# Novel Concepts in Vague Graphs with Application in Hospital’s Management System

Authors: Xiaolong Shi; Saeed Kosari; N. Mehdipoor; A.A. Talebi; G. Muhiuddin
Journal: Journal of Mathematics
 (2022)

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## Abstract

Many problems of practical interest can be modeled and solved by using vague graph (VG) algorithms. Vague graphs, belonging to the fuzzy graphs (FGs) family, have good capabilities when faced with problems that cannot be expressed by FGs. Hence, in this paper, we introduce the notion ofη,γ- HMs of VGs and classify homomorphisms (HMs), weak isomorphisms (WIs), and coweak isomorphisms (CWIs) of VGs by η,γ- HMs. Hospitals are very important organizations whose existence is directly related to the general health of the community. Hence, since the management in each ward of the hospital is very important, we have tried to determine the most effective person in a hospital based on the performance of its staff.


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## Body

## 1. Introduction

Graphs, from ancient times to the present day, have played a very important role in various fields, including computer science and social networks, so that with the help of the vertices and edges of a graph, the relationships between objects and elements in a social group can be easily introduced. But there are some phenomena in our lives that have a wide range of complexities that make it impossible for us to express certainty. These complexities and ambiguities were reduced with the introduction of FSs by Zadeh [1]. Since then, the theory of FSs has become a vigorous area of research in different disciplines including logic, topology, algebra, analysis, information theory, artificial intelligence, operations research, and neural networks and planning [2–6]. The FS focuses on the membership degree of an object in a particular set. But membership alone could not solve the complexities in different cases, so the need for a degree of membership was felt. To solve this problem, Gau and Buehrer [7] 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 Kafmann [8] in 1993, from Zade’s fuzzy relations [9, 10]. But Rosenfeld [11] introduced another elaborated definition including fuzzy vertex and fuzzy edges and several fuzzy analogs of graph theoretic concepts such as paths, cycles, and connectedness. Ramakrishna [12] introduced the concept of VGs and studied some of their properties. Akram et al. [13–16] defined the vague hypergraphs, Cayley-VGs, and regularity in vague intersection graphs and vague line graphs. Rashmanlou et al. [17] investigated categorical properties in intuitionistic fuzzy graphs. Bhattacharya [18] gave some remarks on FGs, and some operations of FGs were introduced by Mordeson and Peng [19]. The concepts of weak isomorphism, coweak isomorphism, and isomorphism between FGs were introduced by Bhutani in [2]. Khan et al. [20] studied vague relations. Talebi [21, 22] investigated Cayley-FGs and some results in bipolar fuzzy graphs. Borzooei [23] introduced domination in VGs. Ghorai and Pal studied some isomorphic properties of m-polar FGs [24]. Jiang et al. [25] defined vertex covering in cubic graphs. Krishna et al. [26] presented a new concept in cubic graphs. Rao et al. [27–29] investigated dominating set, equitable dominating set, and isolated vertex in VGs. Hoseini et al. [30] given maximal product of graphs under vague environment. Jan et al. [31] introduced some root-level modifications in interval-valued fuzzy graphs. Amanathulla et al. [32] defined new concepts of paths and interval graphs. Muhiuddin et al. [33, 34] presented the reinforcement number of a graph and new results in cubic graphs.A VG is a generalized structure of an FG that provides more exactness, adaptability, and compatibility to a system when matched with systems run on FGs. Also, a VG is able to concentrate on determining the uncertainty coupled with the inconsistent and indeterminate information of any real-world problems, where FGs may not lead to adequate results. 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. Thus, in this paper, we studied level graphs of VGs and investigated HMs, WIs, and CWIs of VGs by HMs of level graphs. Likewise, we characterized some VGs by their level graphs.


