# A Study onm-Polar Interval-Valued Intuitionistic Fuzzy Graphs with Application in Management

Authors: Lian Chen; Aysha Khan; Maryam Akhoundi; A. A. Talebi; G. Muhiuddin; S. H. Sadati
Journal: Journal of Mathematics
 (2022)

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## Abstract

Interval-valued intuitionistic fuzzy graphs (IVIFGs) have many fields of work in other sciences, including psychology, life sciences, medicine, and social studies, and can help researchers with optimization and save time and money. A very important type of IVIFG is anm-polar IVIFG that can play a significant role in managing and finding effective people in a company or institution. It can also be used to achieve very important results in the field of psychological sciences and social relations. Hence, in this study, we presented the notion of m-polar h-morphism on m-polar IVIFGs. Some elegant theorems on weak isomorphism (WI) and co-weak isomorphism (CWI) are obtained. Likewise, we survey m-polar h-morphism on strong regular (SR) and highly irregular (HI) m-polar-IVIFGs. Management always plays an important role in an organization and can cause its growth and prosperity. But finding a strong leader is a very important issue that needs to be addressed. Therefore, in the last section, we have tried to express the application of the m-polar IVIFG in the management problem.


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## Body

## 1. Introduction

Graph theory, as a branch of mathematics, examines topics and the relationships between them. Simultaneously with the widespread influence of this concept in various phenomena, there were many ambiguities in determining the vertices and edges of the graph. Facing ambiguous amounts has made it difficult for researchers. With the definition of the fuzzy set (FS) by Atanassov [1] and the fuzzy graph (FG) by Atanassov and Gargov [2] and Akram et al. [3], a new chapter of work on graphs began by the researchers. The vague problems that preoccupied researchers were now easily solved. This idea soon came to the attention of researchers in this field, and then, different types of FGs were introduced one after another. Akram et al. [4] introduced fuzzy logic. Akram et al. [5] studied new results of regularity in vague graphs (VGs). Azeem and Nadeem [6] introduced the concept of intuitionistic fuzzy set (IFS) as a generalization of FSs. The idea of a bipolar FS was proposed by Azeem et al. [7, 8] in 1994. Bhutani [9] investigated new concepts in bipolar FGs. The interval-valued fuzzy graph (IVFG) was defined by Borzooei et al. [10]. Chen et al. [11] introduced interval-valued intuitionistic fuzzy sets (IVIFSs). Ezhilmaran and Sankar [12, 13] defined m-polar interval-valued intuitionistic fuzzy graphs (IVIFGs) and interval-valued intuitionistic fuzzy competition graphs. Ghorai and Pal [14–16] investigated the vague hypergraphs and Cayley VGs. Kaufman [17, 18] studied new concepts in vague graph structures. Kou et al. [19–21] defined dominating set and equitable dominating set in VGs. Nagoor Gani and Malarvizhi [22, 23] presented total dominating set (DS) and global DS in product VGs. Poulik and Ghorai [24] introduced new concepts in IFG. Poulik and Ghorai [25] investigated global restrained DS and total k-DS in VGs. Some properties of FGs and their types have been discussed by Ghorai and Poulik [26–29]. Ramprasad et al. [30–32] have studied some topics using graphs.Rao et al. [33] proposed the m-polar FS as an extension of the bipolar FS. This concept was able to study the fuzzy values of several variables on the vertices and edges of a graph simultaneously. Rashmanlou et al. [34, 35] studied the properties of generalized m-polar FGs and m-polar fuzzy planar graphs.Rosenfeld [36, 37] defined the isomorphism properties on FGs. Seethalakshmi and Gnanajothi [38] introduced WI, CWI, and isomorphism in FGs. Shi and Kosari [39] studied a generic definition of fuzzy morphism between graphs. Shi and Kosari [40] proved that an FG can be associated with a fuzzy group naturally as an automorphism group. Shao [41] presented the notion of f-morphism on IFGs. Talebi [42] studied morphism on m-Polar FG. Talebi [43, 44] has defined morphism on a bipolar IFG and m-polar IFG. Zhang [45] investigated isomorphisms on the L-fuzzy graphs. Some features of isomorphic m-polar FGs were introduced by Zhang [46].Although IFGs are better at expressing uncertain variables than FGs, they do not perform well in many real-world situations such as IT management. Therefore, when the data come from several factors, it is necessary to use them-polar IFG. m-polar IFG has better flexibility in the face of problems that cannot be articulated by FG and IFG. IVIFGs have many applications in real systems, especially where the level of information given in the system varies with time and accuracy. m-polarity is used to illustrate real-world phenomena using m-polar fuzzy models in a variety of fields including technology, social networking, and biological sciences. Hence, in this paper, while reviewing the initial definitions, we examine the f-morphism on the m-polar IVIFG. Some of its properties, such as the equivalence relationship, have been studied. The effect of isomorphism and its properties on strong regular and highly irregular m-polar IVIFG has also been investigated. Leadership always plays an important role in an organization and can cause its growth and prosperity. But finding a strong leader is a very important issue that needs to be addressed. Therefore, in this research work, we have tried to determine the leadership of an economic team with the help of an m-polar IVIFG.


