# Vertex Degrees and Isomorphic Properties in Complement of anm-Polar Fuzzy Graph

Authors: Ch. Ramprasad; P. L. N. Varma; S. Satyanarayana; N. Srinivasarao
Journal: Advances in Fuzzy Systems
 (2017)

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## Abstract

Computational intelligence and computer science rely on graph theory to solve combinatorial problems. Normal product and tensor product of anm-polar fuzzy graph have been introduced in this article. Degrees of vertices in various product graphs, like Cartesian product, composition, tensor product, and normal product, have been computed. Complement and μ-complement of an m-polar fuzzy graph are defined and some properties are studied. An application of an m-polar fuzzy graph is also presented in this article.


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## Body

## 1. Introduction

Akram [1] introduced the notion of bipolar fuzzy graphs describing various methods of their construction as well as investigating some of their important properties. Bhutani [2] discussed automorphism of fuzzy graphs. Chen et al. [3] generalized the concept of bipolar fuzzy set to obtain the notion of an m-polar fuzzy set. The notion of an m-polar fuzzy set is more advanced than fuzzy set and eliminates ambiguity more absolutely. Ghorai and Pal [4] studied some operations and properties of an m-polar fuzzy graph. Mordeson and Peng [5] defined join, union, Cartesian product, and composition of two fuzzy graphs. Rashmanlou et al. [6] discussed some properties of bipolar fuzzy graphs and their results. Sunitha and Vijaya Kumar [7] defined the complement of a fuzzy graph in another way which gives a better understanding about that concept. We have studied product m-polar fuzzy graph, product m-polar fuzzy intersection graph, and product m-polar fuzzy line graph [8].In this article, we study the Cartesian product and composition of twom-polar fuzzy graphs and compute the degrees of the vertices in these graphs. The notions of normal product and tensor product of m-polar fuzzy graphs are introduced and some properties are studied. Also in the present work, we introduce the concept of complement, μ-complement of an m-polar fuzzy graph, and some properties are discussed.These concepts strengthen the decision-making in critical situations. Some applications to decision-making are also studied.In this article, unless and otherwise specified, all graphs considered arem-polar fuzzy graphs.


