Elsevier

Knowledge-Based Systems

Volume 39, February 2013, Pages 1-8
Knowledge-Based Systems

Bipolar fuzzy graphs with applications

https://doi.org/10.1016/j.knosys.2012.08.022Get rights and content

Abstract

The concepts of neighbourly irregular bipolar fuzzy graphs, neighbourly totally irregular bipolar fuzzy graphs, highly irregular bipolar fuzzy graphs and highly totally irregular bipolar fuzzy graphs are introduced and investigated. A necessary and sufficient condition under which neighbourly irregular and highly irregular bipolar fuzzy graphs are equivalent is discussed. The notion of bipolar fuzzy digraphs is introduced. The bipolar fuzzy influence graph of a social group is also described.

Introduction

In 1965, Zadeh [20] introduced the notion of a fuzzy subset of a set. Since then, the theory of fuzzy sets has become a vigorous area of research in different disciplines including medical and life sciences, management sciences, social sciences, engineering, statistics, graph theory, artificial intelligence, signal processing, multiagent systems, pattern recognition, robotics, computer networks, expert systems, decision making and automata theory. In 1994, Zhang [24] initiated the concept of bipolar fuzzy sets as a generalization of fuzzy sets. A bipolar fuzzy set is an extension of Zadeh’s fuzzy set theory whose membership degree range is [−1, 1]. In a bipolar fuzzy set, the membership degree 0 of an element means that the element is irrelevant to the corresponding property, the membership degree (0, 1] of an element indicates that the element somewhat satisfies the property, and the membership degree [−1, 0) of an element indicates that the element somewhat satisfies the implicit counter-property.

In 1975, Rosenfeld [16] discussed the concept of fuzzy graphs whose basic idea was introduced by Kauffmann [11] in 1973. The fuzzy relations between fuzzy sets were also considered by Rosenfeld and he developed the structure of fuzzy graphs, obtaining analougs of several graph theoretical concepts. Bhattacharya [8] gave some remarks on fuzzy graphs. The complement of a fuzzy graph was defined by Mordeson and Nair [13]. Recently, the bipolar fuzzy graphs have been discussed in [1], [2], [3]. In this paper, we introduce the concepts of neighbourly irregular bipolar fuzzy graphs, neighbourly totally irregular bipolar fuzzy graphs, highly irregular bipolar fuzzy graphs and highly totally irregular bipolar fuzzy graphs. We prove a necessary and sufficient condition under which neighbourly irregular and highly irregular bipolar fuzzy graphs are equivalent. We introduce the notion of bipolar fuzzy digraphs. We also describe the bipolar fuzzy influence graph of a social group. We have used standard definitions and terminologies in this paper. For other notations, terminologies and applications not mentioned in the paper, the readers are referred to [4], [5], [6], [7], [9], [10], [13], [14], [15], [16], [17], [18], [19], [22], [23], [25].

Section snippets

Preliminaries

In this section, we review some elementary concepts that are necessary for this paper.

By a graph, we mean a pair G=(V,E), where V is the set and E is a relation on V. The elements of V are vertices of G and the elements of E are edges of G. We write xyE to mean (x,y)E, and if e=xyE, we say x and y are adjacent. Formally, given a graph G=(V,E), two vertices x,yV are said to be neighbours, or adjacent nodes, if xyE. The number of edges, the cardinality of E, is called the size of graph

Bipolar fuzzy graphs

Definition 3.1

By a bipolar fuzzy graph G=<V,E,A,B> of a graph G=(V,E) we mean a pair G=(A,B), where A=(μAP,μAN) is a bipolar fuzzy set on V and B=(μBP,μBN) is a bipolar fuzzy relation on E such thatμBP(xy)min(μBP(x),μAP(y))andμBN(xy)max(μAN(x),μAN(y))for all xyE.

Throughout this paper, G is a crisp graph, and G is a bipolar fuzzy graph.

Definition 3.2

The number of vertices, the cardinality of V, is called the order of a bipolar fuzzy graph G=(A,B) and denoted by |V| (or O(G)), and defined byO(G)=|V|=xV1+μAP(x)+μAN(x)2

Bipolar fuzzy digraphs

A directed graph (or digraph) is a graph whose edges have direction and are called arcs (edges). Arrows on the arcs are used to encode the directional information: an arc from vertex x to vertex y indicates that one may move from x to y but not from y to x.

Let D1=(V1,E1) and D2=(V2,E2) be two digraphs. The Cartesian product of D1 and D2 gives a digraph D1×D2=(V,E) with V=V1×V2 andE={(x,x2)(x,y2)|xV1,x2y2E2}{(x1,z)(y1,z)|x1y1E1,zV2}.In this section, we will write xyE to mean xyE

Conclusions

Fuzzy graph theory is highly utilized in computer science applications. Especially in research areas of computer science including database theory, data mining, neural networks, expert systems, cluster analysis, control theory, and image capturing. In this paper, specific types of bipolar fuzzy graphs have been introduced and discussed. The bipolar fuzzy influence graph of a social group is also developed. The bipolar fuzzy graph is a generalized structure of a fuzzy graph which gives more

Acknowledgement

The author is highly thankful to the Editor-in-Chief and the referees for their constructive comments for improving the work significantly.

References (25)

  • M. Akram et al.

    Metric in bipolar fuzzy graphs

    World Applied Sciences Journal

    (2011)
  • M. Akram et al.

    Intuitionistic fuzzy hypergraphs with applications

    Information Sciences

    (2012)
  • Cited by (148)

    • Domination in bipolar fuzzy soft graphs

      2024, Journal of Intelligent and Fuzzy Systems
    View all citing articles on Scopus
    View full text