Abstract
The theory developed in this article is based on graphs of cubic intuitionistic fuzzy sets (CIFS) called cubic intuitionistic fuzzy graphs (CIFGs). This graph generalizes the structures of fuzzy graph (FG), intuitionistic fuzzy graph (IFG), and interval-valued fuzzy graph (IVFG). Moreover, several associated concepts are established for CIFG, such as the idea subgraphs, degree of CIFG, order of CIFG, complement of CIFG, path in CIFG, strong CIFG, and the concept of bridges for CIFGs. Furthermore, the generalization of CIFG is proved with the help of some remarks. In addition, the comparison among the existing and the proposed ideas is carried out. Finally, an application of CIFG in decision-making problem is studied, and some future study is proposed.
Highlights
Jun et al [1] proposed cubic set (CS) and started a new research area
Motivated by the existing theory, we proposed the framework of cubic intuitionistic fuzzy sets (CIFSs) and cubic intuitionistic fuzzy graphs (CIFGs)
We introduce some basic concepts about fuzzy set, fuzzy graph, intuitionistic fuzzy set, and intuitionistic fuzzy graph, which provide a base for our graphical work on CIFG. roughout this manuscript, X denotes the universe of discourse and M, Ŋ are considered to be two mappings on [0, 1] intervals denoting the membership and nonmembership grades, respectively, of an element
Summary
Jun et al [1] proposed cubic set (CS) and started a new research area. A CS is a mixture of two concepts known as fuzzy set (FS) and interval-valued fuzzy set (IVFS). e concept of CS draws the attentions of researchers and some potential works in this direction have been done; for example, the idea of CS was proposed in semigroup theory by Khan et al [2], as well as some KU-ideal by Yaqoob et al [3], and KU-algebras are developed for CS by Lu and Ye [4]; the similarity measures of CSs have been proposed and applied in decision-making problem. e framework of cubic neutrosophic sets is proposed by Jun et al [5], while some pattern recognition problems are solved using neutrosophic sets by Ali et al [6]. e concept of cubic soft sets was proposed by Muhiuddin and Al-roqi [7], which was further utilized by Muhiuddin et al [8]. e theory of G-algebras is studied by Jun and Khan in [9] and by Jana and Senapati [10] along with the concepts of ideal in semigroups. E concept of CS draws the attentions of researchers and some potential works in this direction have been done; for example, the idea of CS was proposed in semigroup theory by Khan et al [2], as well as some KU-ideal by Yaqoob et al [3], and KU-algebras are developed for CS by Lu and Ye [4]; the similarity measures of CSs have been proposed and applied in decision-making problem. E concept of IFG was a generalization of fuzzy graphs (FGs) proposed by Kauffman and Rosenfeld [19, 20] after Zadeh’s exemplary work in [16]. Motivated by the existing theory, we proposed the framework of cubic intuitionistic fuzzy sets (CIFSs) and cubic intuitionistic fuzzy graphs (CIFGs).
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