# Morphism ofm-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

The main purpose of this paper is to introduce the notion ofm-polar h-morphism on m-polar fuzzy graphs. The action of m-polar h-morphism on m-polar fuzzy graphs is studied. Some elegant theorems on weak and coweak isomorphism are obtained. Also, some properties of highly irregular, edge regular, and totally edge regular m-polar fuzzy graphs are studied.


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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 m-polar fuzzy set. The notion of m-polar fuzzy set is more advanced than fuzzy set and eliminates doubtfulness more absolutely. Ghorai and Pal [4–6] studied some operations and properties of m-polar fuzzy graphs. Rashmanlou et al. [7] discussed some properties of bipolar fuzzy graphs and some of its results are investigated. Ramprasad et al. [8] studied product m-polar fuzzy graph, product m-polar fuzzy intersection graph, and product m-polar fuzzy line graph. Values between 0 and 1 are used to develop a set theory based on fuzziness by Zadeh [9–11].In the present work the authors introduce the concepts ofm-polar h-morphism, edge regular m-polar fuzzy graph, totally edge regular m-polar fuzzy graph, and highly irregular m-polar fuzzy graph in order to strengthen the decision-making in critical situations.


## 2. Preliminaries

Definition 1.
Throughout the paper,0,1m (m copies of the closed interval [0,1]) is considered to be a poset with pointwise order ≤, where m is a natural number, ≤ is given by w≤f⇔foralli=1,2,…,m, piw≤pif, where w,f∈0,1m and pi:0,1m→0,1 is the ith projection mapping. An m-polar fuzzy set (or a 0,1m-set) on X is a mapping W:X→0,1m.Definition 2.
LetW be an m-polar fuzzy set on X. An m-polar fuzzy relation on W is an m-polar fuzzy set F of X×X such that Fuv≤min⁡Wu,Wv for all u,v∈V, that is, for each i=1,2,…,m, for all u,v∈X, pi∘Fuv≤min⁡pi∘Wu,pi∘Wv.Definition 3.
A generalizedm-polar fuzzy graph of a graph G∗=V,E is a pair G=V,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 Fuv≤minWu,Wv for all uv∈V×~V and Fuv=0 for all uv∈V~2-E0=0,0⋯0 is the smallest element in 0,1m. W is called the m-polar fuzzy set of G and F is called m-polar fuzzy edge set of G.Definition 4.
Anm-polar fuzzy graph G=V,W,F of the graph G∗=V,E is said to be strong if pi∘Fuv=min⁡pi∘Wu,pi∘Wv for all uv∈E, i=1,2,…,m.


## 3. A New Theory of Regularity inm-Polar Fuzzy Graphs

Using the existing graph theories a newm-polar fuzzy graph theory is introduced in this section.Definition 5.
LetG=V,W,F be an m-polar fuzzy graph. Then the degree of a vertex u is defined for i=1,2,…,m as (1)dGu=∑uv∈E,u≠vpi∘Fuv=∑uv∈E,u≠vp1∘Fuv,∑uv∈E,u≠vp2∘Fuv,∑uv∈E,u≠vp3∘Fuv,…,∑uv∈E,u≠vpm∘Fuv.Definition 6.
The degree of an edgexy∈E in an m-polar fuzzy graph G=V,W,F is defined as (2)dGxy=dGx+dGy-2p1∘Fxy,p2∘Fxy,p3∘Fxy,…,pm∘Fxy.Definition 7.
The total degree of an edgexy∈E in an m-polar fuzzy graph G=V,W,F is defined as (3)tdGxy=dGx+dGy-p1∘Fxy,p2∘Fxy,p3∘Fxy,…,pm∘Fxy.Definition 8.
The degree of an edgewjwk∈E in a crisp graph G∗ is dG∗wjwk=dG∗wj+dG∗wk-2.Example 9.
