#### Abstract

The main purpose of this paper is to introduce the notion of -polar -morphism on -polar fuzzy graphs. The action of -polar -morphism on -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 -polar fuzzy graphs are studied.

#### 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 -polar fuzzy set. The notion of -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 -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 -polar fuzzy graph, product -polar fuzzy intersection graph, and product -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 of -polar -morphism, edge regular -polar fuzzy graph, totally edge regular -polar fuzzy graph, and highly irregular -polar fuzzy graph in order to strengthen the decision-making in critical situations.

#### 2. Preliminaries

*Definition 1. *Throughout the paper, ( copies of the closed interval ) is considered to be a poset with pointwise order , where is a natural number, is given by , , where and is the th projection mapping. An -polar fuzzy set (or a -set) on is a mapping .

*Definition 2. *Let be an -polar fuzzy set on . An -polar fuzzy relation on is an -polar fuzzy set of such that for all , that is, for each , for all , .

*Definition 3. *A generalized -polar fuzzy graph of a graph is a pair , where is an -polar fuzzy set in and is an -polar fuzzy set in such that for all and for all is the smallest element in . is called the -polar fuzzy set of and is called -polar fuzzy edge set of .

*Definition 4. *An -polar fuzzy graph of the graph is said to be strong if for all , .

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

Using the existing graph theories a new -polar fuzzy graph theory is introduced in this section.

*Definition 5. *Let be an -polar fuzzy graph. Then the degree of a vertex is defined for as

*Definition 6. *The degree of an edge in an -polar fuzzy graph is defined as

*Definition 7. *The total degree of an edge in an -polar fuzzy graph is defined as

*Definition 8. *The degree of an edge in a crisp graph is

*Example 9. *Consider an -polar fuzzy graph of , where as in Figure 1.

Then, we have

*Definition 10. *If every vertex in an -polar fuzzy graph has the same degree , then is called regular -polar fuzzy graph or -polar fuzzy graph of degree

*Definition 11. *If every edge in an -polar fuzzy graph has the same degree , then is called an edge regular -polar fuzzy graph.

*Definition 12. *If every edge in an -polar fuzzy graph has the same total degree , then is called totally edge regular -polar fuzzy graph.

*Example 13. *Consider an -polar fuzzy graph of , where as in Figure 2.

Then, we have

Theorem 14. *Let be an -polar fuzzy graph on a cycle . Then *

*Proof. *Suppose that is an -polar fuzzy graph and is a cycle .

Now, we get Hence,

*Remark 15. *Let be an -polar fuzzy graph on a crisp graph . Thenwhere , for all

Theorem 16. *Let be an -polar fuzzy graph on a -regular crisp graph . Then *

*Proof. *From Remark 15, we haveSince is a regular crisp graph, we have the degree of every vertex in as

That is, so

Theorem 17. *Let be an -polar fuzzy graph on a crisp graph . Then *

*Proof. *From the definition of total edge degree of , we getFrom Remark 15, we have

Theorem 18. *Let be an -polar fuzzy graph. Then the function is a constant function if and only if the following conditions are equivalent.*(i)* is an edge regular -polar fuzzy graph.*(ii)* is a totally edge regular -polar fuzzy graph.*

*Proof. *Suppose that is a constant function. Thenwhere are constants and . Let be an edge regular -polar fuzzy graph. Then, for all ,

Now we have to show that is a totally edge regular -polar fuzzy graph.

Now Thus is a totally edge regular graph.

Now, let be a -totally edge regular -polar fuzzy graph. Then So, we have Hence Then is an -edge regular -polar fuzzy graph.

Conversely, suppose that is an edge regular -polar fuzzy graph and is a totally edge regular -polar fuzzy graph which are equivalent. We have to prove that is a constant function. In a contrary way, we suppose that is not a constant function. Then for at least one pair of edges . Let be an -edge regular -polar fuzzy graph. Then . Hence, for every and for every , Since we have . Hence, is not a totally edge regular -polar fuzzy graph. This is a contradiction to our assumption. Hence, is a constant function. In the same way, we can prove that is a constant function, when is a totally edge regular -polar fuzzy graph.

Theorem 19. *Let be a -regular crisp graph and be an -polar fuzzy graph on . Then, is a constant function if and only if is both regular -polar fuzzy graph and totally edge regular -polar fuzzy graph.*

*Proof. *Let be an -polar fuzzy graph on and let be a -regular crisp graph. Assume that is a constant function. Then where are constants and . From the definition of degree of a vertex, we getSo . Therefore, is regular -polar fuzzy graph.

Now, for ,Hence, is also a totally edge regular -polar fuzzy graph.

Conversely, assume that is both regular and totally edge regular -polar fuzzy graph. Now we have to prove that is a constant function. Since is regular, for all . Also is totally edge regular. Hence, for all . From the definition of total edge degree, we get so Hence is a constant function.

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

*Definition 20. *Let and be two -polar fuzzy graphs of the graphs and , respectively.

A homomorphism from to is a mapping such that, for each ,An isomorphism from to is a bijective mapping which satisfies the following conditions:A weak isomorphism from to is a bijective mapping which satisfies the following conditions: is homomorphism and , A coweak isomorphism from to is a bijective mapping which satisfies the following conditions: is homomorphism and ,

*Definition 21. *The order of an -polar fuzzy graph is defined as

*Definition 22. *The size of an -polar fuzzy graph is defined as

*Definition 23. *Let and be two -polar graphs on and , respectively.

A bijective function is called an -polar morphism or -polar -morphism if there exists two numbers and such that , , , , . In such a case, will be called an -polar -morphism from to . If , we call , an -polar -morphism.

*Example 24. *Consider two -polar fuzzy graphs and as shown in Figure 3.

An -polar fuzzy graph is shown in Figure 3(a) where Another -polar fuzzy graph is shown in Figure 3(b) where Here, there is an -polar -morphism such that , , , , and .

**(a)**

**(b)**

Theorem 25. *The relation -morphism is an equivalence relation in the collection of -polar fuzzy graphs.*

*Proof. *Consider the collection of -polar fuzzy graphs. Define the relation if there exists a -morphism from to where both and . Consider the identity morphism to . It is a -morphism from to and hence is reflexive. Let . Then there exists a morphism from to for some and . Therefore, Consider . Let . Since is bijective, , , for some . Now,Thus there exists morphism from to . Therefore, and hence is symmetric.

Let and . Then there exists a morphism from to , say for some and , and there exists morphism from to , say for some and . So, for ,Let .

Now, Thus there exists morphism from to Therefore, and hence is transitive. So, the relation -morphism is an equivalence relation in the collection of -polar fuzzy graphs.

Theorem 26. *Let and be two -polar fuzzy graphs such that is -polar morphism to for some and . The image of a strong edge in is also a strong edge in if and only if .*

*Proof. *Let be a strong edge in such that is also a strong edge in .

Now as for , we have Hence, The equation holds if and only if .

Theorem 27. *If an -polar fuzzy graph is coweak isomorphic to and if is regular then is also regular.*

*Proof. *As an -polar fuzzy graph is coweak isomorphic to , there exists a coweak isomorphism which is bijective for that satisfies As is regular, for , . Now . Therefore, is regular.

Theorem 28. *Let and be two -polar fuzzy graphs. If is weak isomorphic to and if is strong then is also strong.*

*Proof. *As is an -polar fuzzy graph which is weak isomorphic with, then there exists a weak isomorphism