Abstract
Vague incidence graph (VIG), belonging to the FG family has good capabilities when facing with problems that cannot be expressed by FGs. When an element membership is not clear, neutrality is a good option that can be well-supported by a VIG. The previous definitions limitations in connectivity concept have led us to offer new definitions in VIGs. Hence, in this paper, the VIG and its matrix form are proposed. Vague incidence subgraph (VISG) is defined with several properties. Incidence pairs, paths, and connectivities between pairs in VIGs are introduced. Likewise, different types of strong and cut pair in VIGs are examined with their properties. Universities are one of the most important centers for human education and play an important role in the development of the country. But the point to be very careful is that the employees of a university must do their job in the best possible way. Therefore, we have tried to identify the most effective person in a university according to its performance by presenting an application.
1. Introduction
The FG concept serves as one of the most dominant and extensively employed tools for multiple real-word 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 relationships 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]. The fuzzy set focuses on the membership degree of an object in a particular set. Kaufmann [2] represented FGs based on Zadeh’s fuzzy relation [3, 4]. Rosenfeld [5] described the structure of FGs obtaining analogs of several graph theoretical concepts. The notion of vague set theory, the generalization of Zadeh’s fuzzy set theory, was introduced by Gau and Buehrer [6] in 1993. Akram et al. [7, 8] studied regularity in vague intersection graphs. Samanta and Pal [9, 10] defined fuzzy competition graphs and some remarks on bipolar fuzzy graphs. Ramakrishna [11] introduced the concept of VGs and studied some of their properties. Borzooei et al. [12, 13] investigated domination in VGs. The strong path between nodes in FG are formulated in [14]. Darabian and Borzooei [15] presented new results in vague graphs. Many operations with their properties in FG theory have been clearly explained in [16]. Ghorai and Pal [17] established various types of FG with their several properties. Some basic definitions of paths, circuit, strong and complete FG, and their applications are briefly discussed in [18]. Strong arcs and paths of generalized FGs and their real applications are given in [19]. Dinesh [20] first defined the notion of FIGs. Different typs of nodes and properties of FIG have been discussed in [21, 22]. The geodesic distance and different types of nodes in bipolar fuzzy graphs are introduced in [23]. Poulik and Ghorai [24–26] initiated degree of nodes and indices of bipolar fuzzy graphs with applications in real life systems. Kosari et al. [27, 28] investigated new concepts in VGs. Zeng et al. [29] introduced certain properties of single-valued neutrosophic graphs. Rashmanlou et al. [30, 31] defined product vague graphs and cubic graphs. Hussain et al. [32] studied neutrosophic vague incidence graph. Rao et al. [33] defined domination in vague incidence graph.
VIGs have a wide range of applications in the field of psychological sciences as well as the identification of individuals based on oncological behaviors. With the help of VIGs, the most efficient person in an organization can be identified according to the important factors that can be useful for an institution. Likewise, a VIG is capable of focusing on determining the uncertainly combined with the inconsistent and indeterminate information of any real-world problem, in which FGs may not lead to adequate results. Therefore, in this paper, the VIG and its matrix form are introduced. VISG is defined with several properties. Incidence pairs, paths, and connectivities between pairs in VIGs are proposed. Also, different types of strong and cut pair in VIGs are examined with their properties. Finally, an application of VIG has given.
2. Preliminaries
Definition 1 (see [34]). Let be a graph. Then, is called an incidence graph (IG), so that . If , and , then is an IG even though . The pair is called an incidence pair or simply a pair. If , then or are called neighbor edges.
Definition 2 (see [20]). Let be an IG and be a FSS of and , a FSS of . Let be a FSS of . If , and , then, is called a FI of graph and is named a FIG of .
Definition 3 (see [6]). A VS is a pair on set V which and are taken as real valued functions which can be defined on so that , .
Definition 4 (see [11]). A VG is defined to be a pair , which is a VS on and is a VS on so that for each , , .
