced interval-valued IFs and discussed their properties. Li [27] investigated multiattribute decision-making under IFSs. From application point of view, IFSs have application in medicine [28], pattern classification [29], decision-making [30], etc.
IF relations and are the foundations of Atanassov [31]. Akram et al. defined in a precise way and presented various concepts such as strong , balanced , IF hypergraphs, IF trees, and IF cycles [32–35]. Karunambigai et al. [36] introduced arcs in and classified them. The idea of nth type of was proposed by the authors in [37]. The concept of domination was developed by Parvathi [38]. Akram [39] introduced double domination for . Sun et al. [40] proposed the framework to increase the degree of consensus in the group of decision-making. The authors also made an analysis, and a numerical example is given to illustrate this work. Xing et al. [41] studied the concept of a new Choquet integral for the purpose to obtain the matrix of consensus of all individual experts. The authors also presented an illustration to show the effectiveness of the proposed work. have a variety of applications, for instance, decision support system, communication networks, and water supplier system [42–44].
Mathematical chemistry deals with molecular structure in terms of mathematical techniques. In mathematical chemistry, molecular descriptors play a vital role. Chemical graph theory as a subject develops a connection among chemistry, graph theory, and mathematics. A graph representing atoms and bonds of a compound by vertices and edges is known as a molecular graph. Topological indices (TIs) are scalars connected with a molecular graph used to know the correlation of a chemical structure with many physical properties, biological activity, and chemical reactivity. [45] is the first TI proposed by Wiener in 1947. This TI is distance-based. There are many degree-based TIs discovered to know various properties of drugs and compounds, and some of them are Randic index, harmonic index, Zagreb indices, atomic bond connectivity index, geometric arithmetic index, and augmented Zagreb index. Binu et al. introduced and for FGs with applications [46, 47]. The TIs based on vertex degrees such as Zagreb indices, Randic index, and harmonic index are defined in [48] for FGs with the discussion on two applications. Naeem et al. [49] proposed the concept of under environment with applications in two types of networks. TIs are widely used in chemistry, mathematics, and pharmacy engineering [50–52]. Besides, TIs also have applications in human trafficking, Internet routing, and transport network flow [46–49].
The setting of our research article is given as follows. Preliminary requirements are arranged in Section 2. Throughout Section 3, the concept of with illustrations and some bounds for it is proved. The relationship between the and including supporting examples is presented in Section 4. An application of for is proposed in Section 5. In the last two sections, the conclusion and advantages of our proposed work are given.
2. Preliminaries
To understand this research paper in a better way, we give some definitions along with some results throughout the preliminary section.
Definition 1. (see [32]). A pair under and known as the sets of vertices and edges is an satisfying the following:(1) are functions that stand for the truth membership degree, and indicates the falsity membership degree of every under the condition that .(2) are functions, represents the truth membership degree, and denotes falsity membership degree of with the following:such that .
Definition 2. (see [53]). A partial subgraph of an is also an such that(1) and (2) and
Definition 3. (see [36]). An is known as complete if and .
Definition 4. (see [36]). A vertex sequence with for all is known as an path if one of the following holds for at least one and .(1) and (2) and (3) and
Definition 5. (see [36]). The T strength and F strength of a path are defined by and for every and . The pair containing both the strengths is known as the strength of P provided both of them lie on the common arc.
Definition 6. (see [36]). The T strength and F strength of connectedness between two vertices and are denoted and defined by and for all paths .
Definition 7. (see [36]). An arc is known as a bridge in if it has one of the following:(1) and (2) and
Definition 8. (see [36]). An edge is strong if and , and is weakest if and .
Definition 9. (see [36]). If is a path between any two nodes and such that and , then it is called strongest path between and .
Definition 10. (see [36]). A path from vertex to vertex in an is known as strong provided it has just strong arcs.
Example 1. In Figure 1, the arcs , , and are all strong arcs, while is the only weak arc. The path is a strong path, as well as strongest path between the vertices and .
Definition 11. (see [36]). An arc in an IFG is such that(1) and are known as -strong(2) and are known as -strong(3) and are known as -weak
Definition 12. (see [35]). A connected IFG is an IF tree if has an IF spanning subgraph , which is a tree, where for all arcs not in , .
Example 2. Here, we have taken for all . As and , we see that and . Thus, is an tree, and is spanning tree of . The graph is shown in Figure 2.
Definition 13. (see [49]). The of an IFG, , is defined by , where and are -connectivity and -connectivity indices of , respectively, given as follows:
Theorem 1. (see [49]). Let be a complete IFG, and with such that and , where and . Then, contain less information than . There are certain situations where we need information from both sides. are more useful in such type of situations. So, to fulfill these requirements, we aim to establish the concept of for settings. We have made the extension in the results for of . One thing is interesting to see that we have two , one for truth membership values and other for falsity membership values, which produce a comparison between them. To get single value, we take the sum of both the values of these indices.
