Abstract
The picture fuzzy set is an efficient mathematical model to deal with uncertain real life problems, in which a intuitionistic fuzzy set may fail to reveal satisfactory results. Picture fuzzy set is an extension of the classical fuzzy set and intuitionistic fuzzy set. It can work very efficiently in uncertain scenarios which involve more answers to these type: yes, no, abstain and refusal. In this paper, we introduce the idea of the picture fuzzy graph based on the picture fuzzy relation. Some types of picture fuzzy graph such as a regular picture fuzzy graph, strong picture fuzzy graph, complete picture fuzzy graph, and complement picture fuzzy graph are introduced and some properties are also described. The idea of an isomorphic picture fuzzy graph is also introduced in this paper. We also define six operations such as Cartesian product, composition, join, direct product, lexicographic and strong product on picture fuzzy graph. Finally, we describe the utility of the picture fuzzy graph and its application in a social network.
Highlights
IntroductionA. Zadeh [1] introduced the idea of fuzzy sets which has been successfully applied to solve the various real life decision problems, which are often uncertain
The main objective of this paper is to introduce the idea of picture fuzzy graph and its different operation
First we propose the definition of a picture fuzzy graph based on the picture fuzzy relation
Summary
A. Zadeh [1] introduced the idea of fuzzy sets which has been successfully applied to solve the various real life decision problems, which are often uncertain. “refusal of voting” means bypassing the vote This type of real life scenarios cannot be handled by intuitionistic fuzzy set. Rashmanlou et al [32] introduced several operations such as direct product, lexicographic product and strong product on intuitionistic fuzzy graphs. As uncertainties are well expressed using picture fuzzy set, picture fuzzy graph would be prominent research direction for modeling the uncertain optimization problems This motivated us to introduce the idea picture fuzzy graph its related operations. Six operations on picture fuzzy graphs, viz. Cartesian product, composition, join, direct product, lexicographic and strong product are introduced in this paper. We describe the utility of picture fuzzy graph and its application in social network
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