# Certain Concepts in Vague Graph with an Application

Authors: Xiaolong Shi; Maryam Akhoundi; Mohammadsadegh Monemrad; Swati Agarwal
Journal: Discrete Dynamics in Nature and Society
 (2022)

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## Abstract

Vague graphs (VGs), belonging to the FGs family, have good capabilities when faced with problems that cannot be expressed by FGs. When an element’s membership is not clear, neutrality is a good option that can be well-supported by a VG. The previous definition limitations in irregular-FG have led us to offer new definitions in VGs. So, in this paper, we introduce the concepts of strongly edged irregular vague graphs (SEIVGs), strongly edged totally irregular vague graphs (SETIVGs), and perfectly regular vague graphs (PRVGs) with several examples. A comparative study between SEIVGs and SETIVGs is presented. Finally, an application of vague influence digraph has been presented.


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## Body

## 1. Introduction

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 Zade [1]. The FS focuses on the membership degree of an object in a particular set. 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 [2] introduced false membership degrees and defined a VS as the sum of degrees not greater than 1. The first definition of FGs was proposed by Kauffman [3] in 1993, from Zade’s fuzzy relations [4, 5]. But Rosenfeld [6] introduced another elaborated definition, including fuzzy vertex and fuzzy edges and several fuzzy analogs of graph theoretic concepts such as paths, cycles, connectedness, etc. Ramakrishna [7] introduced the concept of VGs and studied some of their properties. Balakrishnan et al. [8] presented k-neighborhood RGs. Akram et al. [9–11] defined the vague hypergraphs and Cayley vague graphs. Borzooei and Rashmanlou [12] investigated new concepts of regularity in VGs. Santhimaheswari et al. [13] discussed edge irregularity in FGs. Gani and Radha [14] introduced RFGs and TRFGs. Nandhini and Kamaraj [15] studied strongly irregular interval-valued fuzzy graphs. Gani and Latha [16] defined the concept of IFGs, NIFGs, and HIFGs in 2008. Radha and Kumaravel investigated the concept of edge degree, total edge degree, and edge regular fuzzy graphs and discussed the degree of an edge in some fuzzy graphs [17]. Kosari et al. [18] defined a vague graph structure with an application in medical sciences. Rashmanlou et al. [19] introduced some remarks on VGs. Rao et al. [20–22] studied dominating set, equitable dominating set, valid degree, isolated vertex, and some properties of VGs with the novel application. Kou et al. [23] investigated g-eccentric nodes and vague detour g-boundary nodes in VGs. Shi et al. [24, 25] introduced total dominating set, perfect dominating set, and global dominating set in product vague graphs. Banitalebi et al. [26] defined 2-domination in vague graphs. Ghorai et al. [27] presented regular product vague graphs and product vague line graphs.VGs have a wide range of applications in the field of psychological sciences as well as in the identification of individuals based on oncological behaviors. With the help of VGs, the most efficient person in an organization can be identified according to the important factors that can be useful for an institution. Likewise, a VG is capable of focusing on determining the uncertainty, combined with inconsistent and indeterminate information, of any real-world problem, in which FGs may not lead to adequate results. Therefore, in this paper, the new concepts of PRVG, SEI, and SETIVGs are introduced. A comparative study between SEIVGs and SETIVGs has been conducted. Finally, an application of vague influence digraph has been presented.


