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 which is bijective for that satisfies As is strong, . Now, we get By the definition, . Therefore, So is strong.
Theorem 29. If an -polar fuzzy graph is coweak isomorphic with a strong regular -polar fuzzy graph , then is strong regular -polar fuzzy graph.
Proof. As an -polar fuzzy graph is coweak isomorphic to . Then there exists a coweak isomorphism which is bijective for that satisfies Now, we getBut, by the definition, we have So, .
Therefore, is strong. Also for , as is regular. Therefore, is regular.
Theorem 30. Let and be two isomorphic -polar fuzzy graphs; then is strong regular if and only if is strong regular.
Proof. As an -polar fuzzy graph is isomorphic with an -polar fuzzy graph , there exists an isomorphism which is bijective for that satisfies Now, is strong if and only if , if and only if , and if and only if is strong.
is regular if and only if, for , , if and only if , for all , and if and only if is regular.
Definition 31. Let be a connected -polar fuzzy graph. Then is said to be a highly irregular -polar fuzzy graph if every vertex of is adjacent to vertices with distinct degrees.
Example 32. Consider an -polar fuzzy graph of , where as in Figure 4.
By usual calculations, we get We see that every vertex of is adjacent to vertices with distinct degrees.
Theorem 33. For any two isomorphic highly irregular -polar fuzzy graphs, their order and size are the same.
Proof. If is an isomorphism between the two highly irregular -polar fuzzy graphs and with the underlying sets and , respectively, then, for ,So, we get
Theorem 34. If and are isomorphic highly irregular -polar fuzzy graphs, then, the degrees of the corresponding vertices and are preserved.
Proof. If is an isomorphism between the highly irregular -polar fuzzy graphs and with the underlying sets and , respectively, then, for ,Therefore, That is, the degrees of the corresponding vertices of and 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 of -morphism, highly irregular - polar fuzzy graphs is discussed in this paper.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.