Consider anm-polar fuzzy graph G=V,W,F of G∗=V,E, where (4)V=K,L,M,N,E=KL,LM,MN,NK,W=0.2,0.5,0.6K,0.3,0.5,0.7L,0.3,0.6,0.7M,0.4,0.7,0.8N,F=0.2,0.4,0.5KL,0.3,0.4,0.6LM,0.3,0.5,0.6MN,0.2,0.3,0.6NK,as in Figure 1.Figure 1
3-polar fuzzy graphG.Then, we have(5)dGK=0.4,0.7,1.1,dGL=0.5,0.8,1.1,dGM=0.6,0.9,1.2,dGN=0.5,0.8,1.2,dGKL=0.5,0.7,1.2,dGLM=0.5,0.9,1.1,dGMN=0.5,0.7,1.2,dGNK=0.5,0.9,1.1,tdGKL=0.7,1.1,1.7,tdGLM=0.8,1.3,1.7,tdGMN=0.8,1.2,1.8,tdGNK=0.7,1.2,1.7.Definition 10.
If every vertex in anm-polar fuzzy graph G=V,W,F has the same degree l1,l2,…,lm, then G=V,W,F is called regular m-polar fuzzy graph or m-polar fuzzy graph of degree l1,l2,…,lm.Definition 11.
If every edge in anm-polar fuzzy graph G=V,W,F has the same degree l1,l2,…,lm, then G=V,W,F is called an edge regular m-polar fuzzy graph.Definition 12.
If every edge in anm-polar fuzzy graph G=V,W,F has the same total degree l1,l2,…,lm, then G=V,W,F is called totally edge regular m-polar fuzzy graph.Example 13.
Consider anm-polar fuzzy graph G=V,W,F of G∗=V,E, where (6)V=K,L,M,N,E=KL,LM,MN,NK,W=0.2,0.6,0.1K,0.3,0.5,0.1L,0.3,0.5,0.2M,0.4,0.6,0.3N,F=0.2,0.2,0.1KL,0.2,0.2,0.1LM,0.2,0.5,0.1MN,0.2,0.5,0.1NK,as in Figure 2.Figure 2
An edge regularm-polar fuzzy graph G.Then, we havedGKL=dGLM=dGMN=dGNK=0.4,0.7,0.2.Theorem 14.
LetG=V,W,F be an m-polar fuzzy graph on a cycle G∗=V,E. Then (7)∑wj∈VdGwj=∑wjwk∈E,j≠kdGwjwk.Proof.
Suppose thatG=V,W,F is an m-polar fuzzy graph and G∗ is a cycle w1w2w3⋯wnw1.
Now, we get(8)∑j=1ndGwjwj+1=dGw1w2+dGw2w3+⋯+dGwnw1,wherewn+1=w1,=dGw1+dGw2-2p1∘Fw1w2,p2∘Fw1w2,…,pm∘Fw1w2+dGw2+dGw3-2p1∘Fw2w3,p2∘Fw2w3,…,pm∘Fw2w3+⋯+dGwn+dGw1-2p1∘Fwnw1,p2∘Fwnw1,…,pm∘Fwnw1=2∑wj∈VdGwj-2∑j=1np1∘Fwjwj+1,p2∘Fwjwj+1,…,pm∘Fwjwj+1=∑wj∈VdGwj+∑wj∈VdGwj-2∑j=1np1∘Fwjwj+1,p2∘Fwjwj+1,…,pm∘Fwjwj+1=∑wj∈VdGwj+2∑j=1np1∘Fwjwj+1,p2∘Fwjwj+1,…,pm∘Fwjwj+1-2∑j=1np1∘Fwjwj+1,p2∘Fwjwj+1,…,pm∘Fwjwj+1=∑wj∈VdGwj.Hence,(9)∑wj∈VdGwj=∑wjwk∈E,j≠kdGwjwk.Remark 15.
LetG=V,W,F be an m-polar fuzzy graph on a crisp graph G∗. Then(10)∑wjwk∈EdGwjwk=∑wjwk∈EdG∗wjwkp1∘Fwjwk,p2∘Fwjwk,…,pm∘Fwjwk,where dG∗wjwk=dG∗wj+dG∗wk-2, for all wjwk∈E.Theorem 16.
LetG=V,W,F be an m-polar fuzzy graph on a c-regular crisp graph G∗. Then (11)∑wjwk∈EdGwjwk=c-1∑wj∈VdGwj.Proof.
