# Novel Concepts in Bipolar Fuzzy Graphs with Applications

Authors: Chang Wan; Fei Deng; Shitao Li; S. Omidbakhsh Amiri; A. A. Talebi; H. Rashmanlou
Journal: Journal of Mathematics
 (2022)

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## Abstract

Many problems of practical interest can be modeled and solved by using bipolar graph algorithms. Bipolar fuzzy graph (BFG), belonging to fuzzy graphs (FGs) family, has good capabilities when facing with problems that cannot be expressed by FGs. Hence, in this paper, we introduce the notion ofϑ,δ−homomorphism of BFGs and classify homomorphisms (HMs), weak isomorphisms (WIs), and co-weak isomorphisms (CWIs) of BFGs by ϑ,δ−HMs. Also, an application of homomorphism of BFGs has been presented by using coloring-FG. Universities are very important organizations whose existence is directly related to the general health of the society. Since the management in each department of the university is very important, therefore, we have tried to determine the most effective person in a university 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. However, there are some phenomena around 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]. The FS focuses on the membership degree of an object in a particular set. However, membership alone could not solve the complexities in different cases, so the need for a degree of membership was felt. To solve this problem, Zhang [2] defined the concept of bipolar fuzzy sets (BFSs) as a generalization of fuzzy sets (FSs). BFSs are an extension of FSs whose membership degree range is [−1, 1]. The first definition of FGs was proposed by Kafmann [3] in 1993, from Zade’s fuzzy relations [4, 5]. However, 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, and so on. Akram et al. [7, 8] introduced BFGs and cayley-BFGs. Rashmanlou et al. [9] investigated categorical properties in intuitionistic fuzzy graphs. Bhattacharya [10] gave some remarks on FGs, and some operations of FGs were introduced by Mordeson and Peng [11]. The concept of weak isomorphism, co-weak isomorphism, and isomorphism between FGs was introduced by Bhutani in [12]. Liu [13] defined domination number in maximal outer planar graphs. Borzooei [14] introduced domination in vague graphs. Ghorai and Pal [15] studied some isomorphic properties of m-polar FGs. Krishna et al. [16] presented new concept in cubic graph. Shao et al. [17] investigated strong equality of roman and perfect roman domination in trees. Mordeson and Nair [18] introduced the concept of complement of fuzzy graph and studied some operations on fuzzy graphs. The complement of FGs was studied by Sunitha and Vijayakumar [19]. Nagoorgani and Malarvizhi [20] investigated isomorphism properties on FGs. Ezhilmaran et al. [21] studied morphism of bipolar intuitionistic fuzzy graphs. Muhiuddin et al. [22, 23] introduced new concepts of cubic graphs. Rao et al. [24–26] presented dominating set, equitable dominating set, and isolated vertex of vague graphs. Telebi and Rashmanlou [27] described complement and isomorphism on bipolar fuzzy graphs. Shi et al. [28, 29] introduced total dominating set and global dominating set in product vague graphs. Kosari et al. [30] defined vague graph structure with an application in medical sciences. Kou et al. [31] investigated g-eccentric node and vague detour g-boundary nodes in vague graphs. Ramprasad et al. [32] introduced morphism of m-Polar fuzzy graph. Tahmasbpour et al. [33] presented f-morphism on bipolar fuzzy graphs.A BFG 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 BFG 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. With the help of BFGs, the most efficient person in an organization can be identified according to the important factors that can be useful for an institution. Homomorphisms provide a way of simplifying the structure of objects one wishes to study while preserving much of it that is of significance. It is not surprising that homomorphisms also appeared in graph theory, and that they have proven useful in many areas. Hence, in this paper, we defined the notion ofϑ,δ−homomorphism of BFGs and classify homomorphisms (HMs), weak isomorphisms (WIs), and co-weak isomorphisms (CWIs) of BFGs by ϑ,δ−HMs. Finally, we introduced the application of homomorphism of BFGs by using coloring-FG, and an application of bipolar fuzzy influence digraph has also been presented.


## 2. Preliminaries

In this section, we give some necessary concepts of bipolar fuzzy graphs and bipolar fuzzy subgroups.Definition 1.
LetV be a finite nonempty set. A graph G=V,E on V consists of a vertex set V and an edge set E, where an edge is an unordered pair of distinct vertices of G. We will use xy rather than x,y to denote an edge. If xy is an edge, then we say that x and y are adjacent. A graph is called complete if every pair of vertices is adjacent.Definition 2.