3. Wiener Index for Intuitionistic Fuzzy Graphs
The concepts of geodesic path and for are presented in this section. Some relevant illustrations are given for these concepts, and some bounds of are investigated.
Definition 14. A path between two vertices is known as geodesic if:(1) is strong(2)There does not exist any strong path shorter than The T weight and F weight are the sum of truth and falsity membership values assigned to all the arcs in the geodesic.
Example 3. In Figure 1, all arcs except are strong. The strongest path between and is , which is mentioned already in Example 1, and therefore, and . Moreover, the strong paths between and are and . The latter two of them are shortest, and hence, both are geodesic.
Definition 15. The of an IFG, is defined as follows:where is the minimum weight sum and is the maximum weight sum of the geodesics from to and both lie on the same geodesic.
Remark 1. In 3.3, if both and do not lie on the same geodesic, we left it for the user to choose the geodesic according to his will. In that case, one of the two geodesics must be compromised.
Example 4. From Figure 1, we have the following:
Note 1. It may be noted that , which shows that this problem has more truth level.
Theorem 2. Let be an IFG with and for all with . Let and . Then,
Proof. Let be an . Suppose that . Let . If , then . The number of unordered pairs satisfying is , and the number of unordered vertex pairs satisfying is . Then,Now, let . Then, as argued before, we have the following:Thus, we have the following:
Theorem 3. Let and be two IFGs such that is isomorphic to . Then, .
Proof. Suppose that and are two isomorphic . Then, there exists a bijective mapping such that and for each and for every edge . Suppose be the path for each serving as and . Then, there will be the corresponding path in serving as and such that for each edge . Since all properties are preserved between both the graphs, and . Then,Hence, , which implies that .
4. Relationship between the Connectivity Index and Weiner Index
In the forthcoming example, it can be noted that , but this is not always the case. Examples 5 and 6 are given to show this matter.
Example 5. Since every arc in Figure 3 is strong, therefore and . We see that for all and . Thus, and . Then,Thus, . Obviously, , and thus, .
Note 2. In last example, we can observe that(1)Every arc is strong(2)Every pair of vertices is connected by an edge(3)There is no -arc
Example 6. Consider is a graph depicted in Figure 4. After calculations, we get , and . Since each arc is strong,The paths and are strong between and and of equal length. The sum of membership values of and is and . Thus . This implies that and . Similarly, we have and .Hence, .
Note 3. Note that in Example 6, although every arc is strong. This inequality does not happen always.
Theorem 4. Let be an with the following two conditions.(1) has no -arcs(2)For every pair Then,
Proof. Consider is an satisfying (i) and (ii). Since has no -arcs, all arcs will be either - or -strong. In each of the case, we have for all :So, we have the following:
Theorem 5. Let be a complete IFG with . Also, such that for and be such that for . Then,
Proof. Suppose is a complete IFG. Then, does not have any -arc. By the previous theorem, . By Theorem 1, we obtain the required result.
Corollary 1. Let be a complete IFG. Then, .
Theorem 6. If is an tree and , then .
Proof. Assume that is an tree with . Then, a unique strong path exits between every two vertices. Indeed, it is also unique and strongest. For all and , while and . So, we have the following:If is a strong arc, then and , but if or is not strong, then and . Thus, , which implies that . Similarly, we have , which gives . Hence:
Corollary 2. Let be an tree. Then, .
Proof. Suppose is an tree and . Then, has only one arc joining its two vertices. Obviously, this arc is strong, and therefore, . By Theorem 6 with , we have . By combining both and , we get as desired.
Example 7. From Figure 2, we calculate connectivity index as follows:Thus, . All the arcs except are strong. Now, we calculate as follows:So, . Clearly, we can observe that .
Theorem 7. If is an partial IF subgraph of an , then and .
Proof. (i) By the definition of partial IF subgraph, we have for each . If denotes the T strength of a path in and denotes the T strength of a path in , then implies . So, we have for all possible paths in and in between and . Thus, we obtain .
(ii) Again by the definition of partial IF subgraph, we have , which implies .
Example 8. Using Figure 5, we have prepared the following table after routine computations. From Table 1, we see that of has been increased and decreased by deleting -strong edges. We also have made the observation that of is increased by removing -strong edges, but by deleting the -arc , the strength of connectedness between every pair of vertices will remain the same. So, that is the reason we have .
Theorem 8. Let be an . For , let be the path with minimum sum of truth values and maximum total of falsity values between the entire shortest strong paths among . Let .(1)If is an -strong or -strong arc and is not a part of any for with , then (2)If is a -edge, then
Proof. Suppose is an and . (i) Let be an -strong or -strong arc. Then, arc itself is the shortest strong path between and . Therefore, and .
Case 1. If is a -strong arc, then one or more strong paths connecting and having length of 2 or more will exist. Suppose that is not a part of any for with . Then, one of the following will hold according to the path in :(1) and (2) and (3) and (4) and Let . By supposition, is not a part of any . So, we have and . So, either or implies that .
