Abstract
The vague graph (VG), which has recently gained a place in the family of fuzzy graph (FG), has shown good capabilities in the face of problems that cannot be expressed by fuzzy graphs and interval-valued fuzzy graphs. Connectivity index (CI) in graphs is a fundamental issue in fuzzy graph theory that has wide applications in the real world. The previous definitions’ limitations in the connectivity of fuzzy graphs directed us to offer new classifications in vague graph. Hence, in this paper, we investigate connectivity index, average connectivity index, and Randic index in vague graphs with several examples. Also, one of the motives of this research is to introduce some special types of vertices such as vague connectivity enhancing vertex, vague connectivity reducing vertex, and vague connectivity neutral vertex with their properties. Finally, an application of connectivity index in the selected town for building hospital is presented.
1. Introduction
The FG concept serves as one of the most dominant and extensively employed tools for multiple real-world problem representations, modeling, and analysis. To specify the objects and the relations between them, the graph vertices or nodes and edges or arcs are applied, respectively. Graphs have long been used to describe objects and the relationship between them. Many of the issues and phenomena around us are associated with complexities and ambiguities that make it difficult to express certainty. These difficulties were alleviated by the introduction of fuzzy sets by Zadeh [1]. This concept established well-grounded allocation membership degree to elements of a set. Actually, fuzzy set theory is one of the best and most powerful tools for modeling problems in examining the relationship between uncertainties in the real world. Rosenfeld [2] proposed the idea of FG in 1975. Kauffman [3] represented FGs based on Zadeh’s fuzzy relation [4, 5]. Bhutani and Rosenfeld [6] introduced the concept of strong edges. Bhattacharya [7] presented some observations on FGs, and some operations on FGs were described by Mordeson and Peng [8]. The existence of a single degree for a true membership could not resolve the ambiguity on uncertain issues, so the need for a degree of membership was felt. Afterward, to overcome the existing ambiguities, Gau and Buehrer [9] gave false membership degrees and defined a vague set as the sum of degrees not greater than . Ramakrishna in [10] proposed the vague graph concept. New VG concept was introduced and analyzed by Borzooei et al. [11–13]. The vague graph, which has recently gained a place in the family of FG, has shown good capabilities in the face of problems that cannot be expressed by FGs. A vague graph is referred to as a generalized structure of an FG that conveys more exactness, adaptability, and compatibility to a system when coordinated with systems running on FGs. Furthermore, a VG is able to concentrate on determining the uncertainly coupled with the inconsistent and indeterminate information of any real-world problem, where FGs may not lead to adequate results. Connectivity index is one of the most important topics that has many applications in dynamic detection of competition condition in parallel programming, finding phylogenetic trees based on protein domain information, and social networks. The connectivity index problem can also be used to model many real-world situations in the fields of circuit design, telecommunications, network flow, and so on. Binu et al. [14] introduced connectivity index of a fuzzy graph. Mathew and Sunitha [15] defined several types of arc in fuzzy graph. Naeem et al. [16] investigated connectivity indices of intuitionistic fuzzy graphs (IFGs). Poulik and Ghorai [17] studied indices of graphs under bipolar fuzzy environment. Sebastian et al. [18] presented connectivity parameters in generalized fuzzy graphs. Several concepts and results in VGs were proposed and investigated by Akram et al. [19, 20]. Ghorai et al. [21] studied regular product vague graphs and product vague line graphs. References [22, 23] introduced competition-FGs and some remarks on bipolar-FGs. References [24–29] investigated several concepts on FGs and VGs. Kou et al. [30] studied g-eccentric node and vague detour g-boundary nodes in VGs. Rashmanlou et al. [31–33] described categorical properties and paired domination in IFGs. Kosari et al. [34] introduced the restrained K-rainbow reinforcement number of graphs.
In this paper, we investigated connectivity index, average connectivity index, and Randic index in VGs with several examples. Likewise, we introduced some special types of nodes such as vague connectivity enhancing node, vague connectivity reducing node, and vague connectivity neutral node with their properties. Finally, an application of connectivity index in construction is presented.
2. Preliminaries
A FG is a nonempty set V together with a pair of functions and such that , for all . Here, η is a symmetric fuzzy relation on V × V.
Definition 1 (see [9]). A VS is a pair on set where and are taken as real valued functions which can be defined on , so that , for all .
Definition 2 (see [10]). is called a VG on a crisp graph , in which is a VS on and is a VS on so that and , .