## 2. Preliminaries

In this section, we review the main concepts related to the topic under discussion. In the following definitions, we consider D [0,1] as the set of all subintervals of [0,1], and the elements of this collection are shown in capital letters.Definition 1.
(see [13]). An m-polar IVIFS M on V is described as(1)M=φ1MLr,φ1MUr,…,φmMLr,φmMUr,ψ1MLr,ψ1MUr,…,ψmMLr,ψmMUr,for all r∈V and or shortly(2)M=φiMLr,φiMUri=1m,ψiMLr,ψiMUri=1m|r∈V,m∈ℕ,so that the functions φiM:V⟶D0,1 and ψiM:V⟶D0,1 show the degree of m-polar MV and m-polar N-MV of the element r∈V, respectively. Also,(3)0≤φiMLr≤φiMUr≤1,0≤ψiMLr≤ψiMUr≤1,0≤φiMLr+ψiMUr≤1,∀r∈V.Definition 2.
(see [13]). An m-polar-IVIFG with underlying graph G∗ is defined as G=V,M,N, in which(i)
M is an m-polar-IVIFS on V as(4)M=φiMLr,φiMUri=1m;ψiMLr,ψiMUri=1m|r∈V,m∈ℕ.(ii)
N is an m-polar-IVIFR on V×V so that(5)N=φiNLrs,φiNUrsi=1m;ψiNLrs,ψiNUrsi=1m|rs∈E,m∈ℕ,and the functions φiN:E⊂V×V⟶D0,1 and ψiN:E⊂V×V⟶D0,1 are described by(6)φiNLrs≤minφiMLr,φiMLs,φiNUrs≤minφiMUr,φiMUs,φiNLrs≥maxψiMLr,ψiMLs,ψiNUrs≥maxψiMUr,ψiMUs,such that 0≤φiNUrs+ψiNUrs≤1, ∀rs∈E, and 1≤i≤m.Definition 3.
(see [13]). The m-polar-IVIFG is called to be strong if for 1≤i≤m,(7)φiNLrs=minφiMLr,φiMLs,φiNUrs=minφiMUr,φiMUs,ψiNLrs=maxψiMLr,νiMLs,ψiNUrs=maxψiMUr,ψiMUs.
All the basic notations are shown in Table1.Table 1
Some basic notations.
NotationMeaningIVIFGInterval-valued intuitionistic fuzzy graphWIWeak isomorphismCWICo-weak isomorphismSRStrong regularHIHighly irregularIVIFSInterval-valued intuitionistic fuzzy setBMBijective mappingMVMembership valueSEStrong edge


