Advances in Fuzzy Systems

Volume 2017 (2017), Article ID 3817469, 9 pages

https://doi.org/10.1155/2017/3817469

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

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

Correspondence should be addressed to S. Satyanarayana

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.