# A Study on Interval-Valued Fuzzy Graph with Application in Energy Industry Management

Authors: Xiaoli Qiang; Qian-Ru Xiao; Aysha Khan; A. A. Talebi; Arun Kumar Sivaraman; M. Mojahedfar
Journal: Discrete Dynamics in Nature and Society
 (2022)

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## Abstract

In this paper, we define the concept of level graphs on interval-valued fuzzy graphs (IVFGs) and introduce the notions ofα,β−α,β− homomorphism, isomorphism, weak isomorphism, and co-weak isomorphism of IVFGs by α,β− homomorphism. Also, an application of homomorphism of IVFGs has been presented by using coloring-FG.


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## Body

## 1. Introduction

Zadeh [1] introduced the subject of the interval-valued fuzzy set (IVFS) in 1975 as an explanation of fuzzy sets (FSs) [2] so that the values of the membership degrees are shown through intervals of numbers. IVFS is useful in applications such as fuzzy control. One of the most extreme sections of fuzzy control in terms of calculation is defuzzification [3]. The fuzzy graph (FG) theory as a generalization of Euler’s graph theory was defined by Rosenfeld [4] in 1975. For the first time, the fuzzy relations between FSs were supposed by Rosenfeld, and he extended the structure of FGs and used it to achieve various graph theoretical subjects. After that, Bhattacharya [5] presented some remarks on FGs. Mordeson and Peng [6] explained some operations on FGs. Hongmei and Lianhua gave the definition of IVFG and so they introduced the notion of IVFGs [7]. For the first time, Akram and Dudek defined IVFGs [8]. In graph theory, we define a graph homomorphism that has a mapping between two graphs that relates to their structure. Different notions of graph coloring are generalized by homomorphisms. These concepts give essential information about an important class of constraint satisfaction problems [9].Jiang et al. [10] introduced a new notion of vertex covering in cubic graphs. Turksen [11] defined IVFSs based on normal forms. Nagoorgani and Radha [12] presented isomorphism on fuzzy graphs. Jan et al. [13] studied some root-level modifications in IVFGs. Krishna et al. [14] investigated new results in cubic graphs. Talebi et al. [15, 16] defined Cayley fuzzy graphs and some operations on level graphs of bipolar fuzzy graphs. Kosari et al. [17] introduced the vague graph (VG) structure with an application in medical diagnosis. Kou et al. [18] defined some properties of VG. Qiang et al. [19] studied new kinds of domination in VGs. Rao et al. [20–22] presented some properties of vague incidence graphs and equitable domination in VGs. Shi et al. [23, 24] investigated the domination of product VGs with an application in transportation. Xu et al. [25] studied certain concepts of interval-valued intuitionistic fuzzy graphs. Rashmanlou and Pal [26] defined antipodal interval-valued fuzzy graphs. Bera et al. [27] proposed certain types of m-polar interval-valued fuzzy graphs. Zihni et al. [28] introduced interval-valued fuzzy soft graphs. Akram et al. [29–31] introduced the concept of interval-valued fuzzy line graphs, self-centered interval-valued fuzzy graphs, and certain types of interval-valued fuzzy graphs.IVFG, belonging to the FG family, has good capabilities when faced with problems that cannot be explained by FGs and VGs. Since the principle of membership is not clear, neutrality is a good choice that can be well protected by an IVFG. Also, an IVFG is used to illustrate real-world phenomena using interval-valued fuzzy models in a variety of fields, including technology, social networking, and biological sciences. Therefore, in this paper, we introduce the concept of level graphs on IVFGs and define the notions ofα,β− homomorphism, isomorphism, weak isomorphism, and co-weak isomorphism of an IVFG by α,β− homomorphism. Finally, we have tried to find the most effective person in an electrical power department based on the performance of its staff.


