Bipolar fuzzy graphs with applications
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 , where V is the set and E is a relation on V. The elements of V are vertices of and the elements of E are edges of . We write to mean , and if , we say x and y are adjacent. Formally, given a graph , two vertices are said to be neighbours, or adjacent nodes, if . 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 of a graph we mean a pair , where is a bipolar fuzzy set on V and is a bipolar fuzzy relation on E such thatfor all .
Throughout this paper, 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 and denoted by (or ), and defined by
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 and be two digraphs. The Cartesian product of and gives a digraph with andIn this section, we will write to mean
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)
Bipolar fuzzy graphs
Information Sciences
(2011)- et al.
Interval-valued fuzzy graphs
Computers & Mathematics with Applications
(2011) Some remarks on fuzzy graphs
Pattern Recognition Letter
(1987)- et al.
Node connectivity and arc connectivity of a fuzzy graph
Information Sciences
(2010) Fuzzy graphs
- et al.
Transformation of bipolar fuzzy rough set models
Knowledge-Based Systems
(2012) Fuzzy sets
Information and Control
(1965)Similarity relations and fuzzy orderings
Information Sciences
(1971)Is there a need for fuzzy logic?
Information Sciences
(2008)- et al.
Regular bipolar fuzzy graphs
Neural Computing & Applications
(2012)
Metric in bipolar fuzzy graphs
World Applied Sciences Journal
Intuitionistic fuzzy hypergraphs with applications
Information Sciences
Cited by (148)
Providing decision-making approaches for the assessment and selection of cloud computing using bipolar complex fuzzy Einstein power aggregation operators
2024, Engineering Applications of Artificial IntelligenceA bipolar-valued fuzzy set is an intersected interval-valued fuzzy set
2024, Information SciencesBipolar reasoning in feedback pathways
2022, BioSystemsDomination in bipolar fuzzy soft graphs
2024, Journal of Intelligent and Fuzzy SystemsEvaluation of bipolar fuzzy soft sets in decision-making with a new approach
2023, Research Square