#### 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 , , â€‰â€‰=â€‰â€‰, , and â€‰â€‰=â€‰â€‰, as in Figure 1.

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

#### 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.

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.

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.

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 have