## 2. Preliminaries

In this section, we review some concepts of graph theory and VGs.Definition 1.
LetV be a finite nonempty set. A graph G=V,E on V consist of a vertex set V and an edge set E, where an edge is an unordered pair of distinct nodes of G. We will use pq rather than p,q to denote an edge. If pq is an edge, then we say that p and q are neighbor. A graph is called complete graph if each pair of nodes are neighbor.Definition 2.
LetG1=V1,E1 and G2=V2,E2 be graphs. A mapping h:V1⟶V2 is a homomorphism from G1 to G2 if hp and hq are neighbor whenever p and q are neighbor.Definition 3.
Two graphsG1 and G2 are isomorphic if there is a bijective mapping φ:V1⟶V2 so that p and q are neighbor in G1 if and only if φp and φq are neighbor in G2, φ is named isomorphism from G1 to G2. An isomorphism from a graph G to itself is named an automorphism of G. The set of all automorphisms of G forms a group, which is named the automorphism group of G and shown by AutG.Definition 4.
A VSA is a pair tA,fA on set X where tA and fA are taken as real valued functions which can be defined on V⟶0,1 so that tAp+fAp≤1, ∀p∈X.Definition 5.
LetA,B∈VSV. We say that A is contained in B and write A⊆B, if for any p∈V,(1)tAp≤tBp,fAp≥fBp.
LetK∗=η,γ|η,γ∈0,1,η+γ≤1. For any η1,γ1,η2,γ2∈L∗, the orders ≤ and < on K∗ are defined as(2)η1,γ1≤η2,γ2⇔η1≤η2,γ1≥α2,η1,γ1<η2,γ2⇔η1,γ1≤η2,γ2,η1<η2,orγ1>γ2.
It is easy to see that,K∗,≤ constitutes a complete lattice with maximum element 1,0 and minimum element 0,1.Definition 6.
LetA∈VSV. For each η,γ∈K∗, we define Aη,γ=p∈V:tAp≥η,fAp≤γ.
Then,Aη,γ is named η,γ-level set of A. The set p|p∈V,tAp>0orfAp<1 is called the support A and is denoted by A∗.
LetV be a finite nonempty set. Denote by V˜2 the set of all 2-element subsets of V. A graph on V is a pair V,E where E⊆V˜2, V and E are named vertex set and edge set, respectively.Definition 7.
LetV be a finite nonempty set, A∈VSV and B∈VFSV˜2. The triple X=V,A,B is named a VG on V, if for each p,q∈V˜2,(3)tBp,q≤tAp∧tAq,fBp,q≥fAp∨fAq.
IfX=V,A,B is a VG, then, it is easy to see that X∗=A∗,B∗ is a graph and it is called underlying graph of X. The set of all VG on V is denoted by VGV. For given X=V,A,B∈VGV, in this study suppose that A∗=V.Definition 8.
LetX1=V1,A1,B1 and X2=V2,A2,B2 be two VGs. Then,(1)
A mappingφ:V1⟶V2 is a homomorphism from X1 to X2, if(i)
tA1p≤tA2φp,fA1p≥fA2φp,for allp∈V1(ii)
tB1pq≤tB2φpφq,fB1pq≥fB2φpφq,for allpq∈V˜2(2)
A mappingφ:V1⟶V2 is a weak isomorphism from X1 to X2, if φ is a BH from X1 to X2 and tA1p=tA2φp,fA1p=fA2φp,for allp∈V1.(3)
A mappingφ:V1⟶V2 is a coweak isomorphism from X1 to X2, if φ is a BH from X1 to X2 and tB1pq=tB2φpφq,fB1pq=fB2φpφq,for allpq∈V˜2.(4)
An isomorphism fromX1 to X2 is a bijective mapping φ:V1⟶V2 so that(i)
tA1p=tA2φp,fA1p=fA2φp,for allp∈V1(ii)
tB1pq=tB2φpφq,fB1pq=fB2φpφq,for allpq∈V˜2Definition 9.
VGX=V,A,B is called strong vague graph (SVG) if tBpq=tAp∧tAq, fBpq=fAp∨fAq,for allpq∈V˜2,tBpq,fBpq≠0,1 and is called complete vague graph (CVG), if tBpq=tAp∧tAq, fBpq=fAp∨fAq,for allpq∈V˜2. A CVG X=V,A,B with n nodes is denoted by Kn,A.Definition 10.
Suppose thatX=V,A,B and Y=V,A′,B′ be two VGs. Then, X is VSG of Y, if A⊆A′ and B⊆B′.Definition 11.
LetX=V,A,B be VG and W⊆V. Then, the VG Y=W,A′,B′ so that tA′p=tAp,fA′p=fAp,for allp∈W, tB′pq=tBpq,fB′pq=fBpq,for allpq∈W˜2, is named the induced VSG by W and shown by XW.Definition 12.
A familyΓ=λ1,λ2,…,λk of VSs on V is named a k-coloring of VG X=V,A,B if(i)
∨Γ=A.(ii)
λi∧λj=0 for 1≤i,j≤k.(iii)
For each strong edgepq of X, minλip,λiq=0 for 1≤i≤k. We say that a graph is k-colorable if it can be colored with k colors.
All the basic notations are shown in Table1.Table 1
Some basic notations.
NotationMeaningFGFuzzy graphVSVague setFSFuzzy setVGVague graphCVGComplete vague graphSVGStrong vague graphBMBijective mappingHMHomomorphismWIWeak isomorphismIHInjective homomorphismCWICoweak isomorphismBHBijective homomorphismSGSubgraphIVIsolated vertexCGComplete graphVSGVague subgraph