Definition 5 (see [12]). Let be a VG and . (i)A path in is a sequence of distinct nodes so that , and the length of the path is (ii)If be a path of length between and , then and are defined as is said to be the strength of connectedness between two nodes and in , which and .(iii)If and , then the arc in is called a strong arc. A path is strong path if all arcs on the path are strong
Definition 6 (see [12]). For a VG , if and , then the edge is called a strong edge of .
All the basic notations are shown in Table 1.
3. New Concepts of Vague Incidence Graph
Definition 7. is called a VIG of underlying crisp-IG , if: so that , are VSs on and , respectively. is a mapping from to and we have: , , , and , , .
Example 1. Consider an incidence graph so that , and as shown in Figure 1. Clearly, is a VIG of , denoted in Figure 2, which
Definition 8. Let be a VIG of underlying crisp-IG , and has nodes and edges . Then, the matrix form of the VI () is denoted by and is defined by where .
Example 2. Consider the VIG of Figure 2. Here, the number of nodes in is and the number of edges is . So, the matrix form of is given below as
Definition 9. A VIG called a VIPSG of the VIG if , , , , , and , for all and . Also, is called a VISG of the VIG , if , , , , , , , , and , and .
If we delete a node or an edge from a VIG, then, the effects are initiated in next propositions.
Proposition 10. A VISG of a VIG must be a VIPSG of .
Proof. Let be a VISG of the VIG . Then, from the definitions, we have satisfies all the conditions to be a VIPSG of the VIG . So, is a VISG of the VIG .
Proposition 11. If be a VISG of the VIG , then the VG is a VSG of the VG .
Proof. is a VISG of the VIG . Then, from the definitions, we have the corresponding VG satisfies all the conditions to be a VSG of the VG . Hence, is a VSG of the VG .
Definition 12. A VIG is called complete-VIG if and , and . A VIG is called strong if and , pairs in .
If is a complete-VIG and the nodes are neighbor to the edge , then and
Example 3. Consider the VIG as Figure 3. Clearly, is a complete-VIG and also strong-VIG.
Theorem 13. A complete-VIG is a strong VIG.
Proof. Let be a complete-VIG and be a pair in . Then, and , and . Hence, and , pairs in .
Definition 14. Let are the nodes in a VIG . Then, is called an incidence path in . The incidence strength of this path is shown by and is described as and . The incidence strength of connectedness between and in is , where is the maximum value of true part of incidence strength of all the paths between and and is the minimum value of the false part of incidence strengths of all the paths between and .
Example 4. Consider the CVIG of Figure 4.
Here, , , , . So, and . Therefore, the incidence strength of connectedness between and is .
Definition 15. Let be an edge of a VIG . If , , , and , then and are called pairs. is said to be connected if there is an incidence path between every pair.
Theorem 16. Let be a VIG and be a VISG of . Then, for any pair in , and .
Proof. Consider is a VISG of . From the definition of VISG, we have and , for all pairs in . But, , , and , lies on same incidence pair of and or lies on different pairs of and . Now, there arises two cases.
Case 1. Suppose , , and , lies on same pair of and . Then, from the definition of VISG we have and . Then, and .
Case 2. Suppose , , and , lies on the pairs in and in . This means both the pairs and are the pairs of . If and , then and . If or or both, then or or both. Hence, in any case, and .
Definition 17. A pair in is SP if and .
Example 5. Consider the VIG as Figure 5.
Here, , , , , , , , .
Hence, , and , . Therefore, and are SPs.
Theorem 18. Let be a VIG and for a pair , , for all pairs in . Then, the pair is a SP.
Proof. Let be a pair in the VIG . If there is an unique pair in so that and , for all pairs in , then the true part of the strength of all the paths without must be less than and the false part of the strength of all the paths without must be greater than . Then, is a strong pair.
Again, if there are more than one pair in so that and , for all pairs in , then and , for all pairs in .
Now from the definition, we have , , for all pairs in . Hence, is a SP.
Definition 19. Let be a VIG and be a VISG of so that for a pair in , . If, and , for some pair in , then, the pair is called an incidence-CP of .