From Remark15, we have(12)∑wjwk∈EdGwjwk=∑wjwk∈EdG∗wjwkp1∘Fwjwk,p2∘Fwjwk,…,pm∘Fwjwk=∑wjwk∈EdG∗wj+dG∗wk-2p1∘Fwjwk,p2∘Fwjwk,…,pm∘Fwjwk.Since G∗ is a regular crisp graph, we have the degree of every vertex in G∗ as c.
That is,dG∗wj=c, so(13)∑wjwk∈EdGwjwk=c+c-2∑wjwk∈Ep1∘Fwjwk,p2∘Fwjwk,…,pm∘Fwjwk,∑wjwk∈EdGwjwk=2c-1∑wjwk∈Ep1∘Fwjwk,p2∘Fwjwk,…,pm∘Fwjwk,∑wjwk∈EdGwjwk=c-1∑wjwk∈EdGwj.Theorem 17.
LetG=V,W,F be an m-polar fuzzy graph on a crisp graph G∗. Then (14)∑wjwk∈EtdGwjwk=∑wjwk∈EdG∗wjwkp1∘Fwjwk,p2∘Fwjwk,…,pm∘Fwjwk+∑wjwk∈Ep1∘Fwjwk,p2∘Fwjwk,…,pm∘Fwjwk.Proof.
From the definition of total edge degree ofG, we get(15)∑wjwk∈EtdGwjwk=∑wjwk∈EdGwjwk+p1∘Fwjwk,p2∘Fwjwk,…,pm∘Fwjwk=∑wjwk∈EdGwjwk+∑wjwk∈Ep1∘Fwjwk,p2∘Fwjwk,…,pm∘Fwjwk.From Remark 15, we have(16)∑wjwk∈EtdGwjwk=∑wjwk∈EdG∗wjwkp1∘Fwjwk,p2∘Fwjwk,…,pm∘Fwjwk+∑wjwk∈Ep1∘Fwjwk,p2∘Fwjwk,…,pm∘Fwjwk.Theorem 18.
LetG=V,W,F be an m-polar fuzzy graph. Then the function p1∘F,p2∘F,…,pm∘F is a constant function if and only if the following conditions are equivalent.(i)
G is an edge regular m-polar fuzzy graph.(ii)
G is a totally edge regular m-polar fuzzy graph.Proof.
Suppose thatp1∘F,p2∘F,…,pm∘F is a constant function. Then(17)p1∘Fxy,p2∘Fxy,…,pm∘Fxy=k1,k2,…,km∀xy∈E,where k1,k2,…,km are constants and k1,k2,…,km∈0,1. Let G be an edge regular m-polar fuzzy graph. Then, for all wjwk∈E, dGwjwk=r1,r2,…,rm.
Now we have to show thatG is a totally edge regular m-polar fuzzy graph.
Now(18)tdGwjwk=dGwjwk+p1∘Fwjwk,p2∘Fwjwk,…,pm∘Fwjwk=r1,r2,…,rm+k1,k2,…,km=r1+k1,r2+k2,…,rm+km∀wjwk∈E.Thus G is a totally edge regular graph.
Now, letG be a h1,h2,…,hm-totally edge regular m-polar fuzzy graph. Then (19)tdGwjwk=h1,h2,…,hm∀wjwk∈E.So, we have (20)tdGwjwk=dGwjwk+p1∘Fwjwk,p2∘Fwjwk,…,pm∘Fwjwk=h1,h2,…,hm.Hence (21)dGwjwk=h1,h2,…,hm-p1∘Fwjwk,p2∘Fwjwk,…,pm∘Fwjwk=h1-p1∘Fwjwk,h2-p2∘Fwjwk,…,hm-pm∘Fwjwk=h1-k1,h2-k2,…,hm-km.Then G is an h1-k1,h2-k2,…,hm-km-edge regular m-polar fuzzy graph.