## 2. Preliminaries

Definition 1.
(see [2]). A VSAis a pairtA,fAon set X wheretAandfAare taken as real-valued functions which can be defined onV⟶0,1so thattAp+fAp≤1,∀p∈X.Definition 2.
(see [7]). A VG is defined as a pairG=A,B, whereA=tA,fAis a VS onVandB=tB,fBis a VS onE⊆V×Vso that for eachpq∈E,tBpq≤tAp∧tAq,fBpq≥fBp∨fBq.Definition 3.
(see [12]). iLetG=A,Bbe a VG. The degree of a nodepisdGp=dGtp,dGfp, wheredGtp=∑p≠qtBpqanddGfp=∑p≠qfBpq, forpq∈E.
ii The TD of a node p is defined as tdp=tdtp,tdfp, where tdtp=∑tBpq+tAp=dGtp+tAp and tdfp=∑fBpq+fAp=dGfp+fAp, pq∈E.Definition 4.
(see [28]). LetGbe a VG. The degree of an edgepqis defined asdGpq=dGtpq,dGfpq, wheredGtpq=dGtp+dGtq−2tBpq,dGfpq=dGfp+dGfq−2fBpq.Definition 5.
(see [28]). LetGbe a VG. The TD of an edgepqis defined astdGpq=tdGtpq,tdGfpq, wheretdGtpq=dGtp+dGtq−tBpq,tdGfpq=dGfp+dGfq−fBpq. Note that the degree of an edgepqin the underlying edge is defined asdGpq=dGp+dGq−2.Definition 6.
(see [19]). A VGGis called SVG iftBqiqj=mintAqi,tAqjandfBqiqj=maxfAqi,fAqj, for each edgeqiqj∈E. A VGGis called complete-VG iftBqiqj=mintAqi,tAqjandfBqiqj=maxfAqi,fAqj,∀qiqj∈E.Definition 7.
(see [12]). A VGGis said to be:iR1,R2-regular ifdGqi=R1,R2, for allqi∈Vand alsoGis called to be RVG of degreeR1,R2.iiR1,R2-TR iftdGqi=R1,R2,∀qi∈V.iiiR1,R2-ERVG ifdGpq=R1,R2,∀pq∈E.ivR1,R2- TERVG iftdGpq=R1,R2,∀pq∈E.
All the basic notations are shown in Table1.Table 1
: Some basic notations.
NotationMeaningFGFuzzy graphVSVague setVGVague graphTDTotal degreeIVGIrregular vague graphHIFGHighly irregular fuzzy graphNIFGNeighborly irregular fuzzy graphSVGStrong vague graphCVGConnected vague graphRFGRegular fuzzy graphTRFGTotally regular fuzzy graphPRPerfect regularEREdge regularTRTotally regularTERTotally edge regularCFConstant functionAMAdjacency matrixUCGUnderlying crisp graphERVGEdge regular vague graphTIVGTotally irregular vague graphHIVGHighly irregular vague graphTERVGTotally edge regular vague graphPRVG,Perfect regular vague graphSEIVGStrongly edge irregular vague graphSETIVGStrongly edge totally irregular vague graphPERVGPerfect edge regular vague graphSIVGStrongly irregular vague graphNEIVGNeighborly edge irregular vague graph