## 2. Preliminaries

In this section, we present some basic concepts of IVFGs.Definition 1.
A fuzzy graph is a pairG=ζ,ξ with a set X, that ζ is a fuzzy set in X and ξ is a fuzzy relation in X×X, so that(1)ξab≤minζa,ζb,for all ab∈X×X.SupposeX is a finite nonempty set. The set of all 2-element subsets of X is shown by X˜2. A graph G is a pair X,E on X, so that E⊆X˜2, X and E are named vertex set and edge set, respectively.Definition 2.
A pairG=M,N on graph G∗=X,E is called interval-valued fuzzy graph (IVFG) so that M=ζMl,ζMu is an IVFS on X and N=ζNl,ζNu is an IVFS on E to satisfy the following conditions:(1)
X=a1,a2,…,an so that ζMlai,ζMuai shows the degree of membership of the node ai∈X.(2)
ζNl:E⟶0,1, ζNu:E⟶0,1 are the functions that satisfy(2)ζNlab≤minζMla,ζMlb,ζNuab≤minζMua,ζMub,for allab∈E.Definition 3.
An IVFG is called a strong interval-valued fuzzy graph (SIVFG) if(3)ζNlab=minζMla,ζMlb,ζNuab=minζMua,ζMub.for all ab∈E.Definition 4.
LetL∗=α,β|α,β∈0,1,0≤α≤β≤1. For any α1,β1,α2,β2∈L∗, we have(4)α1,β1≤α2,β2⇔α1≤β2β1≤β2,α1,β1<α2,β2⇔α1,β1≠α2,β2α1<α2β1<β2.Definition 5.
LetM∈IVFSX. For any α,β∈L∗, we define(5)Mα,β=a∈X:ζMla≥α,ζMua≥β.
Then,Mα,β is said to be α,β-level set of M.
The setM∗=a|a∈X,ζMla>0orζMua>1 is named the support M.Definition 6.
SupposeG1=M1,N1 and G2=M2,N2 are two IVFGs. Then,(1)
A homomorphism (HM)ψ from G1 to G2 is a mapping ψ:X1⟶X2 with the following conditions:(i)
ζM1la≤ζM2lψa, ζM1ua≤ζM2uψa(ii)
ζN1lab≤ζN2lψaψb, ζN1uab≤ζN2uψaψb, for all a∈X,ab∈E(2)
An isomorphismψ from G1 to G2 is a bijective mapping (BM) ψ:X1⟶X2 with the following conditions:(i)
ζM1la=ζM2lψa, ζM1ua=ζM2uψa(ii)
ζN1lab=ζN2lψaψb, ζN1uab=ζN2uψaψb, for all a∈X,ab∈E(3)
A weak isomorphism (WI)ψ from G1 to G2 is a BM ψ:X1⟶X2 with the following conditions:(i)
ψ is HM(ii)
ζM1la=ζM2lψa,ζM1ua=ζM2uψa, a∈X(4)
A co-weak isomorphism (CWI)ψ from G1 to G2 is a BM ψ:X1⟶X2 with the following conditions:(i)
ψ is HM(ii)
ζN1lab=ζN2lψaψb, ζN1uab=ζN2uψaψb, for all ab∈EDefinition 7.
SupposeG1=M1,N1 and G2=M2,N2 are two IVFGs. Then, G1 is an interval-valued subgraph (IVSG) of G2, if M1⊆M2 and N1⊆N2.Definition 8.
A familyΓ=λ1,λ2,…,λk of IVFSs on X is called a k-coloring of IVFG G=M,N if(i)
∨Γ=M.(ii)
λi∧λj=0, for 1≤i,j≤k.(iii)
For each strong edgeab of X, minλia,λib=0, for 1≤i≤k. We consider that a graph is k-colorable if it can be colored by k colors.All the essential notations are denoted in Table1.Table 1
Some essential notations.
NotationMeaningFGFuzzy graphIVFSInterval-valued fuzzy setFSFuzzy setIVFGInterval-valued fuzzy graphSIVFGStrong interval-valued fuzzy graphBMBijective mappingHMHomomorphismWIWeak isomorphismIHInjective homomorphismCWICo-weak isomorphismBHBijective homomorphismSGSubgraphIVSGInterval-valued subgraphCIVFGComplete interval-valued fuzzy graphCGComplete graphIVIsolated vertex