## 3. Homomorphisms and Isomorphisms of Vague Graphs

In this section, we discuss the homomorphism and isomorphism of VGs by the homomorphism of level graphs in VGs.Theorem 1.
LetV be a finite nonempty set, A∈VSV and B∈VSV˜2. Then,X=V,A,B∈VFGV if and only if Xη,γ=Aη,γ,Bη,γ is a graph for all η,γ∈L∗,Aη,γ≠∅.Proof.
LetX=V,A,B be VG. For each η,γ∈L∗, Aη,γ≠∅, assume that pq∈Bη,γ. Then, tBpq≥η and fBpq≤γ. Because X is VG,(4)λ≤tBpq≤tAp∧tAq,γ≥fBpq≥fAp∨fAq.
It follows thatp,q∈Aη,γ. Therefore, Aη,γ,Bη,γ is a graph.
Conversely, letXη,γ=Aη,γ,Bη,γ is a graph, ∀η,γ∈L∗, Aη,γ≠∅. For each pq∈V˜2, let tBpq=η,fBpq=γ. Then, pq∈Bη,γ. Hence, p,q∈Aη,γ. Thus, tAp≥η,tAq≥η,fAp≤γ,and,fAq≤α,. This implies that tAp∧tAq≥η=tBpq and fAp∨fAq≤γ=fBpq. Therefore, X=V,A,B is VG.Definition 13.
LetX=V,A,B and Y=W,A′,B′ be two VGs, h:V⟶W a mapping. For any η,γ∈L∗, Aη,γ≠∅, if h is a homomorphism from Xη,γ=Aη,γ,Bη,γ to Yη,γ=Aη,γ′,Bη,γ′, then, h is called η,γ− homomorphism mapping from X to Y.Theorem 2.
LetX=V,A,B and Y=W,A′,B′ be two VGs. Then, h:X⟶Y is a homomorphism from XtoY if and only if h is η,γ- homomorphism from XtoY.Proof.
Assume thath:X⟶Y is a homomorphism from X to Y. Let, Aη,γ≠∅, η,γ∈L∗. If p∈Aη,γ, then(5)tA′hp≥tAp≥η,fA′hp≤fAp≤γ.
Hence,hp∈Aη,γ′ implying h is a mapping from Aη,γ to Aη,γ′. For p,q∈Aη,γ, let pq∈Bη,γ. Then,(6)tBpq≥η,fBpq≤γ.
Hence,(7)tB′hphq≥tBpq≥η,fB′hphq≤fBpq≤γ,
which implieshphq∈Bη,γ′. Therefore, h is a homomorphism from Xη,γ to Yη,γ.
Conversely, leth:V⟶W be a η,γ- homomorphism from X to Y. For arbitrary element p∈X, let tAp=c, fAp=d. Then, p∈Ac,d, hence, hp∈Ac,d′, because h is a homomorphism from Ac,d,Bc,d to Ac,d′,Bc,d′. It follows that(8)tA′hp≥c,fA′hp≤d,that is,(9)tA′hp≥tAp,fA′hp≤fAx.
Now for arbitrariesp,q∈V, let tBpq=e, fBpq=t. Then,(10)e=tBpq≤tAp∧tAq,t=fBpq≥fAp∨fAq.
Hence,p,q∈Ae,t and pq∈Be,t. Because h is a homomorphism from Xe,t=Ae,t,Be,t to Ye,t=Ae,t′,Be,t′, we conclude that hp,hq∈Ae,t′ and hpgq∈Be,t′. Therefore,(11)tB′hphq≥e=tBpq,fB′hphq≤t=fBpq.Theorem 3.
LetX=V,A,B and Y=W,A′,B′ be two VGs. Then, h:V⟶W is a WI from X to Y if and only if h is a bijective η,γ- homomorphism from XtoYand(12)tAp=tA′hp,fAp=fA′hp,for allp∈V.Proof.
Leth be a WI from X to Y. From the definition of homomorphism h is a bijective homomorphism from X to Y. By Theorem 2h is a bijective η,γ- homomorphism from X to Y and also by the definition of WI we have(13)tAp=tA′hp,fAp=fA′hp,for allp∈V.
Conversely, from hypothesis,h:A0,1=V⟶A0,1′=W is a bijective mapping and(14)tAp=tA′hp,fAp=fA′hp,for allp∈V.
Forp,q∈V, let tBpq=e, fBpq=t. Then,(15)e=tBpq≤tAp∧tAq,t=fBpq≥fAp∨fAq,