## 3.h-Morphism on m-Polar IVIFG

In this section,h-Morphism on m-polar IVIFG is introduced and some related results are discussed.Definition 4.
SupposeG1=V1,M1,N1 and G2=V2,M2,N2 are two m-polar-IVIFGs.
A homomorphismh from G1 to G2 is a mapping h:V1⟶V2 so that, for each 1≤i≤m, we have the following:(i)
φiM1Lr≤φiM2Lhr,φiM1Ur≤φiM2Uhr,φiM1Lr≥ψiM2Lhr,ψiM1Ur≥ψiM2Uhx,for eachr∈V1..(ii)
φiN1Lrs≤φiN2Lhr,hs,φiN1Urs≤φiN2Uhr,hs,ψiN1Lrs≥ψiN2Lhr,hs,ψiN1Urs≥ψiN2Uhr,hs,for eachrs∈E1.Definition 5.
LetG1 and G2 be two m-polar-IVIFGs. An isomorphism h from G1 to G2 is a bijective mapping (BM) h:V1⟶V2 that holds in the following conditions:(i)
φiM1Lr=φiM2Lhr,φiM1Ur=φiM2Uhr,ψiM1Lr=ψiM2Lhr,ψiM1Ur=ψiM2Uhr,for eachr∈V1,1≤i≤m.(ii)
φiN1Lrs=φiN2Lhr,hs,φiN1Urs=φiN2Uhr,hs,ψiN1Lrs=ψiN2Lhr,hs,ψiN1Urs=ψiN2Uhr,hs,for eachrs∈E1,1≤i≤m.Definition 6.
A WIh from G1 to G2 is a BM h:V1⟶V2 that satisfies the following conditions:(i)
his homomorphism,(ii)
φiM1Lr=φiM2Lhr,φiM1Ur=φiM2Uhr,ψiM1Lr=νiM2Lhr,ψiM1Ur=ψiM2Uhr,∀r∈V1,1≤i≤m.Definition 7.
LetG1 and G2 be two m-polar-IVIFGs. A BM h:V1⟶V2 is named m-polar interval-valued intuitionistic fuzzy morphism or h-morphism of m-polar-IVIFG if there are two numbers l1>0 and l2>0 so that(i)
φiM2Lhr=l1φiM1Lr,φiM2Uhr=l1φiM1Ur,ψiM2Lhr=l1ψiM1Lr,ψiM2Uhr=l1ψiM1Ur,∀r∈V1,1≤i≤m.(ii)
φiN2Lhr,hs=l2φiN1Lrs,φiN2Uhr,hs=l2φiN1Urs,ψiN2Lhr,hs=l2ψiN1Lrs,ψiN2Uhr,hs=l2ψiN1Urs,∀rs∈E1,1≤i≤m..
In this case,h is named an l1,l2m-polar interval-valued intuitionistic h-morphism on G1 to G2. If l1=l2=l, then, h is named m-polar interval-valued intuitionistic l-morphism.Example 1.
Consider twom-polar IVIFG G1=V1,M1,N1 and G2=V2,M2,N2 that G1∗ and G2∗ as drawn in Figure 1.
Anm-polar IVIFG G1=V1,M1,N1 is shown in Figure 1(a) according to Table 2.
Anotherm-polar IVIFG G1=V2,M2,N2 is shown in Figure 1(b) according to Table 3.
Here, there exists anm-polar interval-valued intuitionistic h-morphism so that hx=x′, hy=y′, hz=z′, l1=10, and l2=10.Figure 1
h-morphism of m-polar IVIFGs G1 and G2.
(a)(b)Table 2
The degree of MV and N-MV of the nodes and edges inG1.
G1M1x0.02,0.03,0.01,0.04,0.04,0.05,0.02,0.04y0.01,0.02,0.03,0.05,0.02,0.05,0.01,0.03z0.03,0.04,0.02,0.03,0.01,0.05,0.04,0.06N1xy0.01,0.02,0.01,0.03,0.04,0.05,0.03,0.04xz0.02,0.03,0.01,0.02,0.04,0.06,0.05,0.06yz0.01,0.02,0.02,0.03,0.03,0.06,0.05,0.06Table 3
