# New Results in Vague Incidence Graphs with Application

Authors: Yongsheng Rao; Saeed Kosari; Mehdi Gheisari
Journal: Journal of Function Spaces
 (2022)

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## 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.


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## Body

## 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]).
LetG=V,E be a graph. Then, G∗=V,E,I is called an incidence graph (IG), so that I⊆V×E. If V=p,q, E=pq and I=p,pq, then V,E,I is an IG even though q,pq∉I. The pair p,pq is called an incidence pair or simply a pair. If p,pq,q,pq,q,qr,r,qr∈I, then pq or qr are called neighbor edges.Definition 2 (see [20]).
LetG∗ be an IG and σ be a FSS of V and μ, a FSS of E. Let ψ be a FSS of I. If ψv,pq≤σv∧μpq, ∀v∈V and pq∈E, then, ψ is called a FI of graph G∗ and G=σ,μ,ψ is named a FIG of G∗.Definition 3 (see [6]).
A VSA is a pair tA,fA on set V which tA and fA are taken as real valued functions which can be defined on V⟶0,1 so that tAp+fAp≤1, ∀p∈V.Definition 4 (see [11]).
A VG is defined to be a pairG=A,B, which A=tA,fA is a VS on V and B=tB,fB is a VS on E⊆V×V so that for each pq∈E, tBpq≤tAp∧tAq, fBpq≥fBp∨fBq.Definition 5 (see [12]).
LetG be a VG and m,n∈V.
(i)
A pathp:m=m0,m1,⋯,mt−1,mt=n in G is a sequence of distinct nodes so that tBmi−1,mi>0,fBmi−1,mi>0, i=1,2,⋯,t and the length of the path is t(ii)
Ifp:m=m0,m1,⋯,mt−1,mt=n be a path of length t between m and n, then tBmnk and fBmnk are defined as
(1)tBmnk=suptBm,m1∧tBm1,m2∧⋯,∧tBmk−1,n,fBmnk=inffBm,m1∨fBm1,m2∨⋯∨fBmk−1,n.tBmn∞,fBmn∞ is said to be the strength of connectedness between two nodes m and n in G, which tBmn∞=supk∈NtBmnk and fBmn∞=infk∈NfBmnk.(iii)
IftBmn≥tB∞mn and fBmn≤fB∞mn, 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 strongDefinition 6 (see [12]).
For a VGG, if tBmn≥tBmn∞ and fBmn≤fBmn∞, then the edge ab is called a strong edge of G.All the basic notations are shown in Table1.Table 1
Some basic notations.
NotationMeaningFGFuzzy graphVSVague setIGIncidence graphFSSFuzzy subsetVIVague incidenceVIPSGVague incidence partial subgraphVISGVague incidence subgraphVSGVague subgraphCVIGConnected vague incidence graphSPStrong pairFIFuzzy incidenceCPCut pair