LetG1=V1,E1 and G2=V2,E2 be graphs. A mapping g:V1⟶V2 is a homomorphism from G1 to G2 if gr and gs are neighbor whenever r and s are neighbor.Definition 3.
Two graphsG1=V1,E1 and G2=V2,E2 are isomorphic if ∃ a bijective mapping ψ:V1⟶V2 so that r and s are neighbor in G1 if and only if ψr and ψs are neighbor in G2, ψ 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 called the automorphism group of G and denoted by AutG.Definition 4.
(see [2]). Let V be a nonempty set. A BFS B in V is an object having the form as follows:(1)B=r,μBPr,μBNr|r∈V,where μBP:V⟶0,1 and μBN:V⟶−1,0 are mappings.
For the sake of simplicity, we shall use the symbolB=μBP,μBN for the BFS.(2)B=r,μBPr,μBNr|r∈V.
The family of all BFSs onV is written as BFSV.Definition 5.
LetP∗=p,q:q∈−1,0,p∈0,1. For any q1,p1,q2,p2∈P∗, the orders ≤ and < on P∗ are defined as(3)q1,p1≤q2,p2⇔q1≥q2andp1≤p2,q1,p1<q2,p2⇔q1,p1≤q2,p2andq1>q2orp1<p2.
By Definition5, it is easy to see that P∗,≤ constitutes a complete lattice with minimum element 0,0 and maximum element 1,−1.Definition 6.
LetB=μBP,μBN be a BFS. For each p,q∈P∗, we describe(4)Bp,q=r∈V:μBPr≥p,μBNr≤q.
Then,Bp,q is named q,p-level set. The set r|r∈V,μAPr≠0,μANr≠0 is named the support A and is shown 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 called vertex set and edge set, respectively.Definition 7.
LetV be a finite nonempty set, A∈BFSV, and B∈BFSV˜2. The triple X=V,A,B is named a BFG on V, if for each r,s∈V˜2,(5)μBPr,s≤μAPr∧μAPsandμBNr,s≥μANr∨μANs.Definition 8.
A BFGX=V,A,B is called a strong bipolar fuzzy graph (SBFG) if for each r,s∈V˜2,(6)μBPr,s=μAPr∧μAPs,μBNr,s=μANr∨μANs,that μBPr,s,μBNr,s≠0,0 and is called complete bipolar fuzzy graph (CBFG), if for each r,s∈V˜2, we have(7)μBPr,s=μAPr∧μAPsandμBNr,s=μANr∨μANs.
A complete bipolar fuzzy graphX=V,A,B with n nodes is shown by Kn,A.
IfX=V,A,B is a BFG, 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 BFG onV is denoted by BFGV. For given X=V,A,B∈BFGV, in this study, suppose that A∗=V.Definition 9.
LetX1=V1,A1,B1 and X2=V2,A2,B2 be two BFGs. Then,(1)
A mappingψ:V1⟶V2 is a homomorphism from X1 to X2 if(i)
μA1Pr≤μA2Pψr,μA1Nr≥μA2Nψr,for allr∈V1(ii)
μB1Prs≤μB2Pψrψs,μB1Nrs≥μB2Nψrψs,for allrs∈V˜2(2)
A mappingψ:V1⟶V2 is a weak isomorphism from X1 to X2 if ψ is a bijective homomorphism from X1 to X2 and(8)μA1Pr=μA2Pψr,μA1Nr=μA2Nψr,for allr∈V1.(3)
A mappingψ:V1⟶V2 is a co-weak isomorphism from X1 to X2 if ψ is a bijective homomorphism from X1 to X2 and(9)μB1Prs=μB2Pψrψs,μB1Nrs=μB2Nψrψs,for allrs∈V˜2.(4)
An isomorphism fromX1 to X2 is a bijective mapping ψ:V1⟶V2 so that(i)
μA1Pr=μA2Pψr,μA1Nr=μA2Nψr,for allr∈V1(ii)
μB1Prs=μB2Pψrψs,μB1Nrs=μB2Nψrψs,for allrs∈V˜2Definition 10.
Suppose thatX=V,A,B and Y=V,A′,B′ be two BFGs. Then, X is a BFSG of Y if A⊆A′ and B⊆B′.Definition 11.