which impliesp,q∈Ae,t and pq∈Be,t. Because h is a homomorphism from Ae,t,Be,t to Ae,t′,Be,t′, we have hp,hq∈Ae,t′ and hphq∈Be,t′. Hence,(16)tB′hphq≥e=tBpq,fB′hphq≤t=fBpq,which complete the proof.Theorem 4.
LetX=V,A,B and Y=W,A′,B′ be two VGs. Then, h:V⟶W is a CWI from X to Y if and only if h is a bijective η,γ- homomorphism from X to Y and(17)tBpq=tB′hphq,fBpq=fB′hphq,for allpq∈V˜2.Proof.
Leth:V⟶W be a CWI from X to Y. Then, h is a bijective homomorphism from X to Y. By Theorem 2h is a bijective η,γ- homomorphism from X to Y. Also by the definition of CWI(18)tBpq=tB′hphq,fBpq=fB′hphq,for allpq∈V˜2.
Conversely, from hypothesis, we know thath:A0,1=V⟶A0,1′=W is a bijective mapping and(19)tBpq=tB′hphq,fBpq=fB′hphq.
For arbitrary elementp∈V, suppose that tAp=c, fAp=d. Then, we have p∈Ac,d. Now because h is a homomorphism from Ac,d,Bc,d to Ac,d′,Bc,d′, hp∈Ac,d′. Thus, tA′hp≥c=tAp, fA′hp≤d=fAp, which implies h is a CWI from X to Y.Corollary 1.
LetX=V,A,B∈VGV, Y=W,A′,B′∈VGW. Ifh:V⟶Wis a CWI fromXtoY, then,his an IH fromXη,γtoYη,γ, ∀η,γ∈K∗, Aη,γ≠∅.From the following example, we conclude that the converse of Corollary1 do not need to be true.Example 1.
LetX=V,A,B and Y=W,A′,B′ be two VGs, as shown in Figure 1. Consider the mapping h:V⟶W, defined by hvi=wi, 1≤i≤5. In view of the η,γ- level graphs of X and Y in Figure 1, if Aη,γ≠∅ then, h is an IH from Xη,γ to Yη,γ, but h is not a CWI.Figure 1
VGsX, Y and the mapping h:Vi⟶Wi which is not a CWI.Theorem 5.
LetX=V,A,B∈VGV,Y=W,A′,B′∈VGW, and h:V⟶W be a mapping. For each η,γ∈K∗, Aη,γ≠∅, if h is an isomorphism from Xη,γto a SG ofYη,γ, then,his a CWI fromXto an induced VSG ofY.Proof.
The mappingh is an isomorphism from X0,1=V,B0,1 to a SG Y0,1=W,B0,1′, so h:V⟶W is an IM. For arbitrary p∈V, suppose that tAp=η, fAp=γ. Then, p∈Aη,γ, and so hp∈Aη,γ′. Hence, tA′hp≥η=tAp and fA′hp≤γ=fAp. For p,q∈V, let tBpq=η and fBpq=γ. Then, η≤tAp, η≤tAq, γ≥fAp, γ≥fAq and pq∈Bη,γ. Hence, p,q∈Aη,γ and pq∈Bη,γ. Since h isomorphism from Xη,γ to Yη,γ, we get hp,hq∈Aη,γ′ and hphq∈Bη,γ′. Therefore,(20)tB′hphq≥η=tBpq,fB′hphq≤γ=fBpq.
Now, lettB′hphq=k, fB′hphq=s. Then, hphq∈Ak,s′. Because h is injective and an isomorphism from Xk,s to a SG of Yk,s, we have p,q∈Ak,s and pq∈Bk,s. Therefore,(21)tBpq≥k=tB′hphq,fBpq≤s=fB′hphq.
Now by (20) and (21), we conclude that(22)tB′hphq=tBpq,fB′hphq=fBpq.Corollary 2.
LetX=V,A,BandY=W,A′,B′ be two VGs with V=W, andh:V<display>W a mapping. For η,γ∈KL∗,Aη,γ≠∅, if h is an isomorphism from Xη,γto a SG ofYη,γ, then,his a CWI fromXtoY.Theorem 6.
LetX=V,A,BandY=W,A′,B′be two VGs,h:V⟶Wbe a bijective mapping. If for eachη,γ∈K∗,his an isomorphism fromXη,γtoYη,γ, then,his an isomorphism fromXtoY.Proof.