Conversely, suppose thatG is an edge regular m-polar fuzzy graph and G is a totally edge regular m-polar fuzzy graph which are equivalent. We have to prove that p1∘F,p2∘F,…,pm∘F is a constant function. In a contrary way, we suppose that p1∘F,p2∘F,…,pm∘F is not a constant function. Then (22)p1∘Fwjwk,p2∘Fwjwk,…,pm∘Fwjwk≠p1∘Fwrws,p2∘Fwrws,…,pm∘Fwrwsfor at least one pair of edges wjwk,wrws∈E. Let G be an r1,r2,…,rm-edge regular m-polar fuzzy graph. Then dGwjwk=dGwrws=r1,r2,…,rm. Hence, for every wjwk∈E and for every wrws∈E, (23)tdGwjwk=dGwjwk+p1∘Fwjwk,p2∘Fwjwk,…,pm∘Fwjwk=r1,r2,…,rm+p1∘Fwjwk,p2∘Fwjwk,…,pm∘Fwjwk=r1+p1∘Fwjwk,r2+p2∘Fwjwk,…,rm+pm∘Fwjwk,tdGwrws=dGwrws+p1∘Fwrws,p2∘Fwrws,…,pm∘Fwrws=r1,r2,…,rm+p1∘Fwrws,p2∘Fwrws,…,pm∘Fwrws=r1+p1∘Fwrws,r2+p2∘Fwrws,…,rm+pm∘Fwrws.Since (24)p1∘Fwjwk,p2∘Fwjwk,…,pm∘Fwjwk≠p1∘Fwrws,p2∘Fwrws,…,pm∘Fwrws,we have tdGwjwk≠tdGwrws. Hence, G is not a totally edge regular m-polar fuzzy graph. This is a contradiction to our assumption. Hence, p1∘F,p2∘F,…,pm∘F is a constant function. In the same way, we can prove that p1∘F,p2∘F,…,pm∘F is a constant function, when G is a totally edge regular m-polar fuzzy graph.Theorem 19.
LetG∗ be a h-regular crisp graph and G=V,W,F be an m-polar fuzzy graph on G∗. Then, p1∘F,p2∘F,…,pm∘F is a constant function if and only if G is both regular m-polar fuzzy graph and totally edge regular m-polar fuzzy graph.Proof.
LetG=V,W,F be an m-polar fuzzy graph on G∗ and let G∗ be a h-regular crisp graph. Assume that p1∘F,p2∘F,…,pm∘F is a constant function. Then (25)p1∘Fxy,p2∘Fxy,…,pm∘Fxy=k1,k2,…,km∀xy∈E,where k1,k2,…,km are constants and k1,k2,…,km∈0,1. From the definition of degree of a vertex, we get(26)dGwj=∑wjwk∈Ep1∘Fwjwk,p2∘Fwjwk,…,pm∘Fwjwk=∑wiwj∈Ek1,k2,…,km=hk1,hk2,…,hkmfor  everywj∈V.So dGwj=hk1,hk2,…,hkm. Therefore, G is regular m-polar fuzzy graph.
Now, fori=1,2,…,m,(27)tdGwjwk=∑pi∘Fwjwk+∑pi∘Fwkwl+pi∘Fwjwl=∑wjwk∈Ej≠kk1,k2,…,km+∑wkwl∈Ek≠lk1,k2,…,km+k1,k2,…,km=h-1k1,k2,…,km+h-1k1,k2,…,km+k1,k2,…,km=2h-1k1,k2,…,km∀wjwk∈E.Hence, G is also a totally edge regular m-polar fuzzy graph.
Conversely, assume thatG is both regular and totally edge regular m-polar fuzzy graph. Now we have to prove that p1∘F,p2∘F,…,pm∘F is a constant function. Since G is regular, dGwj=l1,l2,…,lm for all wj∈V. Also G is totally edge regular. Hence, tdGwjwk=h1,h2,…,hm for all wjwk∈E. From the definition of total edge degree, we get (28)tdGwjwk=dGwj+dGwk-p1∘Fwjwk,p2∘Fwjwk,…,pm∘Fwjwk∀wjwk∈E,h1,h2,…,hm=l1,l2,…,lm+l1,l2,…,lm-p1∘Fwjwk,p2∘Fwjwk,…,pm∘Fwjwk,so (29)p1∘Fwjwk,p2∘Fwjwk,…,pm∘Fwjwk=2l1,l2,…,lm-h1,h2,…,hm=2l1-h1,2l2-h2,…,2lm-hm∀wjwk∈E.Hence p1∘F,p2∘F,…,pm∘F is a constant function.


## 4.h-Morphism on m-Polar Fuzzy Graphs

Definition 20.