## 2. Preliminaries

Definition 1.
Them-polar fuzzy graph of a graph G∗=V,E is a pair G=W,F, where W:V→0,1m is an m-polar fuzzy set in V and F:V×~V→0,1m is an m-polar fuzzy set in V×~V such that Fqr≤minWq,Wr for all qr∈V×~V and Fqr=0 for all edges qr∈V~2-E0=0,0,…,0 is the smallest element in 0,1m. W is called the m-polar fuzzy vertex set of G and F is called m-polar fuzzy edge set of G.Sometimes we denote the graphG=W,F by G=(V,W,F) also.Definition 2.
Given two graphsG1, G2 their Cartesian product, G1×G2=V1×V2,W1×W2,F1×F2, is defined as follows: for i=1,2,…,m, we have(i)
pi∘W1×W2q1,q2=minpi∘W1q1,pi∘W2q2 for all q1,q2∈V1×V2,(ii)
pi∘F1×F2(q,q2q,r2)=minpi∘W1q,pi∘F2q2r2 for all q∈V1,q2r2∈E2,(iii)
pi∘F1×F2(q1,mr1,m)=minpi∘F1q1r1,pi∘W2m for all m∈V2,q1r1∈E1,(iv)
pi∘F1×F2(q1,q2r1,r2)=0 for all q1,q2r1,r2∈V1×~V22-E.Definition 3.
Given two graphsG1, G2 their composition, G1G2=V1×V2,W1W2,F1F2, is defined as follows: for i=1,2,…,m, we have(i)
pi∘W1W2q1,q2=minpi∘W1q1,pi∘W2q2 for all q1,q2∈V1×V2,(ii)
pi∘F1F2(q,q2q,r2)=minpi∘W1q,pi∘F2q2r2 for all q∈V1,q2r2∈E2,(iii)
pi∘F1F2(q1,mr1,m)=minpi∘F1q1r1,pi∘W2m for all m∈V2,q1r1∈E1,(iv)
pi∘F1F2(q1,q2r1,r2)=minpi∘W2q2,pi∘W2r2,pi∘F1q1r1 for all q1,q2r1,r2∈E0-E, where E0=E∪q1,q2r1,r2∣q1r1∈E1,q2≠r2.Definition 4.
The normal product ofG1 and G2 is defined as the m-polar fuzzy graph G1•G2=V1×V2,W1•W2,F1•F2 on G∗=V,E, where E=q,q2q,r2∣q∈V1,q2r2∈E2∪q1,mr1,m∣q1r1∈E1,m∈V2∪q1,q2r1,r2∣q1r1∈E1,q2r2∈E2 such that for i=1,2,…,m, we have(i)
pi∘W1•W2q1,q2=minpi∘W1q1,pi∘W2q2 for all q1,q2∈V1×V2,(ii)
pi∘F1•F2(q,q2q,r2)=minpi∘W1q,pi∘F2q2r2 for all q∈V1,q2r2∈E2,(iii)
pi∘F1•F2(q1,mr1,m)=minpi∘F1q1r1,pi∘W2m for all m∈V2,q1r1∈E1,(iv)
pi∘F1•F2(q1,q2r1,r2)=minpi∘F1q1r1,pi∘F2q2r2 for all q1q2∈E1 and r1r2∈E2.Definition 5.
The tensor product ofG1 and G2 is defined as G1⊗G2=V1×V2,W1⊗W2,F1⊗F2 on G∗=V,E, where E=q1,q2r1,r2∣q1r1∈E1,q2r2∈E2 such that for i=1,2,…,m, we have(i)
pi∘W1⊗W2q1,q2=minpi∘W1q1,pi∘W2q2 for all q1,q2∈V1×V2,(ii)
pi∘F1⊗F2(q1,q2r1,r2)=minpi∘F1q1r1,pi∘F2q2r2 for all q1q2∈E1 and r1r2∈E2.Definition 6.
The unionG1∪G2=V1∪V2,W1∪W2,F1∪F2 of the graphs G1=V1,W1,F1 and G2=V2,W2,F2 of G1∗ and G2∗, respectively, is defined as follows: for i=1,2,…,m, we have(i)
(1)pi∘W1∪W2q=pi∘W1qifq∈V1-V2pi∘W2qifq∈V2-V1maxpi∘W1q,pi∘W2qifq∈V1∩V2.(ii)
(2)pi∘F1∪F2qr=pi∘F1qrifqr∈E1-E2pi∘F2qrifqr∈E2-E1maxpi∘F1qr,pi∘F2qrifqr∈E1∩E2.(iii)
(3)pi∘F1∪F2qr=0ifqr∈V1×~V22-E1∪E2.Definition 7.
The join of the graphsG1=V1,W1,F1 and G2=V2,W2,F2 of G1∗ and G2∗, respectively, is defined as G1+G2=V1∪V2,W1+W2,F1+F2 such that for i=1,2,…,m, we have the following:(i)
pi∘W1+W2q=pi∘W1∪W2qifq∈V1∪V2.(ii)
pi∘F1+F2qr=pi∘F1∪F2qrifqr∈E1∪E2.(iii)
pi∘F1+F2qr=minpi∘W1q,pi∘W2rifqr∈E′, where E′ denotes the set of all edges joining the vertices of V1 and V2.(iv)
pi∘F1+F2qr=0 if qr∈V1×~V22-E1∪E2∪E′.