The degree of MV and N-MV of the nodes and edges inG2.
G2M2x′0.2,0.3,0.1,0.4,0.4,0.5,0.2,0.4y′0.1,0.2,0.3,0.5,0.2,0.5,0.1,0.3z′0.3,0.4,0.2,0.3,0.1,0.5,0.4,0.6N2x′y′0.1,0.2,0.1,0.3,0.4,0.5,0.3,0.4x′z′0.2,0.3,0.1,0.2,0.4,0.6,0.5,0.6y′z′0.1,0.2,0.2,0.3,0.3,0.6,0.5,0.6Theorem 1.
h-morphism is an equivalence relation on a set of m-polar IVIFGs.Proof.
Consider a set ofm-polar IVIFGs. Assume G1≈G2 if there is a l1,l2m-polar interval-valued intuitionistic h-morphism from G1 to G2 where both l1≠0 and l2≠0. Consider the identity morphism G1 to G1. It is a 1,1-morphism from G1 to G1, and so, ≈ is reflexive. Let G1≈G2. Then, there is a l1,l2-morphism from G1 to G2 for some l1≠0 and l2≠0. So,(8)φiM2hr=l1φiM1r,∀r∈V1,φiN2hr,hs=l2φiN1rs,∀rs∈E1.
Considerh−1:G2⟶G1. Let m,n∈V2. Since h−1 is bijective, m=hr, n=hs, for some r,s∈V. Now,(9)φiM1h−1m=φiM1h−1hr=φiM1r=1l1φiM2hr=1l1φiM2m,φiN1h−1mh−1n=φiN1h−1hrh−1fs=φiN1rs,=1l2φiN2hrhs=1l2φiN2mn.
Similarly,(10)ψiM1h−1m=1l1ψiM2m,ψiN1h−1mh−1n=1l2ψiN2mn.
Therefore, there is1/l1,1/l2-morphism from G2 to G1. Thus, G2≈G1, and so, ≈ is symmetric. Let G1≈G2 and G2≈G3. Then, there is a l1,l2-morphism from G1 to G2, say h for some l1≠0 and l2≠0, and there exists l3,l4-morphism from G2 to G3, call g for some l3≠0 and l4≠0. So, for i=1,2,…,m,(11)φiM3gr=l3φiM2r,∀r∈V2,φiN3grgs=l4φiN2rs,∀rs∈E2.
Letf:goh:G1⟶G3. Now,(12)φiM3fr=φiM3gohr=φiM3ghr=l3φiM2hr=l3l1φiM1r,φiN3frfs=φiN3gohrgohs=φiN3ghrghs,=l4φiN2hrhs=l4l2φiN1rs.
Similarly,(13)ψiM3fr=l3l1ψiM1r,ψiN3frfs=l4l2ψiN1rs.
So, there existsl3l1,l4l2 morphism h from G1 to G3. Then, G1≈G3 and ≈ is transitive. Thus, h-morphism is an equivalence relation in the set of m-polar IVIFGs.Theorem 2.
SupposeG1 and G2 are two m-polar IVIFGs. Let G1 be l1,l2m-polar IVIFG morphism to G2 for l1≠0 and l2≠0. Then, the image of an SE in G1 is also an SE in G2 if and only if l1=l2.Proof.
Letrs be a SE in G1 so that hrhs is also a SE in G2.
Now asG1≈G2 for i=1,2,…,m, we have(14)l2μiN1Lrs=φiN2Lhrhs=minφiM2Lhr,φiM2Lhs,=minl1φiM1Lr,l1φiM1Ls=l1minφiM1Lr,φiM1Ls,=l1φiN1Lrs,∀rs∈E1.
Hence,(15)l2φiN1Lrs=l1φiN1Lrs,∀rs∈E1.
Similarly in other equations,(16)l2φiN1Urs=l1φiN1Urs,l2ψiN1Lrs=l1ψiN1Lrs,l2ψiN1Urs=l1ψiN1Urs,∀rs∈E1.
The equations are true if and only ifl1=l2.Theorem 3.