## 3. New Concepts of Regularity in Vague Graphs

In this section, PRVGs, SEIVGs, and SETIVGs are introduced and several properties of them are studied.Definition 8.
A PRVG is a VG that is both regular and TR.Definition 9.
A PERVG is a VG that is both ER and TER.Lemma 1.
LetG=A,Bbe a PRVG. Then,tA:V⟶0,1, andfA:V⟶0,1are CFs.Proof.
G is a PR, so G is both R1,R2-regular and S1,S2 TR. Then, we have tdGtq=dGtq+tAq=dGtp+tAp=tdGtp, tdGfq=dGfq+fAq=dGfp+fAp=tdGfp, ∀p,q∈V. Since dGq=dGp=R1,R2 and tdGq=tdGp=S1,S2, we have tAq=tAp, fAq=fAp, hence, if G is PR, then tA,fA should be CF.
It is clear that this is not a sufficient condition, as any IVG with a CFtA,fA is neither TR nor PR. Therefore, we can classify PRVGs as the graphs that are regular with a CF tA,fA.Theorem 1.
A VGGis PR if and only if it holds in the following conditions:i∑k≠itBqjqk=∑k≠jtBqjqk,∑k≠ifBqiqk=∑k≠jfBqjqk,∀i,j∈1,…,q.iitAqi=tAqj,fAqi=fAqj,∀i,j∈1,…,q.Proof.
SupposeG is a PR. Thus, G is RVG. Hence, it satisfies i. From Lemma 1 we conclude that ii is hold, too.
Conversely, we assume thatG is a VG holds in both conditions i and ii. Since i is the definition of regularity, G is R1,R2-regular. From ii we have tA,fA=S1,S2 is a constant function, tdGtq=R1+S1, tdGfq=R2+S2, and ∀q∈V. So, G is both regular and TR. Therefore, G is PR. □Remark 1.
LetGbe a PRVG and lettA,fA=R,S,∀q∈V. Then, the order ofGisOG=RV.Proposition 1.
LetGbe a PRVG and letdq=R1,R2,∀q∈V. Then the size ofGisStG=R1V/2andSfG=R2V/2.Proof.
SinceG is PR, so dGtq=R1, dGfq=R2, and ∀q∈V. Hence, ∑q∈VdGtq=R1V, ∑q∈VdGfq=R2V. However, since dGtq=∑p≠qtBpq, dGfq=∑p≠qfBpq, we have ∑q∈VdGtq=∑q∈V∑p≠qtBpq=2∑pq∈EtBpq=2StG, ∑q∈VdGfq=\\∑q∈V∑p≠qfBpq=2∑pq∈EfBpq=2SfG. Thus, we conclude that the size of a PRVG is R1V/2,R2V/2.Lemma 2.
LetGbe a PERVG. Then,tBandfBare constant functions.Proof.
SinceG is PER, so G is both R1,R2-ER and S1,S2-TER. Hence we get S1=tdGtpq=dGtpq+tBpq=R1+tBpq=tdGtxy=dGttu+tBtu=R1+tBtu, S2=tdGfpq=dGfpq+fBpq=R2+fBpq=tdGftu=dGftu+fBtu=k2+fBtu, hence, tBpq=tBtu and fBpq=fBtu. Since pq and tu were arbitrary chosen edges, the proof is complete.Theorem 2.
A VGGis PER if and only if it holds in the following conditions:(1)i∑z≠ptBpz+∑z≠qtBqz−2tBpq=∑z≠ttBtz+∑z≠utBuz−2tBtu,∑z≠pfBpz+∑z≠qfBqz−2fBpq=∑z≠tfBtz+∑z≠ufBuz−2fBtu,∀pq,tu∈E,iitBpq=tBtu,fBpq=fBtu,∀pq,tu∈E.Proof.
LetG be a PER, then, the proof of parts (i) and (ii) is obvious according to definitions of PER and ERVG. Conversely, assume that G is a VG that satisfies both conditions i and ii, i.e., G is R1,R2-ER and has a constant tB=S1 and fB=S2. Because of i is the definition of ER, ii shows that tdGtpq=R1+S1=tdGttu, tdGfpq=R2+S2=tdGftu, and forpq,tu∈E.Remark 2.
LetGbe a PERVG andtBpq=S1,fBpq=S2, andforpq∈E. Then, the size ofGisSG=S1E,S2E.Lemma 3.
IfGis regular andtB,fB=c,dis a CF, then,Gis PER.Proof.
SupposeG is an RVG and tB,fB=S1,S2 be a CF. Then, dGtq,dGfq=R1,R2, ∀q∈V where R1,R2 is a multiple of S1,S2. So, the degree of an arbitrary edge of G is(2)dGtpq=dGtp+dGtq−2tBpq=2R1−2S1=2R1−S1,Hence, G is ER. Since the TD of an edge in a VG is tdGtpq=dGtpq+tBpq, tdGfpq=dGfpq+fBpq, we have that tdGtpq=2R1−S1+S1=2R1−S1, tdGfpq=2R2−S2+S2=2R2−S2, for all edges in G. Thus, G is TER and therefore PER.