## 3. Degree of Vertices inm-Polar Fuzzy Graph

Definition 8.
LetG=(V,W,F) be an m-polar fuzzy graph. Then the degree of a vertex q in G is defined as dGq=∑qr∈Eq≠rp1∘Fqr,∑qr∈Eq≠rp2∘Fqr,∑qr∈Eq≠rp3∘Fqr,…,∑qr∈Eq≠rpm∘Fqr.Further, a highly irregularm-polar fuzzy graph is defined as an m-polar fuzzy graph G=(V,W,F) in which every vertex of G=(V,W,F) is adjacent to vertices with distinct degrees.Example 9.
Consider the graphG=(V,W,F) of G∗=V,E, where V=K,L,M,N, E=KL,LM,MN,NK, W  =  {0.2,0.5,0.6/K,0.3,0.5,0.7/L,0.3,0.6,0.7/M, 0.4,0.7,0.8/N}, and F  =  {0.2,0.4,0.5/KL, 0.3,0.4,0.6/LM,0.3,0.5,0.6/MN,0.2,0.3,0.6/NK} as in Figure 1.Figure 1
Highly irregular 3-polar fuzzy graphG.In this graph, we havedGK=0.2,0.4,0.5+0.2,0.3,0.6=0.4,0.7,1.1, dGL=0.2,0.4,0.5+0.3,0.4,0.6=0.5,0.8,1.1, dGM=0.3,0.4,0.6+0.3,0.5,0.6=0.6,0.9,1.2, dGN=0.3,0.5,0.6+0.2,0.3,0.6=0.5,0.8,1.2.


## 4. Cartesian Product and Vertex Degree

From the definition of Cartesian product graphs, for every vertexq1,q2∈V1×V2, we have(4)dG1×G2q1,q2=∑q1,q2r1,r2∈Epi∘F1×F2q1,q2r1,r2=∑q1=r1,q2r2∈E2minpi∘W1q1,pi∘F2q2r2+∑q2=r2,q1r1∈E1minpi∘W2q2,pi∘F1q1r1fori=1,2,…,m.Theorem 10.
LetG1=V1,W1,F1 and G2=V2,W2,F2 be two graphs. If pi∘W1≥pi∘F2 and pi∘W2≥pi∘F1 then dG1×G2q1,q2=dG1q1+dG2q2 for i=1,2,…,m.Proof.
By definition, we get(5)dG1×G2q1,q2=∑q1,q2r1,r2pi∘F1×F2q1,q2r1,r2=∑q1=r1,q2r2∈E2minpi∘W1q1,pi∘F2q2r2+∑q2=r2,q1r1∈E1minpi∘W2q2,pi∘F1q1r1=∑q2r2∈E2pi∘F2q2r2+∑q1r1∈E1pi∘F1q1r1Sincepi∘W1≥pi∘F2,pi∘W2≥pi∘F1=dG1q1+dG2q2.Example 11.
Consider two graphsG1, G2 and their Cartesian product, G1×G2, as shown in Figure 2.
Sincepi∘W1≥pi∘F2 and pi∘W2≥pi∘F1, by Theorem 10, we have (6)dG1×G2s1,s2=dG1s1+dG2s2=0.2,0.3,0.1+0.2,0.4,0.1=0.4,0.7,0.2,dG1×G2s1,t2=dG1s1+dG2t2=0.2,0.3,0.1+0.2,0.4,0.1=0.4,0.7,0.2dG1×G2t1,s2=dG1t1+dG2s2=0.2,0.3,0.1+0.2,0.4,0.1=0.4,0.7,0.2,dG1×G2t1,t2=dG1t1+dG2t2=0.2,0.3,0.1+0.2,0.4,0.1=0.4,0.7,0.2.Figure 2
Cartesian product of two 3-polar fuzzy graphsG1 and G2.


