New Results in Vague Incidence Graphs with Application

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.


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][25][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.
Definition 2 (see [20]). Let G * be an IG and σ be a FSS of V and μ, a FSS of E. Let ψ be a FSS of I. If ψðv, pqÞ ≤ σðvÞ∧μ ðpqÞ, ∀v ∈ V and pq ∈ E, then, ψ is called a FI of graph G * and G = ðσ, μ, ψÞ is named a FIG of G * .
Definition 3 (see [6]). A VS A is a pair ðt A , f A Þ on set V which t A and f A are taken as real valued functions which can be defined on V ⟶ ½0, 1 so that t A ðpÞ + f A ðpÞ ≤ 1, ∀ p ∈ V.
Definition 4 (see [11]). A VG is defined to be a pair G = ðA , Definition 5 (see [12]). Let G be a VG and m, n ∈ V.
ððt B ðmnÞÞ ∞ , ðf B ðmnÞÞ ∞ Þ is said to be the strength of connectedness between two nodes m and n in , then the arc mn in G is called a strong arc. A path m − n is strong path if all arcs on the path are strong Definition 6 (see [12]). For a VG G, if t B ðmnÞ ≥ ðt B ðmnÞÞ ∞ and f B ðmnÞ ≤ ð f B ðmnÞÞ ∞ , then the edge ab is called a strong edge of G.
All the basic notations are shown in Table 1.

New Concepts of Vague Incidence Graph
Definition 7. G = ðA, B, CÞ is called a VIG of underlying crisp-IG G * = ðV, E, IÞ, if: Example 1. Consider an incidence graph G * = ðV, E, IÞ so that V = fp, q, r, sg, E = fpq, qr, qs, rs, psg and I = fðp, pqÞ, ðq, pqÞ, ðq, qrÞ, ðr, qrÞ, ðq, qsÞ, ðs, qsÞ, ðr, rsÞ, ðs, rsÞ, ðs, psÞ, ðp, psÞg as shown in Figure 1. Clearly, G = ðA, B, CÞ is a VIG of G * , denoted in Figure 2, which Definition 8. Let G = ðA, B, CÞ be a VIG of underlying crisp-IG G * , and ζ has n nodes p 1 , p 2 , p 3 , ⋯, p n and m edges e 1 , e 2 , e 3 , ⋯, e m . Then, the matrix form of the VI (C) is denoted by ½ψ ij n×m and is defined by where Example 2. Consider the VIG G of Figure 2. Here, the number of nodes in G is 4 and the number of edges is 5. So, the matrix form of C is given below as  Example 3. Consider the VIG G as Figure 3. Clearly, G is a complete-VIG and also strong-VIG. Theorem 13. A complete-VIG is a strong VIG.  Definition 14. Let p = p 1 , p 2 , ⋯, p n−1 = q, p n = r are the n nodes in a VIG G. Then, p 1 , ðp 1 , p 1 p 2 Þ, p 1 p 2 , ðp 2 , p 1 p 2 Þ, p 2 , ⋯, q, ðq, qrÞ, qr, ðr, qrÞ is called an incidence path in G. The incidence strength of this path is shown by ðC t p,qr , C f p,qr Þ and is described as C t p,qr = C t ðp 1 , p 1 p 2 Þ∧C t ðp 2 , p 1 p 2 Þ∧⋯∧ C t ðq, qrÞ∧C t ðr, qrÞ and C  Definition 15. Let ab be an edge of a VIG G. If C t ðp, pqÞ > 0, C t ðq, pqÞ > 0, ψ f ðp, pqÞ > 0, and ψ f ðq, pqÞ > 0, then ðp, pqÞ and ðq, pqÞ are called pairs. G is said to be connected if there is an incidence path between every pair. ðp, pqÞ lies on the pairs ðw 1 , w 1 k 1 Þ in G and ðw 2 , w 2 k 2 Þ in H. This means both the pairs ð w 1 , w 1 k 1 Þ and ðw 2 , w 2 k 2 Þ are the pairs of G. If C t ðw 1 , w 1 k 1 Þ = C t ðw 2 , w 2 k 2 Þ and C f ðw 1 , Example 5. Consider the VIG G as Figure 5.  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: E = fRostamiðROÞ, FalahðFAÞ, KaramiðKAÞ, AsadiðASÞ, MahdaviðMAÞ, BagheriðBAÞg. 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 30% of the power needed to do the university work as university educational director but does not have the 40% knowledge needed to be the boss. The incidence edge Karami-Bagheri shows that there is only 20% interaction and friendship between these two employees and unfortunately they have 60% disagreement and conflict. It is clear that Rostami, Kazemi and Asadi obey Fallah. Fallah's dominance rate over all three is equal to 20%. 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 50%.

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.