Definition 3 (see [11]). Let be a VG.(i)A VG is said to be a PVSG of if , , and , , for each edge .(ii)A VG is said to be a VSG of if , , and , , for each edge .
Definition 4 (see [11]). Let be a VG.(i)A path in is a sequence of distinct nodes where , , . The length of is .(ii)If is a path between and of length , then and . is called the strength of connectedness between any two nodes a and in where and . If and , in , then the VG is called a CVG. If and , , then is named a complete VG.
Definition 5 (see [11]). For a VG , if and , then the arc is named a strong arc of .
Definition 6 (see [11]). Let be a CVG.(i) is called a VT if has a VSSG which is also a tree and for all arcs not in , and .(ii)An edge is said to be a VB if and .(iii) is called a vague cycle if is a cycle and there does not exist unique edge in for which and .All the basic notations are shown in Table 1.
3. Connectivity Index of a Vague Graph
Definition 7. The CI of a VG denoted by is defined aswhere and , respectively, denote the true-CI and false-CI of .
Example 1. Consider the CVG of the graph shown in Figure 1, where and . Here, we suppose that the true-MV and false-MV of every node are and , respectively.
Here,Hence,
Theorem 1. For a PVSGof a VG,and.
Proof. Let be a PVSG of . Hence, , , and , , . Since and lie on one or many edges of and and lie on one or many edges of and , , , and hence we haveSinceSo,Therefore,Therefore, and .
Remark 1. For a VSG of a VG , and .
Example 2. Let be a CVSG of the VG as in Example 2 (see Figure 2). Here,So, . Therefore, Example 2 shows that and .
Theorem 2. If is a VSG of a CVG , where , , then and .
Proof. Let have nodes, i.e., and . Then, i.e., . Suppose . Hence, . Since is VSG of , must be a PVSG of . So,Thus, and .
Theorem 3. If is a complete VG and so that and , , where and , then and .
Proof. Let , , and . Since is complete, and . Then, , , . Since and , then and . Therefore,It shows thatTherefore,Thus, and .
Example 3. Consider the complete VG shown in Figure 3 where . Here, and , for .
, , , , , .
Hence, and .Thus, .
Now we define average connectivity index (ACI) of a VG.
Definition 8. The ACI of a VG denoted by is defined as
Example 4. From Example 2, for VG , the ACI of is
Definition 9. Let be a CVG. A node is called a(i)VCRN if and .(ii)VCEN if and .(iii)VCNN if and .
Example 5. Consider the VG shown in Figure 4. We haveHence,Hence, is a VCEN of and and are VCRN of .
Theorem 4. Let be a complete VG with , and , , . The node is a(i)VCEN if and only if and .(ii)VCRN if and only if and .(iii)VCNN if and only if and .
Proof. (i) Assume that is a VCEN. Then, Definition 9 shows thatSo, and . Hence, and . Thus, and .
Conversely, let and . So, and . Hence, and . Thus, and . Therefore, is a VCEN of . In the same way, (ii) and (iii) can be proved.
Theorem 5. Let be a complete VG with . Suppose , , and is an end node of . Then,(i) and iff is a VCEN.(ii) and iff is a VCRN.(iii) and iff is a VCNN.
Proof. (i) Let and . Now,So, and . Hence, and . Thus, and . Therefore,So, and .
In the same way, (ii) and (iii) can be proved.
Definition 10. Let be a CVG.(i) is named CEVG, if it has at least one VCEN.(ii) is named CRVG, if it has no VCRN.(iii) is named NVG, if all nodes of are VCNN.
Example 6. For the VG in Example 5, the node a is a VCEN of , so is a CEVG, but is neither a CRVG nor a NVG.
Theorem 6. For each pair of numbers , where is a positive true real number and is a positive false real number, there always exist VG so that and , .
Proof. Suppose , , and , . Now, we consider a path in so that and for all edges . Hence, we get
Definition 11. The OND or degree of a node in a VG is described as , where and . If , , then is named -regular.
Now, we introduce the RI of a VG with examples.
Definition 12. The RI of a VG is shown by and described as
Example 7. Consider the VG shown in Figure 5. Here, , , , , and .Therefore, .
Theorem 7. Let be a connected VG and so that , with . Then, and .
Proof. has nodes, i.e., . Then, . So, should be a subset of , so is a VSG of . Hence, , , , and , and .