## 5. Composition Graph and Vertex Degree

From the definition of composition graphs, for every vertexq1,q2∈V1×V2, we have (7)dG1G2q1,q2=∑q1,q2r1,r2∈Epi∘F1F2q1,q2r1,r2=∑q1=r1,q2r2∈E2minpi∘W1q1,pi∘F2q2r2+∑q2=r2,q1r1∈E1minpi∘W2q2,pi∘F1q1r1+∑q2≠r2,q1r1∈E1minpi∘W2r2,pi∘W2q2,pi∘F1q1r1fori=1,2,…,m.Theorem 12.
LetG1=V1,W1,F1 and G2=V2,W2,F2 be two graphs. If pi∘W1≥pi∘F2 and pi∘W2≥pi∘F1 then dG1G2q1,q2=V2dG1q1+dG2q2 for all q1,q2∈V1×V2 for i=1,2,…,m.Proof.
By definition, we get(8)dG1G2q1,q2=∑q1,q2r1,r2pi∘F1F2q1,q2r1,r2=∑q1=r1,q2r2∈E2minpi∘W1q1,pi∘F2q2r2+∑q2=r2,q1r1∈E1minpi∘W2q2,pi∘F1q1r1+∑q2≠r2,q1r1∈E1minpi∘W2r2,pi∘W2q2,pi∘F1q1r1=∑q2r2∈E2pi∘F2q2r2+∑q2=r2,q1r1∈E1pi∘F1q1r1+∑q2≠r2,q1r1∈E1pi∘F1q1r1=dG2q2+V2∑q1r1∈E1pi∘F1q1r1=V2dG1q1+dG2q2.Example 13.
Consider two graphsG1, G2 and their composition, G1G2, as shown in Figure 3.Figure 3
Composition of two 3-polar fuzzy graphsG1 and G2.Sincepi∘W1≥pi∘F2 and pi∘W2≥pi∘F1, by Theorem 12, we have (9)dG1G2s1,s2=V2dG1s1+dG2s2=20.8,0.3,0.2+0.7,0.4,0.2=2.3,1.0,0.6,dG1G2s1,t2=V2dG1s1+dG2t2=20.8,0.3,0.2+0.7,0.4,0.2=2.3,1.0,0.6,dG1G2t1,s2=V2dG1t1+dG2s2=20.8,0.3,0.2+0.7,0.4,0.2=2.3,1.0,0.6,dG1G2t1,t2=V2dG1t1+dG2t2=20.8,0.3,0.2+0.7,0.4,0.2=2.3,1.0,0.6.


## 6. Normal Product and Vertex Degree

From the definition of normal product graphs, for every vertexq1,q2∈V1×V2, we have (10)dG1•G2q1,q2=∑q1,q2r1,r2∈Epi∘F1•F2q1,r1q2,r2=∑q1=r1,q2r2∈E2minpi∘W1q1,pi∘F2q2r2+∑q2=r2,q1r1∈E1minpi∘W2q2,pi∘F1q1r1+∑q2r2∈E2,q1r1∈E1minpi∘F1q1r1,pi∘F2q2r2fori=1,2,…,m.Theorem 14.
LetG1=V1,W1,F1 and G2=V2,W2,F2 be two graphs. If pi∘W1≥pi∘F2,pi∘W2≥pi∘F1 and pi∘F1≤pi∘F2, then dG1•G2q1,q2=V2dG1q1+dG2q2 for i=1,2,…,m.Proof.
By definition, we get(11)dG1•G2q1,q2=∑q1,q2r1,r2∈Epi∘F1×F2q1,r1q2,r2=∑q1=r1,q2r2∈E2minpi∘W1q1,pi∘F2q2r2+∑q2=r2,q1r1∈E1minpi∘W2q2,pi∘F1q1r1+∑q2r2∈E2,q1r1∈E1minpi∘F1q1r1,pi∘F2q2r2=∑q2r2∈E2pi∘F2q2r2+∑q2=r2,q1r1∈E1pi∘F1q1r1+∑q1r1∈E1pi∘F1q1r1Sincepi∘W1≥pi∘F2,pi∘W2≥pi∘F1,pi∘F1≤pi∘F2=V2dG1q1+dG2q2.Example 15.
Consider two graphsG1, G2 and their normal product, G1•G2, is shown in Figure 4.Figure 4
Normal product of two 3-polar fuzzy graphsG1 and G2.Sincepi∘W1≥pi∘F2pi∘W2≥pi∘F1 and pi∘F1≤pi∘F2, by Theorem 14, we have (12)dG1•G2s1,s2=V2dG1s1+dG2s2=20.3,0.05,0.4+0.4,0.1,0.5=1.0,0.2,1.3,dG1•G2s1,t2=V2dG1s1+dG2t2=20.3,0.05,0.4+0.4,0.1,0.5=1.0,0.2,1.3dG1•G2t1,s2=V2dG1t1+dG2s2=20.3,0.05,0.4+0.4,0.1,0.5=1.0,0.2,1.3,dG1•G2t1,t2=V2dG1t1+dG2t2=20.3,0.05,0.4+0.4,0.1,0.5=1.0,0.2,1.3.