## 3. Homomorphisms and Isomorphisms of Interval-Valued Fuzzy Graphs

In this part, we explain HM and isomorphisms on IVFGs by the HM of level graphs in IVFGs.Theorem 1.
LetX be a finite nonempty set, M∈IVFSX, and N∈IVFSX˜2. Then, G=M,N∈IVFGX if and only if Gα,β=Mα,β,Nα,β is a graph, for all α,β∈L∗, Mα,β≠∅.Proof.
SupposeG=M,N∈IVFGX is an IVFG. For any α,β∈L∗, Mα,β≠∅, consider that ab∈Nα,β. Then, ζNlab≥α and ζNuab≥β. Because G is an IVFG,(6)α≤ζNlab≤minζMla,ζMlb,β≤ζNuab≤minζMua,ζMub.
Also, we have(7)α≤ζMla,β≤ζMua⇒a∈Mα,β,α≤ζMlb,β≤ζMub⇒b∈Mα,β.
Therefore,x,y∈Aα,β and Aα,β,Bα,β is a graph.
Conversely, supposeGα,β=Mα,β,Nα,β is a graph, for α,β∈L∗, Mα,β≠∅. For any ab∈X˜2, let ζNlab=α,ζNuab=β. Then, ab∈Nα,β. Hence, a,b∈Mα,β. Thus, ζMla≥α, ζMlb≥α,ζMua≥β, ζMua≥β. This results that(8)minζMla,ζMlb≥α=ζNlab,and(9)minζMua,ζMub≥β=ζNuab.
Thus,G=M,N is an IVFG.Definition 9.
SupposeG1=M1,N1 and G2=M2,N2 are two IVFGs on X1 and X2, respectively, ψ:X1⟶X2 a mapping. For any α,β∈L∗, Mα,β≠∅, if ψ is a HM from G1α,β=M1α,β,N1α,β to G2α,β=M2α,β,N2α,β, then, ψ is called α,β− HM mapping from G1 to G2.Theorem 2.
SupposeG1=M1,N1andG2=M2,N2are two IVFGs. Then,ψ:G1⟶G2is a HM fromG1toG2if and only ifψisα,β−HM fromG1toG2.Proof.
Letψ:G1⟶G2 be a HM from G1 to G2. Assume, α,β∈L∗, Mα,β≠∅. If a∈Mα,β, then,(10)α≤ζM1la≤ζM2lψa,β≤ζM1ua≤ζM2uψa.
Thereforeψa∈M2α,β implying ψ is a mapping from M1α,β to M2α,β. For a,b∈M1α,β, let ab∈N1α,β. Then,(11)α≤ζN1lab,β≤ζN1uab.
Hence,(12)α≤ζN1lab≤ζN2lψaψb,β≤ζN1uab≤ζN2uψaψb.
That impliesψaψb∈N2α,β. Thus, ψ is a HM from G1α,β to G2α,β.
Conversely, supposeψ:X1⟶X2 is a α,β− HM from G1 to G2. For the arbitrary element of a∈G1, let ζM1la=w1, ζM1ua=w2. Then, a∈M1w1,w2, so that, ψa∈M2w1,w2, because ψ is a HM from M1w1,w2,N1w1,w2 to M2w1,w2,N2w1,w2. We have(13)w1≤ζM2lψa,w2≤ζM2uψa.
That is,(14)ζM1la≤ζM2lψa,ζM1ua≤ζM2uψa.
Now for arbitrary ofa,b∈X1, let ζN1lab=t1, ζN1uab=t2. Then,(15)t1=ζN1lab≤minζM1la,ζM1lb,t2=ζN1uab≤minζM1ua,ζM1ub.
Hence,a,b∈M1t1,t2 and a,b∈M1t1,t2. Because ψ is a HM from G1t1,t2=M1t1,t2,N1t1,t2 to G2t1,t2=M2t1,t2,N2t1,t2, we conclude that ψa,ψb∈M2t1,t2 and ψaψb∈N2t1,t2. Therefore,(16)ζN2lψaψb≥t1=ζN1labζN2uψaψb≥t2=ζN1uxy.Theorem 3.
LetG1=M1,N1andG2=M2,N2be IVFGs onX1andX2, respectively. Then,ψ:X1⟶X2is a WI fromG1toG2, if and only if,ψis a bijectiveα,β−HM fromG1toG2and(17)ζM1la=ζM2lψa,ζM1ua=ζM2uψaa∈X.Proof.
Supposeψ is a WI from G1 to G2. From the definition of HM, ψ is a BH from G1 to G2. By Theorem 2, ψ is a bijective α,β− HM from G1 to G2 and also, by the definition of WI, we have(18)ζM1la=ζM2lψa,ζM1ua=ζM2uψaa∈X.
Conversely, by hypothesis,ψ:M10,1=X1⟶M20,1=X2 is a BM and(19)ζM1la=ζM2lψa,ζM1ua=ζM2uψaa∈X.