## 3. New Concepts of Vague Incidence Graph

Definition 7.
G=A,B,C is called a VIG of underlying crisp-IG G∗=V,E,I, if:
(2)A=tAm,fAm∣m∈V,B=tBpq,fBpq∣pq∈E,C=tCm,pq,fCm,pq∣m,pq∈I,so that A, B are VSs on V and E⊆V×V, respectively. C=Ct,Cf is a mapping from V×E to 0,1,0,1 and we have: tBpq≤tAp∧tAq, fBpq≥fAp∨fAq, tCm,pq≤tAm∧tBpq,fCm,ab≥fAm∨fBpq,∀m∈V,pq∈E, and 0≤tAm+fAm≤1, 0≤tBpq+fBpq≤1, 0≤tCm,pq+fCm,pq≤1.Example 1.
Consider an incidence graphG∗=V,E,I so that V=p,q,r,s, E=pq,qr,qs,rs,ps and I=p,pq,q,pq,q,qr,r,qr,q,qs,s,qs,r,rs,s,rs,s,ps,p,ps as shown in Figure 1. Clearly, G=A,B,C is a VIG of G∗, denoted in Figure 2, which
(3)A=p0.2,0.4,q0.3,0.5,r0.4,0.6,s0.5,0.5,B=pq0.2,0.5,qr0.2,0.6,qs0.2,0.6,rs0.3,0.6,ps0.2,0.5,C=p,pq0.1,0.5,q,pq0.2,0.5,q,qr0.1,0.6,r,qr0.2,0.6,q,qs0.2,0.7,s,qs0.1,0.6,r,rs0.3,0.7,s,rs0.2,0.6,s,ps0.2,0.5,p,ps0.1,0.5.Definition 8.
LetG=A,B,C be a VIG of underlying crisp-IG G∗, and ζ has n nodes p1,p2,p3,⋯,pn and m edges e1,e2,e3,⋯,em. Then, the matrix form of the VI (C) is denoted by ψijn×m and is defined by
(4)ψijn×m=p1p2⋮pnψ11ψ12ψ13⋯ψ1mψ21ψ22ψ23⋯ψ2m⋮⋮⋮⋮ψn1ψn2ψn3⋯ψnme1e2e3⋯em,where ψij=tCpi,pipj,fCpi,pipj.Example 2.
Consider the VIGG 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
(5)ψij4×5=qrsp0.1,0.60.2,0.70.2,0.50,00,00.2,0.60,00,00.3,0.70,00,00.1,0.60,00.2,0.60.2,0.50,00,00.1,0.50,00.1,0.5.qrqspqrspsDefinition 9.
A VIGH=A′,B′,C′ called a VIPSG of the VIG G=A,B,C if tA′p≤tAp, fA′p≥fAp, tB′pq≤tBpq, fB′pq≥fBpq, tC′p,pq≤tCp,pq, and fC′p,pq≥fCp,pq, for all p∈V and pq∈E. Also, H is called a VISG of the VIG G, if V′⊆V, E′⊆E, C′⊆C, tA′p=tAp, fA′p=fAp, tB′pq=tBpq, fB′pq=fBpq, tC′p,pq=tCp,pq, and fC′p,pq=fCp,pq, ∀p∈V and ∀pq∈E.Figure 1
Incidence graphG∗.Figure 2
Vague incidence graphG.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 VIGG must be a VIPSG of G.Proof.
LetH be a VISG of the VIG G. Then, from the definitions, we have H satisfies all the conditions to be a VIPSG of the VIG G. So, H is a VISG of the VIG G.Proposition 11.
IfH~ be a VISG of the VIG G~, then the VG H is a VSG of the VG G.Proof.
H~ is a VISG of the VIG G~. Then, from the definitions, we have the corresponding VG H satisfies all the conditions to be a VSG of the VG G. Hence, H is a VSG of the VG G.Definition 12.
A VIGG=A,B,C is called complete-VIG if tCp,pq=mintAp,tBpq and fCp,pq=maxfAp,fBpq, ∀p∈V and ∀pq∈E. A VIG G is called strong if tCp,pq=mintAp,tBpq and fCp,pq=maxfAp,fBpq, ∀ pairs p,pq in G.IfG=A,B,C is a complete-VIG and the nodes p,q are neighbor to the edge pq, then tCp,pq=mintAp,tBpq=tBpq=mintAq,tBpq=tCq,pq and fCp,pq=maxfAp,fBpq=fBpq=maxfAq,fBpq=fCq,pq.Example 3.
Consider the VIGG as Figure 3. Clearly, G is a complete-VIG and also strong-VIG.Theorem 13.
A complete-VIG is a strong VIG.Proof.
LetG be a complete-VIG and p,pq be a pair in G. Then, tCp,pq=mintAp,tBpq and fCp,pq=maxfAp,fBpq, ∀p∈V and pq∈E. Hence, tCp,pq=mintAp,tBpq and fCp,pq=maxfAp,fBpq, ∀ pairs p,pq in G.Definition 14.