Case 2. Let be an -strong arc. If and are not connected by any path on , then . Therefore, and . If and are connected in , then there exits one or more strong paths connecting and of length of 2 or more. So, using the case (1), we get .
(ii) If is a -arc, then no geodesic from to for any will contain the arc . Therefore, the deletion of from has no effect on and for any . So, we must have .
Theorem 9. Let be an with the given condition. If is -strong, then a unique strong cycle of strength exits through . Then, .
Proof. Suppose is an with the given condition. Further, suppose is -strong arc. Then, by the given condition, there exits only one strong cycle through having strength , which implies that and for all arcs in .
Any strong path from to in will have two or more arcs with and for each arc in the path. Thus,Suppose is the strong path from to having strength . Let such that is an arc of the shortest strong path from to with the following:Also, and are only two strong paths exist from . Now, we prove the assertion. If possible, there will be a strong path between other than and . Let the strong cycle be . Now, consider three cases.
Case 3. Figure 6 shows that and are internally disjoint with . Then, together with makes a strong cycle through . This contradicts the assumption that is the only strong cycle through .
Case 4. Figure 7 shows that is not internally disjoint with either or . Suppose and are not internally disjoint. Let . Then, is a strong cycle through different from , which is again contradiction to the assumption that is the only strong cycle through . The case when have common vertices with also provides a contradiction.
Case 5. and are not internally disjoint with (see Figure 8).
Suppose that both and are not internally disjoint with . Let .
Then, the path acts as a strong cycle through other than , which is impossible. Thus, there are only two strong paths and between and . Thus,If such that is not an arc of any strong path from to , then and . Hence, .
5. Application: Transport Network Flow
An application of Wiener index is proposed in this section. The Wiener index is applied to a transport network flow to know different situations created by the removal or closing of certain roads.
Consider a transport network flow as shown in the Figure 9. In it, vertices and edges represent cities and roads. The values given to the vertices and edges are incoming and outgoing flows of traffic. The maximum values for incoming and outgoing flows are 25 and 20, and vertices are fuzzified by dividing their values by 45 (sum of 25 and 20). The edges are also fuzzified by the same technique. The fuzzified network is shown in Figure 10.
In this network, each arc is strong. By simple calculations, we have prepared the following table.
From Table 2, we see that cities become isolated after the removal of arcs or roads , and . The people of these cities will be disconnected from the cities and . The deletion of these edges reduces the overall Wiener index of the network, but still path number for incoming flow is greater than the path number for outgoing flow in other places. From all of these observations, we have arrived at the result that a given network has more incoming flow.
Table 2 shows that . This is why that elimination of makes some routes longer than before. For example, the geodesic route from to was , which is of length 2, but now it has become of 4 in length. Besides its length, its sum of weights has increased also. People who used to travel from city to will now have to spend more money and time. Similarly, the routes from to and from to are also affected by this elimination. All other routes are not affected by this removal, for example, the route from to . There are two geodesics between and of the same length and the same sum of incoming and outgoing flows. So, alternative route with the same geodetic distances is available after the removal of and the people who used to travel between these cities will not be affected much. The removal of is shown in the Figure 11. Similar information can be obtained for other three removals , and . The purpose of this application is to analyze the results produced by the closing or removal of certain routes connecting different cities.
By comparison between the Wiener index and the connectivity index, we can observe from Table 2 that the values of both the indices do not agree with each other for and , although each road is strong. The reason is that there is no road between every two cities; for example, the cities and are not connected directly by a road. Another thing to note is that the values for both the indices are lower than before by deleting strong roads; that is, the behavior of both the indices is same, and the values become higher than before in case of deleting -strong roads, but the values of the Wiener index rise than before while the values of the connectivity index remain the same by deleting -strong roads.
6. Conclusion
Mainly, our goal for the setting of this study was to introduce the concept of for , which deals with two types of degrees named as truth membership degree and falsity membership degree.(i)We have purposed the definition of for structure with numeral examples.(ii)Some bounds of are investigated for settings.(iii)We have discussed the relationship between and with examples. Also, some related results are provided.(iv)An application in transport network flow is given.
In the future, we will do this work for T-spherical fuzzy graphs and picture fuzzy graphs. We will also extend this study for interval-valued and explore its applications.
6.1. Advantages
Our study has the following advantages:(i)The main advantage of our work lies in the fact that we have developed for settings, which are defined by two membership degrees.(ii) are defined by only one degree that is membership degree and therefore failed to give more information in some situations. Instead of , give two types of information that is membership and nonmembership and hence are more suitable in various situations.(iii)Our study and results are in fact of the generalization of the corresponding results of . Our results will be shifted into the corresponding results of by ignoring the second membership grades. In this way, the corresponding results of will be taken as a special case of the results for developed by us.(iv)Our work shows that have more information than .
Data Availability
All data and materials used in this research are included in the study, and there are no other supplementary data.
Conflicts of Interest
The authors declare that they have no conflicts of interest.