From hypothesis,h−1:W⟶V is a bijective mapping and an isomorphism from Yη,γ to Xη,γ. By Theorem 5h is a CWI from X to Y and h−1 is a CWI from Y to X. Therefore, h is an isomorphism from X to Y.Corollary 3.
LetX=V,A,Bbe VG andh:V⟶Va bijective mapping. Then,his an automorphism ofXif and only ifh|Aη,γis an automorphism ofXη,γ, from anη,γ∈K∗,Aη,γ≠∅.Theorem 7.
LetX=V,A,B be VG. Then, Xis a CVG if and only if Xη,γ=Aη,γ,Bη,γis a CG forη,γ∈K∗.Proof.
IfX=V,A,B is a CVG and for η,γ∈K∗, Aη,γ≠∅, p,q∈Aη,γ, then, tAp≥η, tAq≥η, fAp≤γ, fAq≤γ, and so(23)tBpq=tAp∧tAq≥η,fBpq=fAp∨fAq≤γ.
Hence,pq∈Bη,γ. It follows that Xη,γ is a CG. Conversely, suppose that X=V,A,B is not a CVG. Then, there are p,q∈V so that tBpq<tAp∧tAq or fBpq>fAp∨fAq. Let tBpq<tAp∧tAq, and tAp∧tAq=η, for η∈0,1. Then, tAp≥η and tAq≥η. Hence, x,y∈Aη,γ, for a γ∈0,1, but pq∉Bη,γ. This implies that Xη,γ is not a CG. For the case fBpq>fAp∨fAq, it follows similarly.Theorem 8.
LetX=V,A,B∈VGV. Then,Xη,γhas not IV, for eachη,γ∈K∗,Aη,γ≠∅if and only if for eachp∈V,∃q∈Vso thattBpq=tAp,fBpq=fAp.Proof.
Suppose that for eachη,γ∈K∗, Aη,γ≠∅, graph Xη,γ has not IV and there is a node p∈V so that for each q∈V, tBpq<tAp or fBpq>fAp. Let tBpq<tAp and tAp=η, fAp=γ, for η,γ∈K∗. Then, p∈Aη,γ and for each q∈V, q≠p, pqinBη,γ. Therefore, p is an IV in the graph Xη,γ=Aη,γ,Bη,γ, which is a contradiction.
Now suppose that forη,γ∈K∗, Aη,γ≠∅, node p∈Aη,γ is an IV in Xη,γ. If qinAη,γ, then, tBpq≤tAq<η≤tAp or fBpq≥fAq>γ≥fAq, and if q∈Aη,γ, it is trivial that pginBη,γ, hence, tBpq<η≤tAp or fBpq>γ≥fAp. Therefore, for each q∈V, tBpq≠tAp, fBpq≠fAp.
Here, we describe the relationship between coloring graph and homomorphism of graph.Theorem 9.
A VGX=V,A,B is r-colorable ⇔ there exists a homomorphism from X to the Kr,A′.Proof.
Assume thatX be r-colorable with r colors labeled Γ=λ1,λ2,…,λr. Let Vi=v∈V|λiv≠0. We define CVG Kr,A′ with vertices set 1,2,…,r, so that the degree of membership vertex i is tA′i=maxtAv|v∈Vi and the degree of non-membership vertex i is fA′i=minfAv|v∈Vi. Now the mapping h:X⟶Kr,A′ defined by hv=i is a graph homomorphism, because(24)tAv≤maxtAw|w∈Vi=tA′i=tA′hv,fAv≥minfAw|w∈Vi=fA′i=fA′hv.
According to the definition of CVG, foru∈Vi and v∈Vj we have(25)tBuv≤tAu∧tAv≤tA′i∧tA′j=tA′hu∧tA′hv,fBuv≥fAu∨tAv≤tA′i∨tA′j=tA′hu∨tA′hv.
Then,tBuv≤tB′huhv, fBuv≥fB′huhv, for all uv∈V˜2.
Conversely, letg:X⟶Kr,A′ be a homomorphism. For a given k∈VKr,A′, define the set h−1k⊆V to be(26)h−1k=x∈V|hx=k.
Ifv∈h−1k, let λkv=tλkv,fλkv=tAv,fAv, otherwise λkv=0. Therefore, the VG X is r-colorable with coloring set λ1,λ2,…,λr.