Theorem 20. Let be a pair in a VIG so that and . Then, the pair is a SP of .
Proof. Let be a VISG of a VIG so that , where is a pair of . If is disconnected, then should be a CP of . Then, and . Then by definition, is a sp. If is connected, then pairs for some , so that and lies in an incidence path from to in . Then, and Thus, and . Therefore, is a SP of .
Theorem 21. Every pair of a complete VIG is a sp.
Proof. Let be a pair in a complete-VIG . Then and . Now, by Theorem 20, it can be proved that is a SP of . Since is arbitrary in the complete-VIG , so all the pairs of are SP.
4. Application of Vague Incidence Graph to Find the Most Effective Person in a University
Science and education are always very important issues for any country that has a very high status. Because if any society has higher education, it will naturally have higher progress and prosperity. Note that the emergence of science and knowledge is equal to the creation of man. Man has always sought to understand and comprehend. Science and knowledge are very important in human life. The role of science in human life is to teach human beings the path to happiness, evolution, and construction. Science enables man to build the future the way he wants. Science is given as a tool at the will of man and makes nature as man wants and commands. Today, the importance of science and knowledge on humanity is not hidden, and all human schools and heavenly religions emphasize the acquisition of science and knowledge and consider the progress and advancement in the path of science as honorable. Science and knowledge are two wings that human beings can fly to infinity. The value of each human being is determined by how they are used. But one of the educational centers that play an important role in educating people are universities. Therefore, the university staff must fulfill their responsibilities in the best possible way so that there is no disruption in education. Therefore, in this section, we try to introduce the most effective staff of a university with the help of an VIG. To do this, we consider the nodes of this graph as the staff of an university and the edges as the influence of one employee on another employee. For this university, the staff is as follows: . (i)Rostami has been working with Falah for 8 years and values his views on issues(ii)Asadi has been the head of the university for a long time, and not only Falah but also Karami is very satisfied with Asadi’s performance(iii)For a university, the protection of educational tools and student files is very important. Mahdavi is a suitable person for this job(iv)Rostami and Karami have a long history of conflict(v)Falah has a very important role in the research work of the university
Given the above, we consider a VIG. The nodes shows each of the university staff. Each staff member has the desired ability as well as shortcomings in the performance of their duties. Therefore, we use of vague set to express the weight of the nodes. The true membership shows the efficiency of the employee, and the false membership shows the lack of management and shortcomings of each staff. But the edges shows the level of relationships and friendships between employees. If these relationships are stronger, then the student education process will be faster. Hence, the edges can be considered as a vague set so that the true membership shows a friendly relationship between both employees and the false membership shows the degree of conflict and difference between the two officials. Name of employees and level of staff capability are shown in Tables 2 and 3. The adjacency matrix corresponding to Figure 6 is shown in Table 4.
Figure 6 shows that Rostami has of the power needed to do the university work as university educational director but does not have the knowledge needed to be the boss. The incidence edge Karami-Bagheri shows that there is only interaction and friendship between these two employees and unfortunately they have disagreement and conflict. It is clear that Rostami, Kazemi and Asadi obey Fallah. Fallah’s dominance rate over all three is equal to . Clearly, Asadi is the most influential employee of the university because he controls all five of the university staff and also he has the highest amount of knowledge among the university staff, which is equal to .
5. Conclusion
Vague incidence graph has various applications in modern science and technology, especially in the fields of neural networks, computer science, operation research, and decision making. Hence, VIG and VISG are defined and their properties are explained by several examples. Also, different incidence paths and their strengths, connectedness, and properties are introduced. Finally, the strength of connectedness between pairs in VIG and VISG are investigated. In our future work, we will study the concepts of covering, matchings, and independent dominating on VIGs and investigate their properties with some examples.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflict of interest.
Acknowledgments
This work was supported by the National Key R & D Program of China (Grant 2019YFA0706402) and the National Natural Science Foundation of China under Grant 62072129.