Fora,b∈X1, let ζN1lab=t1, ζN1uab=t2. Then,(20)t1=ζN1lab≤minζM1la,ζM1lb,t2=ζN1uab≤minζM1ua,ζM1ub.
That impliesa,b∈M1t1,t2 and ab∈N1t1,t2. Because ψ is a HM from G1t1,t2=M1t1,t2,N1t1,t2 to G2t1,t2=M2t1,t2,N2t1,t2, we have ψa,ψb∈M2 and ψaψb∈N2t1,t2. Thus,(21)ζN2lψaψb≥t1=μN1lab,ζN2uψaψb≥t2=ζN1uab.Theorem 4.
LetG1=M1,N1andG2=M2,N2be IVFGs onX1andX2, respectively. Then,ψ:X1⟶X2is a CWI fromG1toG2, if and only if,ψis a bijectiveα,β−HM fromG1toG2and(22)ζN1lab=ζN2lψaψb,ζN1uab=ζN2uψaψbab∈X˜2.Proof.
Letψ:X1⟶X2 be a CWI from G1 to G2. Then, ψ is a BH from G1 to G2. By Theorem 2, ψ is a bijective α,β− HM from G1 to G2. Also, by the description of CWI, we write(23)ζM1lab=ζN2lψaψb,ζN1uab=ζN2uψaψbab∈X˜2.
Conversely, by hypothesis, we see thatψ:M10,1=X1⟶M20,1=X2 is a BM and(24)ζN1lab=ζN2lψaψb,ζN1uab=ζN2uψaψb.
For arbitrary principlea∈X1, consider ζM1la=s1, ζM1ua=s2. Then, we have a∈M1s1,s2d. Now, because ψ is a HM from G1s1,s2=A1s1,s2,B1s1,s2 to G2s1,s2=M2s1,s2,N2s1,s2, ψa∈M2s1,s2. Therefore,(25)ζM2lψa≥s1=ζM1la,ζM2uψa≥s2=ζM1ua.
That impliesψ is a CWI from G1 to G2.Corollary 1.
LetG1=M1,N1∈IVFGX1 and G2=M2,N2∈IVFGX2. If ψ:X1⟶X2 is a CWI from G1 to G2, then, ψ is an IH from G1α,β to G2G2α,β, for every α,β∈L∗, M1α,β≠∅.Example 1.
SupposeG1=M1,N1 and G2=M2,N2 are two IVFGs, as in Figure 1. Consider the mapping of ψ:X1⟶X2, defined by ψx1i=x2i, 1≤i≤3. α,β− level graphs of G1 and G2 in Figure 1, if Mα,β≠∅, therefore, ψ is an IH from G1α,β to G2α,β, but ψ is not a CWI.(26)G1α,β,0.1,0.2<α,β≤0.1.0.3,G2α,β,0.1,0.2<α,β≤0.1.0.3.Figure 1
IVFGs andα,β− level graphs.Theorem 5.
LetG1=M1,N1∈IVFGX1 and G2=M2,N2∈IVFGX2, and ψ:X1⟶X2 is a mapping. For any α,β∈L∗, M1α,β≠∅, if ψ is an isomorphism from G1α,β to a subgraph (SG) of G2α,β, then, ψ is a CWI from G1α,β to an induced IVSG of G2α,β.Proof.
The mappingψ from G10,1=X,M0,1 to a SG G20,1=X,N0,1 is an isomorphism, so ψ:X1⟶X2 is an IM. For arbitrary a∈X1, we consider ζM1la=α, ζM1ua=β. Then, a∈M1α,β and so ψa∈M2α,β.
Therefore,(27)ζM2lψa≥α=ζM1la,ζM2uψa≥β=ζM1ua.
Fora,b∈X1, let ζN1lab=α, ζN1uab=β that(28)α≤ζM1la,α≤ζM1lb,β≤ζM1ua,β≤ζM1ub.and ab∈N1α,β. Thus, a,b∈M1α,β and ab∈N1α,β.
Sinceψ is an isomorphism from G1α,β to G1α,β, we get ψa,ψb∈M2α,β and ψaψb∈N2α,β.
Therefore,(29)ζN2lψaψb≥α=ζN1lab,ζN2uψaψb≥β=ζN1uab.I.
Now supposeζN2lψaψb=z1, ζN2uψaψb=z2. Then ψaψa∈M2z1,z2. Because ψ is injective and an isomorphism from G1z1,z2 to a SG of G2z1,z2, we have a,b∈M1z1,z2 and ab∈N1z1,z2. Therefore,(30)ζN2lψaψb=z1≤ζN1lab,ζN2uψaψb=z2≤ζN1uab.II.
Now byI and II, we conclude the following statement:(31)ζN2lψaψb=ζN1lab,ζN2uψaψb=ζN1uab.Corollary 2.
LetG1=M1,N1andG2=M2,N2be two IVFGs withX1=X2, andψ:X1⟶X2a mapping. Forα,β∈L∗,Mα,β≠∅, if,ψis an isomorphism fromG1α,βto a SG ofG2α,β, then,ψis a CWI fromG1toG2.Theorem 6.
