Discrete Dynamics in Nature and Society

Volume 2016 (2016), Article ID 6301693, 9 pages

http://dx.doi.org/10.1155/2016/6301693

## Certain Concepts in -Polar Fuzzy Graph Structures

^{1}Department of Mathematics, University of the Punjab, New Campus, Lahore, Pakistan^{2}Department of Mathematics, Faculty of Science, King Abdulaziz University, Al-Faisaliah Campus, Jeddah, Saudi Arabia

Received 7 June 2016; Accepted 14 August 2016

Academic Editor: Juan R. Torregrosa

Copyright © 2016 Muhammad Akram 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

We apply the concept of -polar fuzzy sets to graph structures. We introduce certain concepts in -polar fuzzy graph structures, including strong -polar fuzzy graph structure, -polar fuzzy -cycle, -polar fuzzy -tree, -polar fuzzy -cut vertex, and -polar fuzzy -bridge, and we illustrate these concepts by several examples. We present the notions of -complement of an -polar fuzzy graph structure and self-complementary, strong self-complementary, totally strong self-complementary -polar fuzzy graph structures, and we investigate some of their properties.

#### 1. Introduction

Graph theory has applications in many areas of computer science, including data mining, image segmentation, clustering, image capturing, and networking. A graph structure, introduced by Sampathkumar [1], is a generalization of undirected graph which is quite useful in studying some structures including graphs, signed graphs, and graphs in which every edge is labeled or colored. A graph structure helps to study the various relations and the corresponding edges simultaneously.

A fuzzy set [2] is an important mathematical structure to represent a collection of objects whose boundary is vague. Fuzzy models are becoming useful because of their aim in reducing the differences between the traditional models used in engineering and science. Nowadays fuzzy sets are playing a substantial role in chemistry, economics, computer science, engineering, medicine, and decision-making problems. In 1997, Zhang [3] generalized the idea of a fuzzy set and gave the concept of bipolar fuzzy set on a given set as a map which associates each element of to a real number in the interval In , Chen et al. [4] introduced the idea of -polar fuzzy sets as an extension of bipolar fuzzy sets and showed that bipolar fuzzy sets and -polar fuzzy sets are cryptomorphic mathematical notions and that we can obtain concisely one from the corresponding one in [4]. The idea behind this is that “multipolar information” (not just bipolar information which corresponds to two-valued logic) exists because data for a real world problem are sometimes from agents . For example, the exact degree of telecommunication safety of mankind is a point in because different person has been monitored different times. There are many examples such as truth degrees of a logic formula which are based on logic implication operators , similarity degrees of two logic formula which are based on logic implication operators , ordering results of a magazine, ordering results of a university, and inclusion degrees (accuracy measures, rough measures, approximation qualities, fuzziness measures, and decision preformation evaluations) of a rough set.

Kauffman [5] gave the definition of a fuzzy graph in on the basis of Zadeh’s fuzzy relations [6]. Rosenfeld [7] discussed the idea of fuzzy graph in 1975. Further remarks on fuzzy graphs were given by Bhattacharya [8]. Several concepts on fuzzy graphs were introduced by Mordeson and Nair [9]. In 2011, Akram introduced the notion of bipolar fuzzy graphs in [10]. Alshehri and Akram [11] introduced the concept of bipolar fuzzy competition graphs. In 2015, Akram and Younas studied certain types of irregular -polar fuzzy graphs in [12]. Akram and Adeel studied -polar fuzzy line graphs in [13]. Akram and Waseem introduced certain metrics in -polar fuzzy graphs in [14]. Dinesh [15] introduced the notion of a fuzzy graph structure and discussed some related properties. Akram and Akmal [16] introduced the concept of bipolar fuzzy graph structures. In this paper, we apply the concept of -polar fuzzy sets to graph structures. We introduce certain concepts in -polar fuzzy graph structures, including strong -polar fuzzy graph structure, -polar fuzzy -cycle, -polar fuzzy -tree, -polar fuzzy -cut vertex, and -polar fuzzy -bridge, and we illustrate these concepts by several examples. We present the notions of -complement of an -polar fuzzy graph structure and self-complementary, strong self-complementary, totally strong self-complementary -polar fuzzy graph structures, and we investigate some of their properties.

#### 2. Preliminaries

In this section, we review some basic concepts that are necessary for fully benefit of this paper.

In , Zadeh [2] introduced the notion of a fuzzy set as follows.

*Definition 1 (see [2, 6]). *A* fuzzy set * in a universe is a mapping . A* fuzzy relation* on is a fuzzy set in . Let be a fuzzy set in and fuzzy relation on . We call a fuzzy relation on if .

*Definition 2 (see [16]). * is called a* bipolar fuzzy graph structure* (BFGS) of a graph structure (GS) if is a* bipolar fuzzy set on * and for each is a* bipolar fuzzy set on * such that Note that for all and , , where and are called* underlying vertex set* and* underlying **-edge sets* of , respectively.

*Definition 3 (see [16]). *Let be a BFGS of a GS Let be any permutation on the set and the corresponding permutation on , if and only if

If for some and then , while is chosen such that

And BFGS , denoted by , is called the -*complement of BFGS *

*Definition 4 (see [4]). *An *-polar fuzzy set* (or a -set) on is exactly a mapping

Note that a -set is an -set. An -set on the set is a synonym of a mapping , where is a lattice. So, is considered to be a partial order set with the point-wise order , where is an arbitrary ordinal number, is defined by for each , and is the th projection mapping (). When , an -set on will be called a fuzzy set on .

*Definition 5 (see [14]). *Let be an -polar fuzzy subset of a nonempty set . An *-polar fuzzy relation* on is an -polar fuzzy subset of defined by the mapping such that, for all , , where denotes the th degree of membership of the vertex and denotes the th degree of membership of the edge .

*Definition 6 (see [4, 14]). *An *-polar fuzzy graph* is a pair , where is an -polar fuzzy set in and is an -polar fuzzy relation on such that for all .

We note that for all for all . is called the *-polar fuzzy vertex set* of and is called the *-polar fuzzy edge set* of , respectively. An -polar fuzzy relation on is called symmetric if for all .

#### 3. Certain Concepts in -Polar Fuzzy Graph Structures

*Definition 7. *Let be a graph structure (GS). Let be an -polar fuzzy set on and an -polar fuzzy set on such thatfor all , , and for . Then is called an -*polar fuzzy graph structure* (-PFGS) on where is the -*polar fuzzy vertex set* of and is the -*polar fuzzy i-edge set* of .

*Definition 8. *Let and be two -PFGSs of a GS . Then is called an -*polar fuzzy subgraph structure* of , if is called an -*polar fuzzy induced subgraph structure* of by a set , if is called an -*polar fuzzy spanning subgraph structure* of , if and

*Example 9. *Consider a graph structure such that , , and Let , , and be 4-polar fuzzy sets on , , and , respectively, defined by Tables 1 and 2.

By simple calculations, it is easy to check that is a 4-polar fuzzy graph structure of as shown in Figure 1. Note that we represent as in all tables and the figures.

A -polar fuzzy induced subgraph structure of by and a -polar fuzzy spanning subgraph structure of (Figure 1) are shown in Figure 2.