Since PRVGs are regular, so, PRVGs with a CFtB,fB are necessarily PER. However, it is not the case in general that TRVGs with a constant function tB,fB are PER; let us just consider the following example.Example 1.
LetG be a VG as Figure 1.
LettB,fB=0.1,0.3be a CF. Then, the degree ofaandbis0.2,0.6while the degree ofcanddis0.3,0.9, yet the TD of each node is precisely0.4,0.9. Hence, this counterexample has a constanttB,fBand is TR but is not RVG. Now consider the edgesacandcd. We have thatda,c=0.3,0.9≠0.4,1.2=dc,d. Thus, it is not ER. SincetB,fBis constant, clearly this counterexample cannot be TER as it is not ER.Figure 1
Vague graphG.Theorem 3.
IfGis PR and complete, thenGis PER.Proof.
SinceG is PR, tAp=tAq, fAp=fAq, and ∀p,q∈V. Also G is complete, so tBpq=tAp∧tAq, fBpq=fAp∨fAq, and ∀pq∈E. Combined, these two facts yield that tB,fB is a constant function. Thus, we may apply Lemma 3 and complete the proof.Definition 8.
The AMAGof a VGGis ann×nmatrix defined asAG=aij, whereaij=tBqiqj,fBqiqj. We note thattBqiqjshows the strength of the relationship betweenqiandqj, andfBqiqjdescribes the strength of the non-relationship amongqiandqj.Remark 3.
LetGbe a PERVG withtB=S1andfB=S2and letG∗be the UCG ofG. Then,AG=SAG∗.
Given this simple remark, we can immediately enumerate several spectral properties of PERVGs.Theorem 4.
LetGbe a PERVG andG∗be its UCG. Ifλis an eigenvalue ofG∗, then,Sλis an eigenvalue ofG.Proof.
This follows directly from the fact thatAX=λX implies SAX=SλX.Definition 9.
The energy of VGGis defined as(3)EG=E(tBqlqm,E(fBqlqm=∑l=1nλl,∑l=1nμl.Theorem 5.
LetGbe a PERVG with UCGG∗. If the energy ofG∗isEG, then,EG=SEG∗.Proof.
We know thatEG∗=∑i=1nλi that the λi shows the eigenvalues of G∗. From Theorem 4 the energy of G is calculated as EG=∑i=1nSλi,∑i=1nSμi=S∑i=1nλi,S∑i=1nμi=SEG∗.Definition 10.
LetGbe a VG. Then,iG is called to be an IVG if there is a node that is neighbor to nodes with different degrees.iiG is called to be a TIVG if there is a node that is neighbor to nodes with different TDs.iiiG is called to be a SIVG if every couple of nodes have a different degree.ivG is called to be a HIVG if every node in G, which is neighbor to the nodes having different degrees.Definition 11.
LetGbe a CVG. Then(i)
G is called a NEIVG if its neighbor edges have different degrees.(ii)
G is called a NETIVG if its neighbor edges have different TDs.(iii)
G is called a SEIVG if each edge has a different degree (or) no two edges have the same degree.(iv)
G is called a SETIVG if each edge has a different TD or no two edges have similar TDs.Example 2.
LetG be a VG that is a cycle with length five. With a simple calculation in Figure 2 we have(4)dGu=0.5,1.3,dGv=0.3,1.1,dGw=0.4,1.3,dGx=0.3,1.6,dGy=0.5,1.7.
Degrees of the edges are calculated as follows:(5)dGuv=0.6,1.4,dGvw=0.3,1.2,dGwx=0.3,1.5,dGxy=0.6,1.5,dGyu=0.2,1.4.
Obviously, each edge has a different degree. So,G is SEIVG. TDs of the edges are as follows:(6)tdGtuv=dGtuv+tBuv=0.6+0.1=0.7,tdGfuv=dGfuv+fBuv=1.4+0.5=1.9,tdGtvw=dGtvw+tBvw=0.3+0.2=0.5,tdGfvw=dGfvw+fBvw=1.2+0.6=1.8,tdGtwx=dGtwx+tBwx=0.3+0.2=0.5,tdGfwx=dGfwx+fBwx=1.5+0.7=2.2,tdGtxy=dGtxy+tBxy=0.6+0.1=0.7,tdGfxy=dGfxy+fBxy=1.5+0.9=2.4,tdGtyu=dGtyu+tByu=0.2+0.4=0.6,tdGfyu=dGfyu+fByu=1.4+0.8=2.2.