LetG1=V1,W1,F1 and G2=V2,W2,F2 be two m-polar fuzzy graphs of the graphs G1∗=V1,E1 and G2∗=V2,E2, respectively.
A homomorphismg from G1 to G2 is a mapping g:V1→V2 such that, for each i=1,2,…,m,(30)pi∘W1u≤pi∘W2gu,∀u∈V1,pi∘F1uv≤pi∘F2gugv,∀uv∈E1.An isomorphism g from G1 to G2 is a bijective mapping g:V1→V2 which satisfies the following conditions:(31)pi∘W1u=pi∘W2gu,∀u∈V1,pi∘F1uv=pi∘F2gugv,∀uv∈E1.A weak isomorphism g from G1 to G2 is a bijective mapping g:V1→V2 which satisfies the following conditions:g is homomorphism and pi∘W1u=pi∘W2gu, ∀u∈V1.A coweak isomorphism g from G1 to G2 is a bijective mapping g:V1→V2 which satisfies the following conditions:g is homomorphism and pi∘F1uv=pi∘F2gugv, ∀uv∈E1.Definition 21.
The order of anm-polar fuzzy graph G is defined as (32)OG=∑u∈Vpi∘Wu=∑u∈Vp1∘Wu,∑u∈Vp2∘Wu,…,∑u∈Vpm∘Wu.Definition 22.
The size of anm-polar fuzzy graph G is defined as (33)SG=∑u≠vuv∈Epi∘Fuv=∑u≠vuv∈Ep1∘Fuv,∑u≠vuv∈Ep2∘Fuv,…,∑u≠vuv∈Epm∘Fuv.Definition 23.
LetG1 and G2 be two m-polar graphs on V1,E1 and V2,E2, respectively.
A bijective functionh:V1→V2 is called an m-polar morphism or m-polar h-morphism if there exists two numbers l1>0 and l2>0 such that pi∘W2hu=l1pi∘W1u, ∀u∈V1, pi∘F2huhv=l2pi∘F1uv, ∀uv∈E1, i=1,2,…,m. In such a case, h will be called an l1,l2m-polar h-morphism from G1 to G2. If l1=l2=l, we call h, an m-polar l-morphism.Example 24.
Consider twom-polar fuzzy graphs G1=V1,W1,F1 and G2=V2,W2,F2 as shown in Figure 3.
Anm-polar fuzzy graph G1=V1,W1,F1 is shown in Figure 3(a) where (34)V1=K,L,M,E1=KL,LM,MK,W1=0.3,0.45,0.2K,0.3,0.5,0.3L,0.4,0.46,0.3M,F1=0.2,0.3,0.1KL,0.1,0.2,0.2LM,0.2,0.3,0.1MK.Another m-polar fuzzy graph G2=V2,W2,F2 is shown in Figure 3(b) where (35)V2=K′,L′,M′,E2=K′L′,L′M′,M′K′,W2=0.6,0.9,0.4K′,0.6,1.0,0.6L′,0.8,0.92,0.6M′,F2=0.6,0.9,0.3K′L′,0.3,0.6,0.6L′M′,0.6,0.9,0.3M′K′.Here, there is an m-polar h-morphism such that hK=K′, hL=L′, hM=M′, l1=2, and l2=3.Figure 3
h-morphism of m-polar fuzzy graphs G1 and G2.
(a)
(b)Theorem 25.
The relationh-morphism is an equivalence relation in the collection of m-polar fuzzy graphs.Proof.
Consider the collection ofm-polar fuzzy graphs. Define the relation G1≈G2 if there exists a l1,l2h-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 hence ≈ is reflexive. Let G1≈G2. Then there exists a l1,l2 morphism from G1 to G2 for some l1≠0 and l2≠0. Therefore, (36)pi∘W2hu=l1pi∘W1u,∀u∈V1,pi∘F2huhv=l2pi∘F1uv,∀uv∈E1.Consider h-1:G2→G1. Let m,n∈V2. Since h-1 is bijective, m=hu, n=hv, for some u,v∈V2. Now,(37)pi∘W1h-1m=pi∘W1h-1hu=pi∘W1u=1l1pi∘W2hu=1l1pi∘W2m,pi∘F1h-1mh-1n=pi∘F1h-1huh-1hv=pi∘F1uv=1l2pi∘F2huhv=1l2pi∘F2mn.Thus there exists 1/l1,1/l2 morphism from G2 to G1. Therefore, G2≈G1 and hence ≈ is symmetric.