## 7. Tensor Product and Vertex Degree

From the definition of tensor product, for every vertexq1,q2∈V1×V2 for i=1,2,…,m(13)dG1⊗G2q1,q2=∑pi∘F1⊗F2q1,q2r1,r2=∑q1r1∈E1minpi∘F1q1r1,pi∘F2q2r2.Theorem 16.
LetG1=V1,W1,F1 and G2=V2,W2,F2 be two m-polar fuzzy graphs. If pi∘F2≥pi∘F1, then dG1⊗G2q1,q2=dG1q1 and if pi∘F1≥pi∘F2, then dG1⊗G2q1,q2=dG2q2 for i=1,2,…,m.Proof.
Letpi∘F2≥pi∘F1. Then we get (14)dG1⊗G2q1,q2=∑q1r1∈E1minpi∘F1q1r1,pi∘F2q2r2=∑pi∘F1q1r1=dG1q1.
So,dG1⊗G2q1,q2=dG1q1.
Letpi∘F1≥pi∘F2. Then we get (15)dG1⊗G2q1,q2=∑q1r1∈E1minpi∘F1q1r1,pi∘F2q2r2=∑pi∘F2q2r2=dG2q2.
So,dG1⊗G2q1,q2=dG2q2.Example 17.
Consider two graphsG1, G2 and their tensor product, G1⊗G2, is shown in Figure 5.Figure 5
Tensor product of two 3-polar fuzzy graphsG1 and G2.Sincepi∘F2≥pi∘F1, by Theorem 16, we have (16)dG1⊗G2s1,s2=dG1s1=0.7,0.3,0.5,dG1⊗G2s1,t2=dG1s1=0.7,0.3,0.5,dG1⊗G2t1,s2=dG1t1=0.7,0.3,0.5,dG1⊗G2t1,t2=dG1t1=0.7,0.3,0.5.