If anm-polar IVIFG G1 is CWI to G2 and G1 is regular, then, G2 is also regular.Proof.
SupposeG1 is CWI to G2. Then, there is a BM h:G1⟶G2 so that i=1,2,…,m satisfies the following conditions:(17)φiN1Lrs=φiN2Lhrhs,φiN1Urs=φiN2Uhrhs,ψiN1Lrs=ψiN2Lhrhs,ψiN1Urs=ψiN2Uhrhs,∀rs∈E1.
AsG1 is regular, for r∈V1, ∑r≠sφiN1Lrs=Constant. Now,(18)∑hr≠hsφiN2Lhrhs=∑r≠sφiN1Lrs=Constant.
Similarly,(19)∑hr≠hsφiN2Uhrhs=∑r≠sφiN1Urs=Constant,∑hr≠hsψiN2Lhrhs=∑r≠sψiN1Lrs=Constant,∑hr≠hsψiN2Uhrhs=∑r≠sψiN1Urs=Constant.
Therefore,G2 is regular.Theorem 4.
LetG1 and G2 be two m-polar IVIFGs. If G1 is WI to G2 and G1 is strong, then, G2 is also strong.Proof.
AsG1 is WI with G2, then, there exists a WI h:G1⟶G2 which is BM for i=1,2,…,m and satisfies the following:(20)φiM1Lr=φiM2Lhr,φiM1Ur=φiM2Uhr,ψiM1Lr=ψiM2Lhr,ψiM1Ur=ψiM2Uhr,∀r∈V1,φiN1Lrs≤φiN2Lhrhs,φiN1Urs≤φiN2Uhrhs,ψiN1Lr≥ψiN2Lhrhs,ψiN1Ur≥ψiN2Uhrhs,∀rs∈E1.
SinceG1 is strong, φiN1Lrs=minφiM1Lr,φiM1Ls. Now, we get(21)φiN2Lhrhs≥φiN1Lrs=minφiM1Lr,φiM1Ls,=minφiM2Lhr,φiM2Lhs.
According to the definition,φiN2Lhrhs≤minφiM2Lhr,φiM2Lhs. Therefore, φiN2Lhrhs=minφiM2Lhr,φiM2Lhs. Similarly,(22)φiN2Uhrhs=minφiM2Uhr,φiM2Uhs,ψiN2Lhrhs=maxψiA2Lhr,ψiM2Lhs,ψiN2Uhrhs=maxψiA2Uhr,ψiM2Uhs.
So,G2 is strong.Theorem 5.
If anm-polar IVIFG G1 is CWI with an SR m-polar IVIFG G2, then, G1 is SR m-polar IVIFG.Proof.
Since thatG1 is CWI to G2, there is a CWI h:G1⟶G2 that is BM for i=1,2,…,m and satisfies the following:(23)φiM1Lr≤φiM2Lhr,φiM1Urx≤φiM2Uhr,ψiM1Lr≥ψiM2Lhr,ψiM1Ur≥ψiM2Uhr,∀r∈V1,φiN1Lrs=φiN2Lhrhs,φiN1Urs=φiN2Uhrhs,ψiN1Lr=ψiN2Lhrhs,ψiN1Ur=ψiN2Uhrhs,∀rs∈E1.
Now, we get(24)φiN1Lrs=φiN2Lhr,hs=minφiM2Lhr,φiM2Lhs≥φiM1Lr,φiM1Ls.
But, by the definition, we have(25)φiN1Lrs≤φiM1Lr,φiM1Ls.
So,φiN1Lrs=minφiM1Lr,φiM1Ls. Therefore, G1 is strong. Also for r∈V1, we have the following:(26)∑r≠sφiN1Lrs=∑hr≠hsφiN2Lhrhs=Constant.
Also, it is constant in other cases becauseG2 is regular. Hence, G1 is regular.Theorem 6.
SupposeG1 and G2 are two isomorphic m-polar IVIFGs. Then, G1 is SR if and only if G2 is SR.Proof.
SinceG1 is isomorphic with G2, so there exists an isomorphism h:G1⟶G2 that is BM for i=1,2,…,m and satisfies(27)φiM1Lr=φiM2Lhr,φiM1Ur=φiM2Uhr,ψiM1Lr=ψiM2Lhr,ψiM1Ur=ψiM2Uhr,∀r∈V1,φiN1Lrs=φiN2Lhrhs,φiN1Urs=φiN2Uhrhs,ψiN1Lrs=ψiN2Lhrhs,ψiN1Urs=ψiN2Uhrhs,∀rs∈E1.