LetX=V,A,B be a BFG and W⊆V. Then, the BFG Y=W,A′,B′ so that(10)μA′Pr=μAPr,μA′Nr=μANr,for allr∈W,μB′Prs=μBPrs,μB′Nrs=μBNrs,for allrs∈W˜2,is called the induced BFSG by W and shown by XW.Definition 12.
A familyΓ=μ1,μ2,…,μk of BFSs on V is named a k-coloring of BFG X=V,A,B if(i)
∨Γ=A.(ii)
μi∧μj=0, for 1≤i,j≤k.(iii)
For each strong edgers of X, minμir,μis=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 graphBFSBipolar fuzzy setFSFuzzy setBFGBipolar fuzzy graphCBFGComplete bipolar fuzzy graphSBFGStrong bipolar fuzzy graphBMBijective mappingHMHomomorphismWIWeak isomorphismCWICo-weak isomorphismBHBijective homomorphismSGSubgraphCGComplete graphBFSGBipolar fuzzy subgraph


## 3. Homomorphisms and Isomorphisms of Bipolar Fuzzy Graphs

In this section, we discuss homomorphisms and isomorphisms of bipolar fuzzy graphs by homomorphism of level graphs in bipolar fuzzy graphs.Theorem 1.
LetVbe a finite nonempty set,A∈BFSVandB∈BFSV˜2. Then,X=V,A,B∈BFGVif and only ifXϑ,δ=Aϑ,δ,Bϑ,δis a graph, for allϑ,δ∈P∗,Aϑ,δ≠∅.Proof.
LetX=V,A,B be a BFG. For each ϑ,δ∈P∗, Aϑ,δ≠∅, suppose that rs∈Bϑ,δ. Then, μBPrs≥ϑ and μBNrs≤δ. Because X is a BFG,(11)ϑ≤μBPrs≤μAPr∧μAPsandδ≥μBNrs≥μANr∨μANs.
It follows thatr,s∈Aϑ,δ. Therefore, Aϑ,δ,Bϑ,δ is a graph.
Conversely, letXϑ,δ=Aϑ,δ,Bϑ,δ is a graph, for all ϑ,δ∈P∗, Aϑ,δ≠∅. For each rs∈V˜2, let μBPrs=ϑ,μBNrs=δ. Then, rs∈Bϑ,δ. Hence, r,s∈Aϑ,δ. Thus, μAPr≥ϑ,μAPs≥ϑ,μANr≤δ,andμANs≤δ. This implies that μAPr∧μAPs≥ϑ=μBPrs and μANr∨μANs≤δ=μBNrs. Therefore, X=V,A,B is a BFG.Definition 13.
LetX=V,A,B and Y=W,A′,B′ be two BFGs, g:V⟶W a mapping. Then, for any ϑ,δ∈P∗, Aϑ,δ≠∅, if g is a homomorphism from Xϑ,δ=Aϑ,δ,Bϑ,δ to Yϑ,δ=Aϑ,δ′,Bϑ,δ′, g is named ϑ,δ−homomorphism mapping from X to Y.Theorem 2.
LetX=V,A,BandY=W,A′,B′be two BFGs. Then,g:X⟶Yis a homomorphism fromXtoYif and only ifgisϑ,δ−homomorphism fromXtoY.Proof.
Assume thatg:X⟶Y is a homomorphism from X to Y. Let Aϑ,δ≠∅, ϑ,δ∈P∗. If r∈Aϑ,δ, then(12)μA′Pgr≥μAPr≥ϑ,μA′Ngr≤μANr≤δ.
Hence,gr∈Aϑ,δ′ implying g is a mapping from Aϑ,δ to Aϑ,δ′. For r,s∈Aϑ,δ, let rs∈Bϑ,δ. Then,(13)μBPrs≥ϑ,μBNrs≤δ.
Hence,(14)μB′Pgrgs≥μBPrs≥ϑ,μB′Ngrgs≤μBNrs≤δ,which implies grgs∈Bϑ,δ′. Therefore, g is a homomorphism from Xϑ,δ to Yϑ,δ.