Suppose we have two graphsG1=M1,N1 and G2=M2,N2 that they are IVFGs, and ψ:X1⟶X2 is a BM. If for any α,β∈L∗, ψ is an isomorphism from G1α,β to G2α,β, then, ψ is an isomorphism from G1 to G2.Proof.
By hypothesis,ψ−1:X2⟶X1 is a BM and an isomorphism from G2α,β to G1α,β.
By Theorem 3.7,ψ is a CWI from G1 to G2 and ψ−1 is a CWI from G2 to G1. Therefore, ψ is an isomorphism from G1 to G2.Theorem 7.
SupposeG=M,Nis an IVFG. Then,Gis a complete interval-valued fuzzy graph (CIVFG), if and only if,Gα,β=Mα,β,Nα,βis a complete graph (CG) forα,β∈L∗.Proof.
IfG=M,N is a CIVFG and for α,β∈L∗,Mα,β≠∅,a,b∈Mα,β, then(32)α≤ζMla,α≤ζMlb,β≤ζMua,β≤ζMuband so(33)ζNlab=minζMla,ζMlb≥α,ζNuab=minζMua,ζMub≥β.
Hence,a,b∈Nα,β. It shows that Gα,β is a CG.
Conversely, considerG=M,N is not a CIVFG. Then, there are a,b∈X so that(34)ζNlab<minζMla,ζMlborζNuab<minζMua,ζMub.
LetζNlab<minζMla,ζMlb, and/or α=minζMla,ζMlb, for α∈0,1. Then, ζMla≥α,ζMlb≥α. Hence, a,b∈Mα,β, for a β∈0,1, but ab∉Nα,β. This implies that Gα,β is not a CG. For the case ζNuab<minζMua,ζMub, it follows similarly.Theorem 8.
LetG=M,N∈IVFGX. Then,Gα,βdoes not have an isolated vertex (IV), for anyα,β∈L∗,Mα,β≠∅if and only if for eacha∈X,there isb∈Xso thatζNlab=ζMla,ζNuab=ζMub.Proof.
Suppose that for eachα,β∈L∗,Mα,β≠∅, graph Gα,β does not have an IV and there is a vertex a∈X so that for each b∈X, ζNlab<ζMla or ζNuab<ζMua. Let ζNlab<ζMla and ζMla=α, ζMub=β, for α,β∈L∗.
Then,a∈Mα,β and for each a∈X, a≠b, ab∉Nα,β. Thus, a is an IV in the graph Gα,β=Mα,β,Nα,β, which is a contradiction.
Now let that forα,β∈L∗, Mα,β≠∅, vertex a∈Mα,β is an IV in Gα,β. If y∉Mα,β, then,(35)ζNlab≤ζMlb<α≤ζMlaorζNuab≤ζMub<β≤ζMua.and if b∈Mα,β, it is trivial that ab∉Nα,β, hence,(36)ζNlab<α≤ζMlaorζNuab<β≤ζMua.
Thus, for anyb∈X, ζNlab≠ζMla, ζNuab≠ζMua.Theorem 9.
A VGG=M,Nisr-colorable⇔there exists a HM fromXto theKr,M′.Proof.
LetG be r-colorable with r colors, so that Γ=λ1,λ2,…,λr. Let Vi=v∈V|λiv≠0. We denote CIVFG Kr,M′ with vertices set 1,2,…,r, so that the degree of membership vertex i is ζM′li=minζMlv|v∈Vi and the degree of nonmembership vertex i is ζM′ui=minζMuv|v∈Vi. Now the mapping ψ:G⟶Kr,M′ defined by ψv=i is a graph homomorphism, because(37)ζMlv≤minζMlw|w∈Vi=ζM′li=ζM′lψv,ζMuv≤minζMuw|w∈Vi=ζM′ui=ζM′uψv,and by the definition of CIVFG, for u∈Vi and v∈Vj, we have(38)ζNluv≤minζMlu,ζMlv≤minζM′li,ζM′lj=minζM′lψu,ζM′lψv,ζNuuv≤minζMuv,ζMuv≤minζM′ui,ζM′uj=minζM′uψu,ζM′uψv.
Then,ζNluv≤ζN′lψuψv, ζNuuv≤ζN′uψuψv, for all uv∈V˜2.
Conversely, letϕ:G⟶Kr,M′ is a HM. For a given k∈VKr,M′, we define the set ψ−1k⊆V to be(39)ψ−1k=x∈V|ψx=k.
Ifv∈ψ−1k, let λkv=ζλklv,ζλkuv=ζAlv,ζMuv, otherwise λkv=0. Thus, the IVFG G is r-colorable with coloring set λ1,λ2,…,λr.