Therefore,G1 is strong if and only if(28)φiN1Lrs=minφiM1Lr,φiM1Ls,φiN1Urs=minφiM1Ur,φiM1Us,ψiN1Lrs=maxψiM1Lr,ψiM1Ls,ψiN1Urs=maxψiM1Ur,ψiM1Us,if and only if(29)φiN2Lhrhs=minφiM2Lhr,φiM2Lhs,φiN2Uhrhs=minφiM2Uhr,φiM2Uhs,ψiN2Lhrhs=maxψiM2Lhr,ψiM2Lhs,ψiN2Uhrhs=maxψiM2Uhr,ψiM2Uhs,if and only if G2 is strong.
G1 is regular if and only if, for r∈V1,(30)∑r≠sφiN1Lrs=Constant,∑r≠sφiN1Urs=Constant,∑r≠sψiN1Lrs=Constant,∑r≠sψiN1Urs=Constant,if and only if(31)∑hr≠hsφiN2Lhrhs=Constant,∑hr≠hsφiN2Uhrhs=Constant,∑hr≠hsψiN2Lhrhs=Constant,∑hr≠hsψiN2Uhrhs=Constant,for all hr∈V2, if and only if G2 is regular.Definition 8.
SupposeG=V,M,N is a connected m-polar IVIFG. Then, G is said to be a HI m-polar IVIFG whenever each node of G is neighbor to nodes with different degrees.Theorem 7.
For any two isomorphic HIm-polar IVIFGs, the order and the size are the same.Proof.
Supposeh:G1⟶G2 is an isomorphism between two HI m-polar IVIFGs G1 and G2 with the underlying sets V1 and V2, respectively; therefore, for i=1,2,…,m,(32)φiM1Lr1=φiM2Lhr1,φiM1Ur1=φiM2Uhr1,ψiM1Lr1=ψiM2Lhr1,ψiM1Ur1=ψiM2Uhr1,∀r1∈V1,φiN1Lr1s1=φiN2Lhr1hs1,φiN1Ur1s1=φiN2Uhr1hs1,ψiN1Lr1s1=ψiN2Lhr1hsy1,ψiN1Ur1s1=ψiN2Uhr1hs1,∀r1s1∈E1.
So, we get(33)OG1=∑r1∈V1φiM1Lr1,∑r1∈V1φiM1Ur1i=1m;∑r1∈V1ψiM1Lr1,∑r1∈V1ψiM1Ur1i=1m,OG1=∑r1∈V1φiM2Lhr1,∑r1∈V1φiM2Uhr1i=1m;∑r1∈V1ψiM2Lhr1,∑r1∈V1ψiM2Uhr1i=1m=∑r2∈V2φiM2Lr2,∑r2∈V2φiM2Ur2i=1m;∑r2∈V2ψiM2Lr2,∑r2∈V2ψiM2Ur2i=1m=OG2,SG1=∑r1s1∈E1φiN1Lr1s1,∑r1s1∈E1φiN1Ur1s1i=1m;∑r1s1∈E1ψiN1Lr1s1,∑r1s1∈E1ψiN1Ur1s1i=1m=∑r1s1∈E1φiN2Lhr1hs1,∑r1s1∈E1φiN2Uhr1hs1i=1m;∑r1s1∈E1ψiN2Lhr1hs1,∑r1s1∈E1ψiN2Uhr1hs1i=1m=∑r2s2∈E2φiN2Lr2s2,∑r2s2∈E2φiN2Ur2s2i=1m;∑r2s2∈E2ψiN2Lr2s2,∑r2s2∈E2ψiN2Ur2s2i=1m=SG2.Theorem 8.
IfG1 and G2 are isomorphic HI m-polar IVIFGs, then the degrees of the corresponding nodes r and hr are maintained.Proof.
Leth:G1⟶G2 be an isomorphism from the HI m-polar IVIFGs G1 and G2 with the sets V1 and V2, respectively. Therefore, for i=1,2,…,m,(34)φiN1Lrs=φiN2Lhrhs,φiN1Urs=φiN2Uhrhs,ψiN1Lrs=ψiN2Lhrhs,ψiN1Urs=ψiN2Uhrhs,rs∈V1.
Therefore,(35)dG1r=∑rs∈V1r≠sφiN1Lrs,∑rs∈V1r≠sφiN1Ursi=1m;∑rs∈V1r≠sψiN1Lrs,∑rs∈V1r≠sψiN1Ursi=1m=∑rs∈V1r≠sφiN2Lhrhs,∑rs∈V1r≠sφiN2Uhrhsi=1m∑rs∈V1r≠sψiN2Lhrhs,∑rs∈V1r≠sψiN2Uhrhsi=1m=dG2hr.
It follows that the degrees of the corresponding nodes ofG1 and G2 are maintained.