Conversely, letg:V⟶W be a ϑ,δ−homomorphism from X to Y. For arbitrary element r∈X, let μAPr=c, μANr=d. Then, r∈Ac,d. Hence, gr∈Ac,d′ because g is a homomorphism from Ac,d,Bc,d to Ac,d′,Bc,d′. It follows that(15)μA′Pgr≥c,μA′Ngr≤d,that is,(16)μA′Pgr≥μAPr,μA′Ngr≤μANr.
Now, for arbitrariesr,s∈V, let μBPrs=e, μBNrs=f. Then,(17)e=μBPrs≤μAPr∧μAPs,f=μBNrs≥μANr∨μANs.
Hence,r,s∈Ae,f and rs∈Be,f. Because g is a homomorphism from Xe,f=Ae,f,Be,f to Ye,f=Ae,f′,Be,f′, we conclude that gr,gs∈Ae,f′ and grgs∈Be,f′. Therefore,(18)μB′Pgrgs≥e=μBPrs,μB′Ngrgs≤f=μBNrs.Theorem 3.
LetX=V,A,BandY=W,A′,B′be two BFGs. Then,g:V⟶Wis a WI fromXtoYif and only ifgis a bijectiveϑ,δ−homomorphism fromXtoYand(19)μAPr=μA′Pgr,μANr=μA′Ngrfor allr∈V.Proof.
Letf be a WI from X to Y. From the definition of homomorphism, g is a bijective homomorphism from X to Y. By Theorem 2, f is a bijective ϑ,δ−homomorphism from X to Y, and also by the definition of WI, we have(20)μAPr=μA′Pgr,μANr=μA′Ngrfor allr∈V.
Conversely, from hypothesis,g:A0,1=V⟶A0,1′=W is a bijective mapping and(21)μAPr=μA′Pgr,μANr=μA′??Ngrfor allr∈V.
Forr,s∈V, let μBPrs=e, μBNrs=f. Then,(22)e=μBPrs≤μAPr∧μAPs,f=μBNrs≥μANr∨μANs,which implies r,s∈Ae,f and rs∈Be,f. Because g is a homomorphism from Ae,f,Be,f to Ae,f′,Be,f′, we have gr,gs∈Ae,f′ and grgs∈Be,f′. Hence,(23)μB′Pgrgs≥e=μBPrs,μB′Ngrgs≤f=μBNrs,which completes the proof.Theorem 4.
LetX=V,A,BandY=W,A′,B′be two BFGs. Then,g:V⟶Wis a co-weak isomorphism fromXtoYif and only ifgis a bijectiveϑ,δ−homomorphism fromXtoYand(24)μBPrs=μB′Pgrgs,μBNrs=μB′Ngrgsfor allrs∈V˜2.Proof.
Letg:V⟶W be a co-weak isomorphism from X to Y. Then, g is a bijective homomorphism from X to Y. By Theorem 2, f is a bijective ϑ,δ−homomorphism from X to Y. Also, by the definition of co-weak isomorphism,(25)μBPrs=μB′Pgrgs,μBNrs=μB′Ngrgsfor allrs∈V˜2.
Conversely, from hypothesis, we know thatf:A0,1=V⟶A0,1′=W is a bijective mapping and(26)μBPrs=μB′Pgrgs,μBNrs=μB′Ngrgs.
For arbitrary elementr∈V, suppose that μAPr=c, μANr=d. Then, we have r∈Ac,d. Now, because g is a homomorphism from Ac,d,Bc,d to Ac,d′,Bc,d′, gr∈Ac,db′. Thus, μA′Pgr≥c=μAPr and μA′Ngr≤d=μANr, which implies g is a co-weak isomorphism from X to Y.Corollary 1.
LetX=V,A,B∈BFGV,Y=W,A′,B′∈BFGW. Ifg:V⟶Wis a co-weak isomorphism fromXtoY, thengis an injective homomorphism fromXϑ,δtoYϑ,δ, for allϑ,δ∈P∗,Aϑ,δ≠∅.
From the following example, we conclude that the converse of Corollary1 does not need to be true.Example 1.
LetX=V,A,B and Y=W,A′,B′ be two BFGs as shown in Figure 1. Consider the mapping g:V⟶W, defined by gvi=wi, 1≤i≤4. In view of the ϑ,δ−level graphs of X and Y in Figure 1, it is easy to see that if Aϑ,δ≠∅, then g is an injective homomorphism from Xϑ,δ to Yϑ,δ, but g is not a co-weak isomorphism.Figure 1
BFGsX, Y and the mapping g:Vi⟶Wi which is not a co-weak isomorphism.Theorem 5.