## 4. Application

In graph theory, the subject of coloring is an important case of graph labeling, and there are more examples of coloring graph. In this part, we have expressed an application of the coloring of vertices in an IVFG, and we present this concept in the form of an example.Example 2.
LetG=M1,N1 be an IVFG (see Figure 2). In this example, we consider four sport groups such as A,B,C,D so that they are vertices of the graph. We show each one of the sports groups with interval-valued. The interval-valued of the vertices are the good activity rate in the world cup games, so that are(40)ζM1lA,ζM1uA=0.1,0.3,ζM1lB,ζM1uB=0.3,0.5,ζM1lC,ζM1uC=0.2,0.5,ζM1lD,ζM1uD=0.3,0.8.
Suppose thatAB, BC, AC, CD, and BD are the edges of graph G1.(41)ζN1lAB,ζN1uAB=0.1,0.3,ζN1lBC,ζN1uBC=0.2,0.4,ζN1lAC,ζN1uAC=0.2,0.4,ζN1lCD,ζN1uCD=0.3,0.5,ζN1lBD,ζN1uBD=0.1,0.3.
Here we want to see how many days they will need to exercise between these sport groups. LetX be a set of sport groups, X=A,B,C,D and E= \{AB, BC, AC, CD, BD \}. Now, form graph G1 with vertices set E, where a,b∈E are neighbors, if and only if, Xa∩Xb≠∅. For example, XAB=A,B and XBD=B,D. So XAB∩XBD=B≠∅, and so AB, BD are neighbors. By Theorem 9, there is a HM from G1 to complete graph with n=3. Therefore, these sport groups are needed to play 3 days, AB,CD,BC,AC,BD (see Figure 3).
In the next example, we want to identify the most effective employee of an electrical power department with the help of an interval-valued fuzzy influence digraph.Figure 2
Interval-valued fuzzy graphG1=A1,B1.Figure 3
The colored graph of example2.Example 3.
Electricity is one of the discoveries that has changed the daily lives of everyone on the planet. Electricity is a staple of modern technology, and without it, most of the things we use every day will work. Other major areas where electricity is used are the production, supply, and distribution of food. Farmers use electricity for almost everything, including lights, pumps, electric motors, and so on. Electricity is very useful, especially in homes, offices, shopping malls, schools, supermarkets, and any other place we can think of. Today, it has become such a vital element of a society that depends on electricity that it is difficult for human beings to even imagine life without electricity. Access to electricity has become the most important thing in human society and has created great economic, political, and cultural changes in the world. Perhaps what was more important than electricity itself, and multiplied the importance of electricity, was the emergence of devices that received their driving forces from electricity and raised the level of well-being and quality of human life in society. It can be boldly said that the largest volume of living items in today’s society are electrical appliances. Most of these devices in human life play the role of converting electrical power into mechanical, thermal, and cooling power. Therefore, considering the importance of electricity in everyday human life, we have tried to consider an electrical power department and find the most efficient person in it with the help of an interval-valued fuzzy influence. Because worthy people should be considered to lead this organization, and with their leadership, electricity consumption should be saved and this very valuable blessing should not be wasted. Therefore, we consider the vertices of the interval-valued influence graph as the heads of each ward of the electrical power department and the edges of the graph as the degree of interaction and influence of each other. For this department, the set of staff isM=Maleki,Eskandari,Farhadi,Tahmasbi,Ashkani,Bozorgi.(1)