## 4. Application

Nowadays, the issue of coloring is very important in the theory of fuzzy graphs because it has many applications in controlling intercity traffic, coloring geographical maps, as well as finding areas with high population density. Therefore, in this section, we have tried to present an application of the coloring of vertices in a VG.Example 2.
LetX=V,A1,B1 be a VG (See Figure 2). We modeled a FG by considering countries A,B,C,D as vertices of graph. The membership and nonmembership value of the vertices are the good and bad activity of a country with respect technology so that are tA1A,fA1A=0.1,0.2, tA1B,fA1B=0.4,0.5, tA1C,fA1C=0.2,0.5, tA1D,fA1D=0.2,0.8, respectively. There is an edge if they share a boundary. Let AB, BC, AC, CD, and BD are edges of graph X. The membership and nonmembership value of the edges are the political relationship in a good and bad attitude such that tB1AB,fB1AB=0.1,0.5, tB1BC,fB1BC=0.2,0.8, tB1AC,fB1AC=0.1,0.5, tB1CD,fB1CD=0.2,0.8, tB1BD,fB1BD=0.2,0.8, respectively. We now want to see how many days we will need to hold a conference between these countries. Let S be a set of countries; S=A,B,C,D and P=ABBCACCDBD Suppose that Sp be countries have boundary for p∈P. Now, form FG G with vertices set P, where a,b∈P are neighbor if and only if Sa∩Sb≠∅. For instance, SAB=A,B and SBC=C,B. So SAB∩SBC=B≠∅ and hence AB, BC are neighbor. By Theorem 9 there is a homomorphism from G to complete graph with n=3. Then, 3 days are required to hold a conference between these countries, AB,CD,BC,AC,BD. The colored graph of the example 2 is shown in Figure 3.
In the next example, we want to identify the most effective employee of a hospital with the help of a vague influence digraph.Figure 2
Vague graphX=V,A1,B1.Figure 3
The colored graph of the example2.Example 3.
Hospitals are very important organizations whose existence is directly related to the general health of the community. Researchers in each country examine factors that contribute to the success of strategic planning to improve the management status of these health organizations. The lives and health of many people are in the hands of health systems. From the safe delivery of a healthy baby to the respectful care of an elderly person, the health department has a vital and ongoing responsibility to individuals throughout their lives. The health industry has undergone many political, social, economic, environmental, and technological changes since the early 1980s. These changes have created challenges for managers of healthcare organizations, especially hospitals that cannot be managed with operational plans. Thus, hospital managers have resorted to strategic planning since the 1980s to achieve excellence. Since the management in each ward of the hospital is very important, so in this section, we have tried to determine the most effective person in a hospital based on the performance of its staff. Therefore, we consider the vertices of the VIG as the heads of each ward of the hospital, and the edges of the graph as the degree of interaction and influence of each other. For this hospital, the set of staff isF=Taheri, Ameri, Talebi, Taleshi, Najafi, Kamali, Badri.(i)
Ameri has been working with Taleshi for 14 years and values his views on issues.(ii)
Taheri has been responsible for audiovisual affairs for a long time, and not only Ameri, but also Taleshi, are very satisfied with Taheri’s performance.(iii)