Now, and , i.e., and are sum of the true-MVs and false-MVs of the edges incident in in , respectively. Then, is s positive real number. So, and , i.e., and are sum of the T-MVs and false-MVs of the edges incident in in , respectively. Then, is a positive real number. Thus,So,Therefore, and .
Example 8. Consider VG of Figure 6 and VG of Figure 5. Clearly, , , , and , , . Hence, is a VSG of the VG .Therefore, and .
Remark 2. Let be a connected VG and so that , . Then, and .
Proof. Assume has nodes, i.e., and . Let . So, . Then, must be a subset of , and so is a of . So, and .
Theorem 8. Let be a complete VG so that is a constant function. Then, , where and , .
Proof. Since is constant and , , , then , , . Since is a complete VG, and , . Again, is complete and , and thus there are nodes and edges and each node in is neighbor to nodes. Therefore,Hence,Therefore, .
Example 9. Let be a VG as shown in Figure 7. Here, , , and , , . Hence, is a constant function.and , , for all edges . Now, and . So, is a complete VG. Using Theorem 8,
4. Connectivity Index in the Selected Town for Building Hospital
Today, the treatment and medical services issue sits among the most important issues for every country. Governments are always trying to provide the best possible services and medical facilities for patients. In the past, unfortunately, as a result of the lack of medical services and the lack of hospitals and clinics, patients were not transferred to these medical centers for the required time and so they lost their lives. This crisis was because of the governments’ constraints for revenue and financial problems. So, governments decided to build hospitals in the cities so that patients could seek treatment without stress as soon as possible. But determining the right place to build a hospital is very important because firstly, it must be built in a place where there is no traffic and crowds nearby, and secondly, patients living in neighboring cities can easily access it. Therefore, in this paper, by using the concept of connectivity index, we try to determine the most appropriate places to establish a hospital. For this purpose, we consider six cities in China (Guangdong state), namely, Shaoguan, Qingyuan, Heyuan, Zhaoqing, Foshan, and Guangzhou, and the attributing symbols for each city are given in the graph as , , , , , and , respectively. Location of cities is shown in Figure 8. Also, distance between cities is shown in Table 2.
In this vague graph, the nodes representing cities and edges also represent the quality of roads as well as the amount of traffic of cars during most hours of the day. Weight of nodes and edges is shown in Tables 3 and 4.
The vertex shows that the city of Qingyuan has of the necessary facilities and equipment to build a hospital, of which the necessary manpower is also part of this equipment. Clearly, it does not have 30% of the equipments for construction. The AB edge shows that the Shaoguan-Qingyuan route has quality and the necessary road and transportation standards (road quality and traffic signs), and the amount of car traffic on this route is equal to . For the vague graph shown in Figure 9, we have
In the same way, we have
Hence,
Clearly, the vertices and are VCRN and the vertex is VCEN. If we remove the node , then the true-CI is strictly increased and false-CI is strictly decreased. Also, by deleting vertex , vertex is automatically deleted (because it is completely disconnected).
Therefore, if the government wants to build a hospital in one of these six cities for the treatment of patients, it should avoid the cities of Zhaoqing and Guangzhou. Also, the cities of Qingyuan and Foshan are the best choices for building a hospital because firstly, they have the most facilities and human resources for construction, and secondly, the communication roads between other cities with these two cities are of better quality and have the least traffic and congestion compared to other routes.
5. Conclusion
The vague graphs can amplify flexibility and precision to model complex real-time problems better than a fuzzy graph. They have several applications in many decision-making processes among solution choice, weather forecasting, prognosis risks in business, and so on, and one of the most important features of VGs that has many applications in real problems is the concept of connectivity index. Connectivity index has many applications in psychology, medical sciences, social groups, and computer networks. Therefore, in this paper, we examined connectivity index, average connectivity index, and Randic index in VGs with several examples. Likewise, some special types of nodes such as vague connectivity enhancing vertex, vague connectivity reducing vertex, and vague connectivity neutral vertex are presented. Finally, an application of connectivity index in construction has been introduced. In our future work, we will define the domination of the VGs in terms of strong edges and examine their properties. Also, we will study domination in terms of independent sets, and since many of the phenomena surrounding us are hybrid, we also discuss the domination concept on its fuzzy operations.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by the National Key R&D Program of China (No. 2018YFB1005100), the National Natural Science Foundation of China (No. 62172116), and the Guangzhou Academician and Expert Workstation (No. 20200115-9).