Eskandari has been working with Tahmasbi for 8 years and values his views on issues.(2)
Farhadi has been responsible for maintaining the electrical department’s property for a long time, and everyone is satisfied with his performance.(3)
In an electrical power department, it is very important to control the staff and check their treatment of clients. Maleki is the best person to do this.(4)
Tahmasbi and Ashkani have a long history of conflict.(5)
Due to Bozorgi’s long experience in finance, he is the best person for the job.According to the above values, we consider an interval-valued influence graph. The nodes show each of the department’s staff that the lower bound indicates the ability of staff to perform related tasks and the upper bound shows the weakness and inability of each staff member. But the edges show the level of friendship and interest among employees. If this friendship is more, then the amount of influence on each other will naturally be more. For the weight of the edges, the lower bound means the amount of friendship and the upper bound means the amount of conflict. Names of employees and level of staff capability are shown in Tables2 and 3. The adjacency matrix corresponding to Figure 4 is shown in Table 4.Table 2
Name of employees in an electrical department and their services.
NameServicesFarhadiWarehouse keeperMalekiSecurity guardBozorgiFinancial affairs managerEskandariPresident of the companyTahmasbiManaging directorAshkaniBoard of directorsTable 3
The level of staff capability.
FarhadiMalekiBozorgiEskandariTahmasbiAshkaniζMl0.20.50.50.40.30.3ζMu0.40.50.50.50.30.4Figure 4
Interval-valued fuzzy influence digraph.Table 4
Adjacency matrix corresponding to Figure4.
FarhadiMalekiBozorgiEskandariTahmasbiAshkaniFarhadi0,00,00,00,00,00,0Maleki0,00,00.3,0.40,00,00,0Bozorgi0.2,0.40,00,00,00,00.2,0.4Eskandari0.2,0.30.4,0.50.4,0.50,00,00,0Tahmasbi0.2,0.20,00,00,00,00,0Ashkani0.1,0.30,00,00,00.2,0.30,0Figure4 shows that Ashkani has 30% of the power needed to do the work of the board of directors of the electrical power department, but unfortunately he does not have 40% of the necessary power to do so. The directional edge of Eskandari-Farhadi shows that there is 20% friendship among these two employees, but unfortunately, they have 30% conflict. Clearly, Eskandari has dominion over both Maleki and Bozorgi, and his dominance over both is 40%. It is clear that Eskandari is the most influential employee of the electrical power department because he controls both the security guard and the financial affairs manager, who have 50% of the power in the department.


## 5. Conclusion

IVFG, belonging to FG family, has good abilities when faced with problems that cannot be explained by FGs. It has a wide range of applications in the field of psychological sciences as well as in the reconnoiter of individuals founded on oncological behaviors. So, in this survey, we tried to show the new notion of level graph in IVFG. Also, we introduced the notion ofα,β− HM, isomorphism, WI, and CWI of IVFG by α,β− HM. Finally, an application of HM of IVFGs has been presented by using coloring-FG. In our future work, we will introduce the new concepts of connectivity in IVFGs and investigate some of their properties. Also, we will study the new results of the complete dominating set, regular complete dominating set, and global dominating set on IVFGs.



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*Source: 8499577-2022-08-21.xml*