Advances in Fuzzy Systems

Volume 2017 (2017), Article ID 4715421, 9 pages

https://doi.org/10.1155/2017/4715421

## Morphism of -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, K L University, Vaddeswaram 522502, India^{4}Department of Mathematics, Tirumala Engineering College, Narasaraopet 522034, India

Correspondence should be addressed to S. Satyanarayana

Received 24 May 2017; Accepted 19 July 2017; Published 23 August 2017

Academic Editor: Zeki Ayag

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

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.