## 8.μ-Complement of an m-Polar Fuzzy Graph

Now for anm-polar fuzzy graph G, we introduce the notion of its μ-complement.Definition 18.
The complement of anm-polar fuzzy graph G=(V,W,F) is also an m-polar fuzzy graph G-=V,W-,F-, where W-=W and F- is defined as follows: (17)pi∘F-qr=∧pi∘Wq,pi∘Wr-pi∘Fqr∀q,r∈Vfori=1,2,…,m.Definition 19.
LetG=V,W,F be an m-polar fuzzy graph. If G is isomorphic to G- we say G is self-complementary. Similarly, if G is weak isomorphic to G- then we say G is self-weak complementary.Definition 20.
Theμ-complement of G is defined as Gμ=V,Wμ,Fμ, where Wμ=W and Fμ is given by (18)pi∘Fμqr=minpi∘Wq,pi∘Wr-pi∘Fqrifpi∘Fqr>00ifpi∘Fqr=0for i=1,2,…,m.Theorem 21.
LetG=(V,W,F) be a self-weak complementary and highly irregular graph. Then, we have ∑q≠rpi∘Fqr≤0.5∑q≠r∧pi∘Wq,pi∘Wr.Proof.
LetG=(V,W,F) be a self-weak complementary, highly irregular graph of G∗=V,E. Then, there exists a weak isomorphism g from G to G- such that for all q,r∈V, we get pi∘Wq=pi∘W-gq=pi∘Wgq,pi∘Fqr≤pi∘F-gqgr. From the definition of complement, from the above inequality, for all q,r∈V, we get that pi∘Fqr≤pi∘F-gqgr=∧pi∘Wq,pi∘Wr-pi∘Fgqgr and pi∘Fqr+pi∘Fgqgr≤∧pi∘Wq,pi∘Wr. Hence, ∑q≠rpi∘Fqr+∑q≠rpi∘Fgqgr≤∑q≠r∧pi∘Wq,pi∘Wr. Thus, we get(19)2∑q≠rpi∘Fqr≤∑q≠r∧pi∘Wq,pi∘Wror∑q≠rpi∘Fqr≤0.5∑q≠r∧pi∘Wq,pi∘Wr.Theorem 22.
LetG=(V,W,F) be a graph which is self-complementary and highly irregular. Then, we have ∑q≠rpi∘Fqr=0.5∑q≠r∧pi∘Wq,pi∘Wr, for all qr∈V~2,i=1,2,…,m.Proof.
LetG=(V,W,F) be a self-complementary, highly irregular graph of G∗=V,E. Then, there exists an isomorphism g from G to G- such that pi∘Wq=pi∘W-q for all q∈V and pi∘Fqr=pi∘F-gqgr for all qr∈V~2.
Letqr∈V~2. Then, for i=1,2,…,m, we have(20)pi∘F-gqgr=∧pi∘Wgq,pi∘Wgr-pi∘Fgqgr;that is, pi∘Fqr=∧pi∘Wgq,pi∘Wgr-pi∘Fgqgr. Therefore, (21)∑q≠rpi∘Fqr+∑q≠rpi∘Fgqgr=∑q≠r∧pi∘Wgq,pi∘Wgr=∑q≠r∧pi∘Wq,pi∘Wr2∑q≠rpi∘Fqr=∑q≠r∧pi∘Wq,pi∘Wri.e. ∑q≠rpi∘Fqr=0.5∑q≠r∧pi∘Wq,pi∘Wr.Theorem 23.
LetG1=V1,W1,F1 and G2=V2,W2,F2 be two graphs. If G1 and G2 are isomorphic, then their μ-complements, G1μ and G2μ, are also isomorphic.Proof.
LetG1≅G2 and let g be an isomorphism from G1 to G2. Then for i=1,2,…,m(22)pi∘W1q=pi∘W2gq,∀q∈V1,pi∘F1qr=pi∘F2gqgr,∀qr∈E1.If pi∘F1qr>0, then pi∘F2gqgr>0, and (23)pi∘F1μqr=minpi∘W1q,pi∘W1r-pi∘F1qr=minpi∘W2gq,pi∘W2gr-pi∘F2gqgr=pi∘F2μgqgr.
Ifpi∘F1qr=0, then pi∘F2gqgr=0, and pi∘F1μqr=0=pi∘F2μgqgr.
Thus,pi∘F1μqr=pi∘F2μgqgr for all qr∈E1. Therefore, g from G1μ to G2μ is an isomorphism; that is, G1μ≅G2μ.Theorem 24.
LetG1=V1,W1,F1 and G2=V2,W2,F2 be two graphs such that V1∩V2=ϕ. Then G1+G2μ≅G1μ∪G2μ.Proof.
LetI:V1∪V2→V1∪V2 be the identity map. To show that G1+G2μ≅G1μ∪G2μ, it is enough to prove that pi∘W1+pi∘W2μq=pi∘W1μq∪pi∘W2μq and pi∘F1+pi∘F2μqr=pi∘F1μqr∪pi∘F2μqr for all q,r∈V and for i=1,2,…,m. Then for any q,r∈V, we have(24)pi∘W1+pi∘W2μq=pi∘W1qifq∈V1pi∘W2qifq∈V2=pi∘W1μqifq∈V1pi∘W2μqifq∈V2=pi∘W1μ∪pi∘W2μq.Moreover,(25)pi∘F1+pi∘F2μqr=minpi∘W1+pi∘W2q,pi∘W1+pi∘W2r-pi∘F1+pi∘F2qr=pi∘W1q∧pi∘W1r-pi∘F1qrifqr∈E1pi∘W2q∧pi∘W2r-pi∘F2qrifqr∈E2pi∘W1q∧pi∘W2r-pi∘W1q∧pi∘W2rifqr∈E′=pi∘F1μqrifqr∈E1pi∘F2μqrifqr∈E20ifqr∈E′=pi∘F1μ∪pi∘F2μqr.Hence, the result is proved.Theorem 25.
LetG1=V1,W1,F1 and G2=V2,W2,F2 be two graphs such that V1∩V2=ϕ. Then G1∪G2μ≅G1μ∪G2μ.Proof.
We shall prove that the identity mapI:V1∪V2→V1∪V2 is the required isomorphism. Then for any i,1≤i≤m we have the following: Ifq∈V1, pi∘W1∪pi∘W2μq=pi∘W1∪pi∘W2q=pi∘W1q=pi∘W1μq=pi∘W1μ∪pi∘W2μq.Ifq∈V2, then pi∘W1∪pi∘W2μq=pi∘W1∪pi∘W2q=pi∘W2q=pi∘W2μq=pi∘W2μ∪pi∘W2μq.Ifqr∈E1, then pi∘F1∪pi∘F2μqr=pi∘W1q∧pi∘W1r-pi∘F1qr=pi∘F1μqr=pi∘F1μ∪pi∘F2μqr.Ifqr∈E2, then pi∘F1∪pi∘F2μqr=pi∘W2q∧pi∘W2r-pi∘F2qr=pi∘F2μqr=pi∘F1μ∪pi∘F2μqr.
Hence, the result is proved.