LetG1≈G2 and G2≈G3. Then there exists 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, say q for some l3≠0 and l4≠0. So, for i=1,2,…,m,(38)pi∘W3qx=l3pi∘W2x,x∈V2,pi∘F3qxqy=l4pi∘F2xy,∀xy∈E2.Let r:q∘p:G1→G3.
Now,(39)pi∘W3ru=pi∘W3q∘hu=pi∘W3qhu=l3pi∘W2hu=l3l1pi∘W1u,pi∘F3rurv=pi∘F3q∘huq∘hv=pi∘F3qhuqhv=l4pi∘F2huhv=l4l2pi∘F1uv.Thus there exists l3l1,l4l2 morphism r from G1 to G3. Therefore, G1≈G3 and hence ≈ is transitive. So, the relation h-morphism is an equivalence relation in the collection of m-polar fuzzy graphs.Theorem 26.
LetG1 and G2 be two m-polar fuzzy graphs such that G1 is l1,l2m-polar morphism to G2 for some l1≠0 and l2≠0. The image of a strong edge in G1 is also a strong edge in G2 if and only if l1=l2.Proof.
Letxy be a strong edge in G1 such that hxhy is also a strong edge in G2.
Now asG1≈G2 for i=1,2,…,m, we have (40)l2pi∘F1xy=pi∘F2hxhy=pi∘W2hx∧hy=l1pi∘W1x∧l1pi∘W1y=l1pi∘W1x∧pi∘W1y=l1pi∘F1xy∀xy∈E1.Hence, (41)l2pi∘F1xy=l1pi∘F1xy,∀xy∈E1.The equation holds if and only if l1=l2.Theorem 27.
If anm-polar fuzzy graph G1 is coweak isomorphic to G2 and if G1 is regular then G2 is also regular.Proof.
As anm-polar fuzzy graph G1 is coweak isomorphic to G2, there exists a coweak isomorphism h:G1→G2 which is bijective for i=1,2,…,m that satisfies (42)pi∘W1u≤pi∘W2hu,∀u∈V1,pi∘F1uv=pi∘F2huhv,∀uv∈E1.As G1 is regular, for u∈V, ∑u≠v,v∈V1pi∘F1uv=constant. Now ∑h(u)≠h(v)pi∘F2huhv=∑u≠v,v∈V1pi∘F1uv=constant. Therefore, G2 is regular.Theorem 28.
LetG1 and G2 be two m-polar fuzzy graphs. If G1 is weak isomorphic to G2 and if G1 is strong then G2 is also strong.Proof.
AsG1 is an m-polar fuzzy graph which is weak isomorphic withG2, then there exists a weak isomorphism h:G1→G2 which is bijective for i=1,2,…,m that satisfies (43)pi∘W1u=pi∘W2hu,∀u∈V1,pi∘F1uv≤pi∘F2huhv,∀uv∈E1.As G1 is strong, pi∘F1uv=minpi∘W1u,pi∘W1v. Now, we get (44)pi∘F2huhv≥pi∘F1uv=min⁡pi∘W1u,pi∘W1v=min⁡pi∘W2hu,pi∘W2hv.By the definition, pi∘F2huhv≤min⁡pi∘W2hu,pi∘W2hv. Therefore, pi∘F2huhv=min⁡pi∘W2hu,pi∘W2hv. So G2 is strong.Theorem 29.
If anm-polar fuzzy graph G1 is coweak isomorphic with a strong regular m-polar fuzzy graph G2, then G1 is strong regular m-polar fuzzy graph.Proof.
As anm-polar fuzzy graph G1 is coweak isomorphic to G2. Then there exists a coweak isomorphism h:G1→G2 which is bijective for i=1,2,…,m that satisfies (45)pi∘W1u≤pi∘W2hu,∀u∈V1,pi∘F1uv=pi∘F2huhv,∀uv∈E1.Now, we get(46)pi∘F1uv=pi∘F2huhv=min⁡pi∘W2hu,pi∘W2hv≥min⁡pi∘W1u,pi∘W1v.But, by the definition, we have (47)pi∘F1uv≤min⁡pi∘W1u,pi∘W1v.So, pi∘F1uv=minpi∘W1u,pi∘W1v.