Clearly, each edge inG has a different TD. Thus, G is SETIVG. Hence, G is both SEIVG and SETIVG.Figure 2
Vague graphG.Example 3.
SEIVGs need not be SETIVGs. From Figure3 we have(7)dGu=0.5,1,dGv=0.4,1.3,dGw=0.3,1.1,dGuv=0.3,1.1,dGvw=0.5,1,dGwu=0.4,1.3,tdGuv=0.6,1.7,tdGvw=0.6,1.7,tdGwu=0.6,1.7.
We note thatG is SEIVG because each edge has a different degree. Furthermore, G is not SETIVG, since all the edges have same TDs.
Hence, SEIVG need not be SETIVG.Figure 3
Vague graphG.Theorem 6.
LetGbe a CVG andtB,fBis a CF. IfGis SEIVG, then,Gis SETIVG.Proof.
Let us assume thattB,fB is a CF. Let tBpq,fBpq=S1,S2, and ∀pq∈E, where S1,S2 is constant. Let pq and tu be arbitrary edges in E. Assume G is SEIVG. Then, dGpq≠dGtu, where pq and tu are any pair of edges in E. So, dGpq≠dGtu, dGtpq+S1≠dGttu+S1, dGfpq+S2≠dGf+S2tu, dGtpq+tBpq≠dGttu+tBtu, dGfpq+fBpq≠dGftu+fBtu, tdGtpq≠tdGttu, and tdGfpq≠tdGftu, where pq and tu are any pair of edges in E.
Hence,G is SETIVG.Theorem 7.
LetGbe a CVG andtB,fBis a CF. IfGis SETIVG, then,Gis SEIVG.Proof.
Let us assume thattB,fB is a CF. Let tBpq,fBpq=S1,S2, ∀pq∈E, where S1,S2 is constant. Let pq and tu be arbitrary edges in E. Let us assume G is SETIVG. Then, tdGtpq≠tdGftu, tdGfpq≠tdGftu, where pq and tu are any pair of edges in E. tdGtpq≠tdGftu, tdGfpq≠tdGftu. Hence, dGtpq+tBpq≠dGttu+tBtu and dGfpq+fBpq≠dGftu+fBtu. Therefore, dGtpq+S1≠dGttu+S1, dGfpq+S2≠dGf+S2tu, where pq and tu are any pair of edges in E. So, G is SEIVG.Remark 4.
LetGbe a CVG. IfGis both SEIVG and SETIVG, then,tB,fBneed not be a CF.Example 4.
In Example 2,Gis both SEIVG and SETIVG. ButtB,fBis not CF.Theorem 8.
Suppose thatGis a CVG. IfGis SEIVG, then,Gis NEIVG.Proof.
LetG be a CVG. Let us assume that G is SEIVG. Then, each edge in G has a different degree. Hence, neighbor edges have different degrees. So, G is NEIVG.Theorem 9.
LetGbe a CVG. IfGis SETIVG, then,Gis NETIVG.Proof.
Let us assume thatG is a CVG that is SETIVG, then, each edge in G has a different TD. So, neighbor edges have different degrees. Thus, G is NETIVG.Remark 5.
The converse of Theorems 3.26 and 3.27 are not established.Example 5.
SupposeGis a VG that is a path with four nodes.
From Figure4, dGu=0.1,0.5, dGv=0.2,1, dGw=0.2,1, dGx=0.1,0.5, dGuv=0.1,0.5, dGvw=0.2,1, and dGwx=0.1,0.5. Here, dGuv≠dGvw and dGvw≠dGwx. Hence, G is NEIVG. But G is not SEIVG, since dGuv=dGwx. Also, tdGuv=0.2,1, tdGvw=0.3,1.5, and tdGwx=0.2,1. Note that tdGuv≠tdGvw and tdGvw≠tdGwx. Hence, G is NETIVG. But G is not SETIVG, since tdGuv=tdGwx.Figure 4
Vague graphG.Theorem 10.
LetGbe a CVG on andtB,fBis a constant function. IfGis SEIVG, then,Gis an IVG.Proof.
LetG be a CVG, tB,fB be a CF, and tBpq,fBpq=S1,S2, ∀pq∈E, where S1,S2 is constant. Suppose that G is SEIVG. Then, every pair of edges has different degrees. Let pq and qt be adjacent edges in G having distinct degrees. Then, dGtpq≠dGtqt and dGfpq≠dGfqt. So, dGtp+dGtq−2tBpq≠dGtq+dGtt−2tBqt and dGfp+dGfq−2fBpq≠dGfq+dGft−2fBqt. Hence, dGtp+dGtq−2S1≠dGtq+dGtt−2S1 and dGfp+dGfq−2S2≠dGfq+dGft−2S2. Therefore, dGtp+dGtq≠dGtq+dGtt and dGfp+dGfq≠dGfq+dGft, that we get dGp≠dGt. So, ∃ a node q, which is a neighbor to nodes u and t and has different degrees. Hence, G is an IVG.