Letp=p1,p2,⋯,pn−1=q,pn=r are the n nodes in a VIG G. Then, p1,p1,p1p2,p1p2,p2,p1p2,p2,⋯,q,q,qr,qr,r,qr is called an incidence path in G. The incidence strength of this path is shown by Cp,qrt,Cp,qrf and is described as Cp,qrt=Ctp1,p1p2∧Ctp2,p1p2∧⋯∧Ctq,qr∧Ctr,qr and Cp,qrf=Cfp1,p1p2∨Cfp2,p1p2∨⋯∨Cfq,qr∨Cfr,qr. The incidence strength of connectedness between p and qr in G is ICONNGp,qr=ICONNGtp,qr,ICONNGfp,qr,, where ICONNGtp,qr=∨Cp,qrt is the maximum value of true part of incidence strength of all the paths between p and qr and ICONNGfp,qr=∧Cp,qrf is the minimum value of the false part of incidence strengths of all the paths between p and qr.Example 4.
Consider the CVIGG of Figure 4.Figure 3
Complete VIGG.Figure 4
VIGG.Here,Ctp,pq∧Ctq,pq∧Ctq,qr=0.1, Cfp,pq∨Cfq,pq∨Cfq,qr=0.7, Ctp,pr∧Ctr,pr∧Ctr,qr=0.1, Cfp,pr∨Cfr,pr∨Cfr,qr=0.7. So, ICONNGtp,qr=∨0.1,0.1=0.1 and ICONNGfp,qr=∧0.7,0.7=0.7. Therefore, the incidence strength of connectedness between p and qr is ICONNGp,qr=0.1,0.7.Definition 15.
Letab be an edge of a VIG G. If Ctp,pq>0, Ctq,pq>0, ψfp,pq>0, and ψfq,pq>0, then p,pq and q,pq are called pairs. G is said to be connected if there is an incidence path between every pair.Theorem 16.
LetG=A,B,C be a VIG and H=A′,B′,C′ be a VISG of G. Then, for any pair p,pq in G, ICONNGtp,pq≥ICONNHtp,pq and ICONNGfp,pq≤ICONNHfp,pq.Proof.
ConsiderH is a VISG of G. From the definition of VISG, we have C′tp,pq=Ctp,pq and C′fp,pq=Cfp,pq, for all pairs p,pq in H. But, ICONNGtp,pq, ICONNHtp,pq, and ICONNGfp,pq, ICONNHfp,pq lies on same incidence pair of H and H~ or lies on different pairs of H and H~. Now, there arises two cases.
Case 1. SupposeICONNGtp,pq, ICONNHtp,pq, and ICONNGfp,pq, ICONNHfp,pq lies on same pair w,wk of H and G. Then, from the definition of VISG we have C′tw,wk=Ctw,wk and C′fw,wk=Cfw,wk. Then, ICONNGtp,pq=ICONNHtp,pq and ICONNGfp,pq=ICONNHfp,pq.
Case 2. SupposeICONNGtp,pq, ICONNHtp,pq, and ICONNGfp,pq, ICONNHfp,pq lies on the pairs w1,w1k1 in G and w2,w2k2 in H. This means both the pairs w1,w1k1 and w2,w2k2 are the pairs of G. If Ctw1,w1k1=Ctw2,w2k2 and Cfw1,w1k1=Cfw2,w2k2, then ICONNGtp,pq=ICONNHtp,pq and ICONNGfp,pq=ICONNHfp,pq. If Ctw1,w1k1≠Ctw2,w2k2 or Cfw1,w1k1≠Cfw2,w2k2 or both, then ICONNGtp,pq>ICONNHtp,pq or ICONNGfp,pq<ICONNHfp,pq or both. Hence, in any case, ICONNGtp,pq≥ICONNHtp,pq and ICONNGfp,pq≤ICONNHfp,pq.Definition 17.
A pairp,pq in G=A,B,C is SP if Ctp,pq≥ICONNGtp,pq and Cfp,pq≤ICONNGfp,pq.Example 5.
Consider the VIGG as Figure 5.
Here,Ctp,pq=0.1, Cfp,pq=0.5, Ctq,qr=0.1, Cfq,qr=0.6, ICONNGtp,pq=0.1, ICONNGfp,pq=0.7, ICONNGtq,qr=0.1, ICONNGfq,qr=0.6.
Hence,Ctp,pq=ICONNGtp,pq=0.1, Cfp,pq=0.5<ICONNGfp,pq=0.7 and Ctq,qr=ICONNGtq,qr=0.1, Cfq,qr=ICONNGfb,bc=0.6. Therefore, p,pq and q,qr are SPs.Theorem 18.
LetG=A,B,C be a VIG and for a pair p,pq, Ctp,pq,Cfp,pq=∨Ctw,wk,∧Cfw,wk, for all pairs w,wk in G. Then, the pair p,pq is a SP.Proof.