## 9.m-Polar Fuzzy Graphs and Their Applications

One of the important applications ofm-polar fuzzy graphs is guidance in decision-making. For example, let us consider a scenario where a college wants to select a principal. Let W = Kasim, Leo, Modi, Nandhu be the set of doctorates shortlisted and let F = Krishna, Srinu, Mano, Ramesh be the set of college management members. The management has to make a selection taking into account the set of qualities Q = management skills, attitude, patents, and research. Each member from set F can assign a value between 0 and 1 to each element listed in the set Q for each candidate mentioned in set W, such as W(K)=0.3,0.4,0.6,0.2, W(L)=0.9,0.5,0.7,0.3, W(M)=0.4,0.7,0.3,0.5, and W(N)=0.3,0.6,0.7,0.6. Then we have a 4-polar fuzzy graph, in which each member gives his opinion to w that belongs to W on the basis of his qualities. So W(K), W(L), W(M), and W(N) denote the degree of management skills, attitude, patents, and research of each person given by the members of college management and edges denote the identical qualities of two persons (see Figure 6).Figure 6
4-polar fuzzy graph in decision-making.The membership degree of the edgeKL indicates that Kasim is 30% suitable, Leo is 40% suitable, Modi is 60% suitable, and Nandhu is 20% suitable. Thus, the edge KL specifies that Modi is the suitable person for the post of principal. In the same way, LM specifies Leo, MN specifies Leo, and NK also specifies Leo. Hence, according to all, Leo has a higher value of desirability among the shortlisted candidates and should be appointed as the principal.


## 10. Conclusions

Numerous uses can be harnessed from the theory of a fuzzy graph in the areas of number theory, algebra, topology, operation research, and so on. Not only researchers but also the common man can benefit fromm-polar fuzzy data. It solves the day to day problems that are faced in the society related to probabilistic data. The novel approach discussed here enables decision makers to formulate better- preferred option with the help of unique case pattern than the conventional fuzzy graph approaches. The concepts complement, μ-complement of an m-polar fuzzy graph, Cartesian product, composition, tensor product, and normal product are applied to m-polar fuzzy graphs and many results have been obtained.



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*Source: 3817469-2017-08-22.xml*