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Advances in Fuzzy Systems
Volume 2017 (2017), Article ID 3817469, 9 pages
https://doi.org/10.1155/2017/3817469
Research Article

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

1Department of Mathematics, Vasireddy Venkatadri Institute of Technology, Nambur 522 508, India
2Department of Mathematics, VFSTR University, Vadlamudi 522 237, India
3Department of CSE, KL University, Vaddeswaram 522502, India
4Department of Mathematics, Tirumala Engineering College, Narasaraopet 522034, India

Correspondence should be addressed to S. Satyanarayana; moc.liamg@1snaytas.s

Received 22 May 2017; Accepted 5 July 2017; Published 22 August 2017

Academic Editor: Mehmet Onder Efe

Copyright © 2017 Ch. Ramprasad et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Computational intelligence and computer science rely on graph theory to solve combinatorial problems. Normal product and tensor product of an -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 -complement of an -polar fuzzy graph are defined and some properties are studied. An application of an -polar fuzzy graph is also presented in this article.

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 -polar fuzzy set. The notion of an -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 -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 -polar fuzzy graph, product -polar fuzzy intersection graph, and product -polar fuzzy line graph [8].

In this article, we study the Cartesian product and composition of two -polar fuzzy graphs and compute the degrees of the vertices in these graphs. The notions of normal product and tensor product of -polar fuzzy graphs are introduced and some properties are studied. Also in the present work, we introduce the concept of complement, -complement of an -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 are -polar fuzzy graphs.

2. Preliminaries

Definition 1. The -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 edges is the smallest element in . is called the -polar fuzzy vertex set of and is called -polar fuzzy edge set of .

Sometimes we denote the graph by also.

Definition 2. Given two graphs , their Cartesian product, , is defined as follows: for , we have(i) for all ,(ii) for all ,(iii) for all ,(iv) for all

Definition 3. Given two graphs , their composition, , is defined as follows: for , we have(i) for all ,(ii) for all ,(iii) for all ,(iv) for all , where

Definition 4. The normal product of and is defined as the -polar fuzzy graph on , where such that for , we have(i) for all ,(ii) for all ,(iii) for all ,(iv) for all and

Definition 5. The tensor product of and is defined as on , where such that for , we have(i) for all ,(ii) for all and

Definition 6. The union of the graphs and of and , respectively, is defined as follows: for , we have(i) (ii) (iii)

Definition 7. The join of the graphs and of and , respectively, is defined as such that for , we have the following:(i)(ii)(iii), where denotes the set of all edges joining the vertices of and .(iv) if

3. Degree of Vertices in -Polar Fuzzy Graph

Definition 8. Let be an -polar fuzzy graph. Then the degree of a vertex in is defined as .

Further, a highly irregular -polar fuzzy graph is defined as an -polar fuzzy graph in which every vertex of is adjacent to vertices with distinct degrees.

Example 9. Consider the graph of , where , ,   =  , , and   =  , as in Figure 1.

Figure 1: Highly irregular 3-polar fuzzy graph .

In this graph, we have , , ,

4. Cartesian Product and Vertex Degree

From the definition of Cartesian product graphs, for every vertex , we have

Theorem 10. Let and be two graphs. If and then for .

Proof. By definition, we get

Example 11. Consider two graphs , and their Cartesian product, , as shown in Figure 2.
Since and , by Theorem 10, we have

Figure 2: Cartesian product of two 3-polar fuzzy graphs and .

5. Composition Graph and Vertex Degree

From the definition of composition graphs, for every vertex , we have

Theorem 12. Let and be two graphs. If and then for all for .

Proof. By definition, we get

Example 13. Consider two graphs , and their composition, , as shown in Figure 3.

Figure 3: Composition of two 3-polar fuzzy graphs and .

Since and , by Theorem 12, we have

6. Normal Product and Vertex Degree

From the definition of normal product graphs, for every vertex , we have

Theorem 14. Let and be two graphs. If and , then for .

Proof. By definition, we get

Example 15. Consider two graphs , and their normal product, , is shown in Figure 4.

Figure 4: Normal product of two 3-polar fuzzy graphs and .

Since and , by Theorem 14, we have

7. Tensor Product and Vertex Degree

From the definition of tensor product, for every vertex for

Theorem 16. Let and be two -polar fuzzy graphs. If , then and if , then for .

Proof. Let . Then we get So, .
Let . Then we get So, .

Example 17. Consider two graphs , and their tensor product, , is shown in Figure 5.

Figure 5: Tensor product of two 3-polar fuzzy graphs and .

Since , by Theorem 16, we have

8. -Complement of an -Polar Fuzzy Graph

Now for an -polar fuzzy graph , we introduce the notion of its -complement.

Definition 18. The complement of an -polar fuzzy graph is also an -polar fuzzy graph , where and is defined as follows:

Definition 19. Let be an -polar fuzzy graph. If is isomorphic to we say is self-complementary. Similarly, if is weak isomorphic to then we say is self-weak complementary.

Definition 20. The -complement of is defined as , where and is given by for .

Theorem 21. Let be a self-weak complementary and highly irregular graph. Then, we have .

Proof. Let be a self-weak complementary, highly irregular graph of . Then, there exists a weak isomorphism from to such that for all , we get . From the definition of complement, from the above inequality, for all , we get that and Hence, . Thus, we get

Theorem 22. Let be a graph which is self-complementary and highly irregular. Then, we have , for all .

Proof. Let be a self-complementary, highly irregular graph of . Then, there exists an isomorphism from to such that for all and for all
Let . Then, for , we havethat is, Therefore, i.e.

Theorem 23. Let and be two graphs. If and are isomorphic, then their -complements, and , are also isomorphic.

Proof. Let and let be an isomorphism from to . Then for If , then , and If , then , and
Thus, for all Therefore, from to is an isomorphism; that is, .

Theorem 24. Let and be two graphs such that . Then .

Proof. Let be the identity map. To show that , it is enough to prove that and for all and for . Then for any , we haveMoreover,Hence, the result is proved.

Theorem 25. Let and be two graphs such that . Then .

Proof. We shall prove that the identity map is the required isomorphism. Then for any we have the following: If , .If , then