In a hospital, the preservation of medical records is a very important task. Kamali is the most suitable person for this responsibility.(iv)
Talebi and Kamali have a long history of conflict.(v)
Talebi has an important role in the radiology department of the laboratory.
Given the abovementioned, we consider this a VIG. The vertices represent each of the hospital staff. Note that each staff member has the desired ability as well as shortcomings in the performance of their duties. Therefore, we use of VS to express the weight of the vertices. The true membership indicates the efficiency of the employee and the false membership shows the lack of management and shortcomings of each staff. But the edges describe the level of relationships and friendships between employees such that the true membership shows a friendly relationship between both employees and the false membership shows the degree of conflict between the two officials. Names of employees and levels of staff capability are shown in Tables2 and 3. The adjacency matrix corresponding to Figure 4 is shown in Table 4.
Figure4 shows that Najafi has 90% of the power needed to do the hospital work as the medical equipment expert, but does not have the 10% knowledge needed to be the boss. The directional edge of Taleshi–Ameri shows that there is 30% friendship among these two employees, and unfortunately, they have 40% conflict. Clearly, Badri has dominion over both Kamali and Najafi, and his dominance over both is 60%. It is clear that Badri is the most influential employee of the hospital because he controls both the head of the medical equipment and the medical records archive expert, who have 90% of the power in the hospital.Table 2
Names of employees in a hospital and their services.
NameServicesTaheriHead of audiovisual departmentAmeriEnvironment health expertTalebiHead of radiology departmentTaleshiNetwork expertNajafiMedical equipment expertKamaliMedical records archive expertBadriHead of hospitalTable 3
The level of staff capability.
TaheriAmeriTalebiTaleshiNajafiKamaliBadritA0.40.50.60.70.90.90.8fA0.40.30.30.20.10.10.2Figure 4
Vague influence digraph.Table 4
Adjacency matrix corresponding to Figure4.
TaheriAmeriTalebiTaleshiNajafiKamaliBadriTaheri0,00,00.4,0.50.3,0.50,00,00,0Ameri0,00,00,00.3,0.40,00,00,0Talebi0,00,00,00.5,0.40,00,00,0Taleshi0,00,00,00,00,00,00,0Najafi0,00.4,0.40,00.6,0.20,00,00,0Kamali0,00,00,00,00.7,0.20,00,0Badri0,00,00,00.5,0.30.6,0.40.6,0.40,0


## 5. Conclusion

VGs 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 VGs, the most efficient person in an organization can be identified according to the important factors that can be useful for an institution. Hence, in this paper, we introduced the notion ofη,γ- homomorphism of VGs and classify HMs, WIs, and CWIs of VGs by η,γ- homomorphisms. We also investigated the level graphs of VGs to characterize some VGs. Finally, we presented two applications of VGs in coloring problem and also finding effective person in a hospital. In our future work, we will introduce new concepts of connectivity in VGs and investigate some of their properties. Also, we will study the new results of connected perfect dominating set, regular perfect dominating set, and independent perfect dominating set on VGs.



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