LetX=V,A,B∈BFGV,Y=W,A′,B′∈BFGW, andg:V⟶Wbe a mapping. For eachϑ,δ∈P∗,Aϑ,δ≠∅, ifgis an isomorphism fromXϑ,δto a subgraph ofYϑ,δ, thengis a co-weak isomorphism fromXto an induced BFSG ofY.Proof.
The mappingg is an isomorphism from X0,1=V,B0,1 to a subgraph Y0,1=W,B0,1′. So, g:V⟶W is an injective mapping. For arbitrary r∈V, suppose that μAPr=ϑ, μANr=δ. Then, r∈Aϑ,δ, and so gr∈Aϑ,δ′. Hence, μA′Pgr≥ϑ=μAPr and μA′Ngr≤δ=μANr. For r,s∈V, let μBPrs=ϑ and μBNrs=δ. Then, ϑ≤μAPr, ϑ≤μAPs, δ≥μANr, δ≥μANs, and rs∈Bϑ,δ. Hence, r,s∈Aϑ,δ and rs∈Bϑ,δ. Since g isomorphism from Xϑ,δ to Yϑ,δ, we get gr,gs∈Aϑ,δ′ and gr,gs∈Bϑ,δ′. Therefore,(27)μB′Pgrgs≥ϑ=μBPrs,μB′Ngrgs≤δ=μBNrs.I.
Now, letμB′Pgrgs=t, μB′Ngrgs=s. Then, grgs∈At,s′. Because g is injective and an isomorphism from Xp,q to a subgraph of Yp,q, we have p,q∈Ap,q and xy∈Bp,q. Therefore,(28)μBPs≥p=μB′Pgrgs,μBNrs≤q=μB′Ngrgs.II.
Now, byI and II, we conclude that(29)μB′Pgrgs=μBPrs,μB′Ngrgs=μBNrs.Corollary 2.
LetX=V,A,BandY=W,A′,B′be two BFGs withV=W, andg:V⟶Wa mapping. Forϑ,δ∈P∗,Aϑ,δ≠∅, ifgis an isomorphism fromXϑ,δto a subgraph ofYϑ,δ, thengis a co-weak isomorphism fromXtoY.Theorem 6.
LetX=V,A,BandY=W,A′,B′be two BFGs,f:V⟶Wbe a bijective mapping. If for eachϑ,δ∈P∗,gis an isomorphism fromXϑ,δtoYϑ,δand thengis an isomorphism fromXtoY.Proof.
From hypothesis,g−1:W⟶V is a bijective mapping and an isomorphism from Yϑ,δ to Xϑ,δ. By Theorem 5, g is a co-weak isomorphism from X to Y and g−1 is a co-weak isomorphism from Y to X. Therefore, g is an isomorphism from X to Y.Corollary 3.
LetX=V,A,Bbe a BFG andg:V⟶Va bijective mapping. Then,gis an automorphism ofXif and only iff|Aϑ,δis an automorphism ofXϑ,δ, from anϑ,δ∈P∗,Aϑ,δ≠∅.Theorem 7.
LetX=V,A,Bbe a BFG. Then,Xis a complete bipolar fuzzy graph if and only ifXϑ,δ=Aϑ,δ,Bϑ,δis a complete graph (CG) forϑ,δ∈P∗.Proof.
Proof. IfX=V,A,B is a complete bipolar fuzzy graph and for ϑ,δ∈P∗, Aϑ,δ≠∅, r,s∈Aϑ,δ, then μAPr≥ϑ, μAPs≥ϑ, μANr≤δ, μANs≤δ, and so(30)μBPrs=μAPr∧μAPs≥ϑ,μBNrs=μANr∨μANs≤δ.
Hence,rs∈Bϑ,δ. It follows that Xϑ,δ is a CG.
Conversely, suppose thatX=V,A,B is not a complete bipolar fuzzy graph. Then, there are r,s∈V so that μBPrs<μAPr∧μAPs or μBPrs<μAPr∨μAPs. Let μBPrs<μAPr∧μAPs and μAPr∧μAPs=ϑ, for ϑ∈0,1. Then, μAPr≥ϑ and μAPs≥ϑ. Hence, r,s∈Aϑ,δ, for a δ∈0,1, but rs∉Bϑ,δ. This implies that Xϑ,δ is not a CG. For the case μBNrs>μANr∨μANs, it follows similarly.Theorem 8.