Letp,pq be a pair in the VIG G. If there is an unique pair w,wk in G so that Ctp,pq=∨Ctw,wk and Cfp,pq=∧Cfw,wk, for all pairs w,wk in G, then the true part of the strength of all the paths without p,pq must be less than Ctp,pq and the false part of the strength of all the paths without p,pq must be greater than Cfp,pq. Then, p,pq is a strong pair.
Again, if there are more than one pairp,pq in G so that Ctp,pq=∨Ctp,pq and Cfp,pq=∧Cfw,wk, for all pairs w,wk in G, then Ctp,pq=Ctw,wk and Cfp,pq=Cfw,wk, for all pairs w,wk in G−p,pq.
Now from the definition, we haveCtp,pq≥ICONNGtw,wk, Cfp,pq≤ICONNGfw,wk, for all pairs w,wk in G. Hence, p,pq is a SP.Definition 19.
LetG be a VIG and H be a VISG of G so that for a pair p,pq in G, H=G−p,pq. If, ICONNGtw,wk>ICONNHtw,wk and ICONNGfw,wk<ICONNHfw,wk, for some pair w,wk in G, then, the pair p,pq is called an incidence-CP of G.Theorem 20.
Letp,pq be a pair in a VIG G=A,B,C so that Ctp,pq=∧tAp,tBpq and Cfp,pq=∨fAp,fBpq. Then, the pair p,pq is a SP of G.Proof.
LetH=A′,B′,C′ be a VISG of a VIG G=A,B,C so that H=G−p,pq, where p,pq is a pair of G. If H is disconnected, then p,pq should be a CP of G. Then, ICONNGtp,pq>ICONNHtp,pq and ICONNGfp,pq<ICONNHfp,pq. Then by definition, p,pq is a sp. If H is connected, then ∃ pairs p,pr for some p≠r, so that p,pr and q,pq lies in an incidence path from p to pq in H. Then, ICONNHtp,pq≤∧Ctp,pr,Ctq,pq≤∧tAp,tBpr,tAq,tBpq≤∧tAp,tBpq=Ctp,pq and ICONNHfp,pq≥∧Cfp,pr,Cfq,pq≥∧fAp,fBpr,fAq,fBpq≥∧fAp,fBpq=Cfp,pq. Thus, Ctp,pq≥ICONNHtp,pq and Cfp,pq≥ICONNHfp,pq. Therefore, p,pq is a SP of G.Theorem 21.
Every pair of a complete VIG is a sp.Proof.
Letp,pq be a pair in a complete-VIG G=A,B,C. Then Ctp,pq=∧tAp,tBpq and Cfp,pq=∨fAp,fBpq. Now, by Theorem 20, it can be proved that p,pq is a SP of G. Since p,pq is arbitrary in the complete-VIG G, so all the pairs of G are SP.Figure 5
Vague incidence graphG with strong pairs.


## 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:E=RostamiRO,FalahFA,KaramiKA,AsadiAS,MahdaviMA,BagheriBA.
(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 universityGiven 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 Tables2 and 3. The adjacency matrix corresponding to Figure 6 is shown in Table 4.Table 2
Name of employees in an university and their services.
NameServicesAsadiHead of the universityRostamiUniversity educational directorFalahDirector of researchBagheriFinancial affairs managerKaramiDirector of student nutritionMahdaviHead of securityTable 3
The level of staff capability.
RostamiFalahAsadiMahdaviBagheriKaramitA0.30.30.50.40.30.2fA0.40.20.20.30.50.5Figure 6
Vague incidence graph.Table 4
Adjacency matrix corresponding to Figure6.
RostamiFalahKaramiAsadiMahdaviBagheriRostami0,00.2,0.40.2,0.50,00,00,0Falah0.2,0.50,00.2,0.50.2,0.40,00,0Karami0.1,0.50.1,0.60,00.1,0.60,00.2,0.6Asadi0.3,0.50.3,0.30.2,0.50,00.4,0.30.3,0.5Mahdavi0,00,00,00.3,0.40,00.2,0.7Bagheri0,00,00.2,0.70.2,0.50.2,0.60,0Figure6 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%.


## 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.



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