Therefore,G1 is strong. Also for u∈V1, ∑u≠v,v∈V1pi∘F1uv=∑pi∘F2huhv=constant as G2 is regular. Therefore, G1 is regular.Theorem 30.
LetG1 and G2 be two isomorphic m-polar fuzzy graphs; then G1 is strong regular if and only if G2 is strong regular.Proof.
As anm-polar fuzzy graph G1 is isomorphic with an m-polar fuzzy graph G2, there exists an isomorphism h:G1→G2 which is bijective for i=1,2,…,m that satisfies (48)pi∘W1u=pi∘W2hu,∀u∈V1,pi∘F1uv=pi∘F2huhv,∀uv∈E1.Now, G1 is strong if and only if pi∘F1uv=minpi∘W1u,pi∘W1v, if and only if pi∘F2huhv=minpi∘W2hu,pi∘W2hv, and if and only if G2 is strong.
G
1 is regular if and only if, for u∈V1, ∑u≠v,v∈V1pi∘F1uv=constant, if and only if ∑p(u)≠p(v)pi∘F2huhv=Constant, for all hu∈V2, and if and only if G2 is regular.Definition 31.
LetG=V,W,F be a connected m-polar fuzzy graph. Then G is said to be a highly irregular m-polar fuzzy graph if every vertex of G is adjacent to vertices with distinct degrees.Example 32.
Consider anm-polar fuzzy graph G=V,W,F of G∗=V,E, where (49)V=K,L,M,N,E=KL,LM,MN,NK,W=0.3,0.4,0.5K,0.2,0.5,0.9L,0.5,0.5,0.5M,0.4,0.3,0.8N,F=0.1,0.3,0.4KL,0.2,0.5,0.5LM,0.3,0.2,0.5MN,0.1,0.2,0.4NK,as in Figure 4.
By usual calculations, we get(50)dGK=0.2,0.5,0.8,dGL=0.3,0.8,0.9,dGM=0.5,0.7,1.0,dGN=0.4,0.4,0.9.We see that every vertex of G is adjacent to vertices with distinct degrees.Figure 4
A highly irregularm-polar fuzzy graph.Theorem 33.
For any two isomorphic highly irregularm-polar fuzzy graphs, their order and size are the same.Proof.
Ifh:G1→G2 is an isomorphism between the two highly irregular m-polar fuzzy graphs G1 and G2 with the underlying sets V1 and V2, respectively, then, for i=1,2,…,m,(51)pi∘W1u=pi∘W2hu,∀u∈V1,pi∘F1uv=pi∘F2huhv,∀uv∈E1.So, we get (52)OG1=∑x1∈V1pi∘W1x1=∑x1∈V1pi∘W2hx1=∑x2∈V2pi∘W2x2=OG2,SG1=∑x1y1∈E1pi∘F1x1y1=∑x1y1∈E1pi∘F2hx1hy1=∑x2y2∈E2pi∘F2x2y2=SG2.Theorem 34.
IfG1 and G2 are isomorphic highly irregular m-polar fuzzy graphs, then, the degrees of the corresponding vertices u and hu are preserved.Proof.
Ifh:G1→G2 is an isomorphism between the highly irregular m-polar fuzzy graphs G1 and G2 with the underlying sets V1 and V2, respectively, then, for i=1,2,…,m,(53)pi∘F1uv=pi∘F2huhv∀u,v∈V1.Therefore, (54)dG1u=∑u,v∈V1pi∘F1uv=∑u,v∈V1pi∘F2huhv=dG2hu.That is, the degrees of the corresponding vertices of G1 and G2 are the same.


## 5. Conclusion

Any dissimilar fuzzy graph hypothesis needs large data for training to be able to help in decision-making which is crucial to utilitarian research in science and technology. The new method developed in this paper based on the pattern of unique cases helps us to make a better choice in contrast to the established fuzzy graph solutions. The concept ofh-morphism, highly irregular m- polar fuzzy graphs is discussed in this paper.



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