LetX=V,A,B∈BFGV. Then,Xϑ,δhas not IV, for eachϑ,δ∈P∗,Aϑ,δ≠∅if and only if for eachr∈V, there existss∈Vso thatμBPrs=μAPrandμBNrs=μANr.Proof.
Suppose that for eachϑ,δ∈P∗, Aϑ,δ≠∅, graph Xϑ,δ has not IV and there is a vertex r∈V so that for each s∈V, μBPrs<μAPr or μBNrs>μANr. Let μBPrs<μAPr and μAPr=ϑ, μANr=δ, for ϑ,δ∈P∗. Then, r∈Aϑ,δ and for each s∈V, s≠r, rs∉Bϑ,δ. Therefore, r is an IV in the graph Xϑ,δ=Aϑ,δ,Bϑ,δ, which is a contradiction.
Now, suppose that forϑ,δ∈P∗, Aϑ,δ≠∅, vertex r∈Aϑ,δ is an IV in Xϑ,δ. If s∉Aϑ,δ, then μBPrs≤μAPs<ϑ≤μAPr or μBNrs≥μANs>δ≥μANr, and if s∈Aϑ,δ, it is trivial that rs∉Bϑ,δ. Hence, μBPrs<ϑ≤μAPr or μBNrs>δ≥μANr. Therefore, for each s∈V, μBPrs≠μAPr, μBNrs≠μANr.Theorem 9.
A BFGX=V,A,Bisr-colorable⇔there exists a homomorphism fromXtoKr,A′.Proof.
Assume thatX be r-colorable with r colors labeled Γ=μ1,μ2,…,μr. Let Vi=v∈V|μiv≠0. We define complete bipolar fuzzy graph Kr,A′ with vertices set 1,2,…,r, so that the degree of positive membership vertex i is μA′Pi=maxμAPv|v∈Vi and the degree of negative membership vertex i is μA′Ni=minμANv|v∈Vi. Now, the mapping g:X⟶Kr,A′ defined by gv=i,v∈Vi is a graph homomorphism because for v∈Vi(i)
μAPv≤maxμAPw|w∈Vi=μA′Pi=μA′Pgv,(31)μANv≥minμANw|w∈Vi=μA′Ni=μA′Ngv.(ii)
According to the definition of complete bipolar fuzzy graph, foru∈Vi and v∈Vj, we have(32)μBPuv≤μAPu∧μAPv≤μA′Pi∧μA′Pj=μA′Pgu∧μA′Pgv,μBNuv≥μANu∨μANv≥μA′Ni∨μA′Nj=μA′Ngu∨μA′Ngv,then μBPuv≤μB′Pgugv and μBNuv≥μB′N(gugv, for all uv∈V˜2.
Conversely, letg:X⟶Kr,A′ be a homomorphism. For a given k∈VKr,A′, define the set g−1k⊆V to be(33)g−1k=x∈V|gx=k.
Ifv∈g−1k, let μkv=μμkPv,μμkv=tAv,fAv; otherwise, μkv=0. Therefore, the fuzzy bipolar graph X is r-colorable with coloring set μ1,μ2,…,μr.


## 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 BFG.Example 2.
We obtain a BFGX=V,A,B on the vertex set v1,v2,v3,v4,v5,v6 by joining two vertices with respect to the effect they have one another (see Figure 2). Let v1v2, v1v3, v1v4, v2v4, v2v6, v2v5, v3v4, and v6v4 be edges of graph X. The positive membership and negative membership values μAP,μAN of the vertices are the good and bad quality, respectively. Also, the positive membership and negative membership values μBP,μBN of the edges are compatible and incompatible materials, respectively. We want to see that how to put the materials in such a way that they do not have any effect on each other. Now, by Theorem 9, there is a homomorphism from X to CG with n=3. Therefore, we need at least 3 parts (3-colors) to put the materials.
In the next example, we want to identify the most effective employee of a university with the help of a bipolar influence digraph.Figure 2
Bipolar fuzzy graph.X=V,A,B.Example 3.