## 4. Application

Every year, thousands of entrepreneurs start new businesses, hoping to make significant profits in the coming years; unfortunately, according to statistics, half of these businesses are closed and destroyed in less than 5 years due to a lack of leadership skills in the organization. There is a difference between leadership and management of an organization. The manager focuses on the operational process of the organization and the production of the product, determines the strategy, and identifies the solution to the problems. In contrast, the main goal of any leader should be to align the employees of the organization with the vision of the organization.Power and politics are two important factors in leadership. Power means a person’s ability to influence the behavior of others to achieve a goal. Politics is the study of power and influence. When people form a group with a specific purpose, some of them definitely gain power and access to sources of power. The members of the group seek to gain access to these resources and strive to obtain them. As soon as individuals use their power in action and find a way to exercise it, they actually engage in political behavior. They learn how to use their sources of power to achieve their goals. Leaders cannot influence in the upward direction and in the horizontal plane. But this is not true of power.Consider an entrepreneur appointing a certain number of managers to oversee the specific activities that need to be done and the goals that need to be achieved. Each of them oversees a team that specializes in tasks and focuses on achieving those goals. These managers include the CEO and managers in the areas of finance, human resources, purchasing, production, sales, and services. This entrepreneur intends to choose leadership from these managers. In field surveys, the entrepreneur found the following:(1)
Although financial managers and human resources are under the CEO, there is a lot of competition between them for the CEO(2)
Purchasing, production, and sales managers operate under the supervision of the financial manager(3)
The human resources manager oversees the purchasing, sales, and service managers(4)
The activities of the service manager are under the orders of the production manager(5)
The purchasing manager is responsible for supplying raw materials for the production and service units(6)
Decisions of production and service managers are taken in coordination with the sales managerAnm-polar IVIFG can help an entrepreneur determine leadership. For this purpose, based on the surveys conducted for each of the managers, according to their abilities in terms of power and politics in their working group, the 2-polar IVIF values are considered according to Table 4. The numbers in Table 4 show that, for example, the sales manager has 50 to 60 percent power in his work and 25 to 30 percent does not have this power. Also, 70 to 75 percent has a policy, and 15 to 20 percent does not have the necessary policy.Table 4
2-polar IVIF values of managers.
Managers2-polar IVIF values of managersCEO0.55,0.65,0.60,0.70,0.10,0.20,0.15,0.20Finance0.65,0.75,0.70,0.80,0.05,0.10,0.10,0.15Human resources0.65,0.75,0.70,0.80;0.10,0.15,0.10,0.20Purchasing0.55,0.60,0.40,0.50;0.10,0.20,0.20,0.25Production0.55,0.65,0.60,0.75;0.15,0.25,0.10,0.20Sales0.60,0.65,0.70,0.75;0.15,0.20,0.05,0.20Services0.50,0.60,0.60,0.65;0.25,0.30,0.15,0.20If we consider managers as vertices, then we have a 2-polar IVIFG shown in Figure2. Since the relationship between managers is considered to be the strongest, so all edges are strong. The values are shown in Table 5.Figure 2
2-polar IVIFG related to managers.Table 5
2-polar IVIF of relationships between managers.
Managers2-polar IVIF valuesCEO—finance0.55,0.65,0.60,0.70;0.05,0.10,0.10,0.15CEO—human resources0.55,0.65,0.60,0.70;0.10,0.15,0.10,0.20Finance—purchasing0.55,0.60,0.40,0.50;0.05,0.10,0.10,0.15Finance—production0.55,0.65,0.60,0.75;0.05,0.10,0.10,0.15Finance—sales0.60,0.65,0.70,0.75;0.05,0.10,0.05,0.15Human resources—purchasing0.55,0.65,0.60,0.75;0.10,0.15,0.10,0.20Human resources—sales0.60,0.65,0.70,0.75;0.10,0.15,0.05,0.20Human resources—services0.50,0.65,0.60,0.65;0.10,0.15,0.10,0.20Purchasing—production0.55,0.60,0.40,0.50;0.10,0.20,0.10,0.20Purchasing—services0.50,0.60,0.40,0.50;0.10,0.20,0.15,0.20Production—services0.50,0.60,0.60,0.65;0.15,0.25,0.10,0.20Sales—production0.55,0.60,0.60,0.75;0.15,0.20,0.05,0.20Sales—services0.50,0.60,0.60,0.65;0.15,0.20,0.05,0.20As can be seen in Figure2, financial and human resource managers have the most power and policy over other managers and can be leadership options. But since the financial manager has the least degree of hesitation compared to the human resources manager, the best option for leadership is the financial manager.


## 5. Conclusion

m-polar IVIFG is one of the types of FGs that has better flexibility in dealing with problems that cannot be expressed with FGs and IFGs. It has a wide range of applications in the field of psychological sciences as well as the identification of individuals based on oncological behaviors. In this study, we presented the notion of m-polar h-morphism on m-polar IVIFGs. Some helpful theorems on WI and CWI are obtained. Furthermore, we investigated m-polar h-morphism on SR and HI m-polar-IVIFGs. In the last section, we have tried to express the application of the m-polar IVIFG in the management problem. The results show that the isomorphism in m-polar IVIFG is an equivalence relation so that some structural properties are equivalent in two m-polar IVIFG. Some parameters, such as order, size, and strong edges, remain the same in two isomorphic m-polar IVIFGs. Some of these features are the same in both WI and CWI. As future work topics, we try to study some features of other types of fuzzy graphs such as m-polar fuzzy graph structure and m-polar cubic fuzzy graph.



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