## 4. Application of Vague Influence Digraph to Find the Most Effective Person in a Hospital

The issue of treatment has been one of the most important issues for any country, because any society that benefits from healthy people will naturally make significant progress. But, there are very important issues that can play a significant role in the treatment and improvement of patients, one of which is the amount of labor and also the ability of staff in a hospital or treatment center, because the more capable and faster the workforce is, the more patients will be accepted and also the treatment process of patients will be done more quickly.In this section, we want to identify the most effective person in a hospital. To do this, we consider the vertices of the vague influence graph as the heads of each ward of the hospital, and the edges of the graph as the degree of interaction and influence each has on other. For this hospital, the set of staff isF=Ali,Hassan,Jafar,Mohammad,Ehsan,Nima,St.(a)
Hassan has been working with Mohammad for 10 years and values his views on business issues.(b)
Ali has been the head of the hospital for a long time, and not only Hassan but also Mohammad are very satisfied with Ali’s performance.(c)
For a hospital, the protection and maintenance of medical equipment and facilities is very important. Nima is a suitable person for this job.(d)
Jafar and Nima have a long history of conflict.(e)
Jafar has a very important role in the laboratory of the hospital.Given the above, we consider a vague influence graph. The vertices represent each of the hospital staff members. Note that each staff member has the desired ability as well as shortcomings in the performance of their duties. Therefore, we use a vague set to express the weight of the vertices. The true membership indicates the efficiency of the employee, while the false membership shows the lack of management and shortcomings of each staff member. But the edges describe the level of relationships and friendships among employees. If these relationships were stronger, patients would be treated faster. Therefore, the edges can be considered as a vague set so that the true membership indicates a friendly relationship between both employees, and the false membership shows the degree of conflict and difference between the two officials. Names of employees and levels of staff capability are shown in Tables2 and 3. The adjacency matrix corresponding to Figure 5 is shown in Table 4.Figure 5 shows that Ali has 90% of the power needed to do the hospital work as the head but does not have the 10% knowledge needed to be the boss. The directional edge of Mohammad-Hassan shows that there is 60% interaction and friendship between these two employees, and unfortunately, they have 30% disagreements and conflicts. Clearly, Mohammad has dominion over both Ali and Hassan, and his dominance over both is 60%. It is clear that, Mohammad is the most influential employee of the hospital because he controls both the head of the hospital and the manager of the pharmacy, who have 90% of the power in the hospital.Table 2
Name of employees in a hospital and their services.
NameServicesAliHead of the hospitalHassanPharmacy managerJafarHead of the laboratoryMohammadFinancial affairs managerEhsanHead of nursesNimaHead of securityStaff(St)StaffTable 3
The level of staff capability.
AliHassanJafarMohammadEhsanNimaSttA0.90.90.80.70.60.50.4fA0.10.10.20.20.30.30.4Figure 5
Vague influence digraph.Table 4
Adjacency matrix corresponding to Figure5.
AliHassanJaffarMohammadEhsanNimaStAli0,00.8,0.20,00,00,00,00,0Hassan0,00,00.7,0.20,00,00,00.3,0.5Jafar0,00,00,00,00,00,00.3,0.5Mohammad0.6,0.30.6,0.30,00,00,00,00.3,0.4Ehsan0,00,00,00,00,00.5,0.30.2,0.5Nima0,00,00,00,00,00,00.3,0.5St0,00,00,00,00,00,00,0


## 5. Conclusion

VGs have a wide range of applications in the field of psychological sciences as well as in the identification of individuals based on oncological behaviors. With the help of VGs, the most efficient person in an organization can be identified according to the important factors that can be useful for an institution. Today, VGs play an important role in social networks and allow users to find the most effective person in a group or organization. Hence, in this paper, certain notions of SEIVG and SETIVG are introduced. A comparative study between SEIVGs and SETIVGs has been conducted. Likewise, the concept of PRVG that is both regular and TR is given. Finally, an application of vague influence digraph has been introduced. In our future work, we will introduce the connectivity index, Winer index, and Randic index in VGs and investigate some of their properties.



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