The emergence of science and knowledge is equal to the creation of man, and man has always sought to understand and comprehend. Science and knowledge have a special place 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. Science and knowledge are two wings with which man can fly indefinitely. All the tools and instruments that we use today and cause the fundamental difference between past and present life are the result of effort and science and knowledge that man has discovered and used. Thanks to science and knowledge that many patients are saved from death, earthquake-proof buildings are built, and man can see the whole planet from above. Science and knowledge are the result of discovering hidden secrets in the heart of nature and secrets that human beings have endured many hardships to discover so that we can now easily use them. Although knowledge plays a very important role in human life and causes evolution and progress, sometimes it may also bring dangers to the human race, and this is if man uses what he has learned in the wrong way, science and knowledge need to know how to use it properly so that man is always on the right path. So, universities should hire the best teachers and staff to do the work of the students and provide the necessary conditions for their education. Therefore, in this section, we try to identify the most effective employees in a university according to their performance. Hence, we consider the vertices of the bipolar influence graph as the head of each ward of the university and the edges of the graph as the degree of interaction and influence of each other. For this university, the set of staff isB=Alavi,Rasooli,Tabari,Omrani,Razavi,Salehi,Taghavi:(a)
Rasooli has been working with Omrani for 11 years and values his views on issues.(b)
Alavi has been the head of library for a long time, and not only Rasooli but also Omrani is very satisfied with Alavi’s performance.(c)
In a university, preserving educational documents as well as taking care of university services is a very important task. Omrani is the most suitable person for this responsibility.(d)
Tabari and Salehi have a long history of conflict.(e)
Tabari has an important role in the informatics department.
Given the above, we consider a bipolar influence graph. The vertices represent each of the university staff. Note that each staff member has the desired ability as well as shortcomings in the performance of their duties. Therefore, we use of BFS to express the weight of the vertices. The positive membership indicates the efficiency of the employee, and the negative membership shows the lack of management and shortcomings of each staff. However, the edges describe the level of relationships and friendships between employees that the positive membership shows a friendly relationship between both employees and the negative membership shows the degree of conflict between the two officials. Name of employees and level of staff capability are shown in Tables2and 3. The adjacency matrix corresponding to Figure 3 is shown in Table 4.
Figure3 shows that Salehi has 90% of the power needed to do the university work as the head of welfare services but does not have the 20% knowledge needed to be the boss. The directional edge Rasooli–Omrani shows that there is 60% friendship among these two employees, and unfortunately they have 30% conflict. Clearly, Razavi has dominion over both Salehi and Rasooli, and his dominance over both is 60%. It is clear that Razavi is the most influential employee of the university because he controls both the head of welfare services and head of postgraduate education, who have 90% of the power in the university.Table 2
Name of employees in a university and their services.
NameServicesAlaviHead of libraryRasooliHead of postgraduate educationTabariHead of informatics departmentOmraniHead of securityRazaviHead of universitySalehiHead of welfare servicesTaghaviHead of research departmentTable 3
The level of staff capability.
AlaviRasooliTabariOmraniRazaviSalehiTaghaviμBP0.40.90.60.70.80.90.5μBN−0.4−0.3−0.2−0.3−0.2−0.2−0.4Figure 3
Bipolar influence digraph.Table 4
Adjacency matrix corresponding to Figure3.
AlaviRasooliTabariOmraniRazaviSalehiTaghaviAlavi0,00,00.4,−0.20.3,−0.30,00,00,0Rasooli0,00,00,00.6,−0.30,00,00.4,−0.2Tabari0,00,00,00.5,−0.20,00,00,0Omrani0,00,00,00,00,00,00,0Razavi0,00.6,−0.10,00.5,−0.10,00.6,−0.10,0Salehi0,00.7,−0.20,00,00,00,00,0Taghavi0,00,00,00.3,−0.20,00,00,0


## 5. Conclusion

BFGs 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 BFGs, 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 ofϑ,δ−homomorphism of BFGs and classify homomorphisms, weak isomorphisms, and co-weak isomorphisms of BFGs by ϑ,δ−homomorphisms. We also investigated the level graphs of BFGs to characterize some BFGs. Finally, we presented two applications of BFGs in coloring problem and also finding effective person in a university. In our future work, we will introduce new concepts of connectivity in BFGs and investigate some of their properties. Also, we will study new results of global dominating set, restrain dominating set, connected perfect dominating set, regular perfect dominating set, and independent perfect dominating set on BFGs.



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