Abstract
The purpose of the study is to explore graph theory based on cubic Pythagorean fuzzy sets. The concept of cubic Pythagorean fuzzy graphs (CuPFGs) is introduced in this research work. In addition, we define certain fundamental operations on CuPFGs including semistrong product, lexicographical product, and symmetric difference of two CuPFGs and demonstrate some of their key characteristics. Meanwhile, to investigate the preference of decision makers, cubic Pythagorean fuzzy preference relation is defined. Moreover, through a practical example, the applicability of our proposed work in multicriteria decision-making is described, and to clarify the organization of the proposed method, a frame diagram is presented.
1. Introduction
To deal with uncertain and vague information, Zadeh [1] initiated the concept of fuzzy set theory. Fuzzy sets (FSs) have many applications in the field of science and technology. Atanassov [2] put forward the idea of intuitionistic fuzzy set (IFS), a new tool of presenting ambiguous and uncertain information by associating degree of belongingness and nonbelongingness with condition . After successful implementation of IFS in numerous fields, Gorzalczany [3] introduced interval-valued fuzzy sets, where the ambiguous information is presented in the form of interval. Later, Atanassov and Gargov [4] proposed the concept of interval-valued intuitionistic fuzzy sets, in which the degree of belongingness and nonbelongingness is expressed in the form of intervals.
Preceding, the idea of IFS, Yager [5] introduced the concept Pythagorean fuzzy set (PFS) with more flexible constraint . PFSs have more capacity than IFS and IVFS to model the ambiguity and to handle complex coarsens. The applications of IFS and PFS have gained a lot of attention in areas like multicriteria decision-making (MCDM) and image processing. Furthermore, Yager [6] developed the Pythagorean membership grades in decision-making.
In many real-life situations, one has to mention the degree of belongingness both in interval value and simple fuzzy value. Such types of situations cannot be handled by FS, IFS, PFS etc. Thus, to adequately treat such types of situations, Jun et al. [7] proposed the idea of cubic sets (CSs), which is the mixture of both interval-valued fuzzy sets and fuzzy sets. Furthermore, they investigated some properties and operations on CSs. Since, CSs are centered only on degree of belongingness, it may face trouble when degree of nonbelongingness comes into account. Thus, to investigate such type of obscurity, Kaur and Garg [8, 9] proposed the concept of cubic intuitionistic fuzzy set (CuIFS) which includes two parts, one representing the degree of belongingness and nonbelongingness in IVFS and other representing the degree of belongingness and nonbelongingness in simple FS. CuIFS is a more generalized form of IFS and IVIFS because it provides a more flexible environment to describe those situations where degree of belongingness or nonbelongingness fluctuates during the procedure of decision-making and it also enlarges the level of precision.
Since CuIFS fails to deal with information where sum of squares of upper interval-values of degree of belongingness and nonbelongingness and sum of squares of simple degree of belongingness and nonbelongingness are less than 1, to deal with such problem, Abbas et al. [10] introduced the concept of CuPFSs and their application to MCDM with unknown weight information. Recently, Khan et al. [11] studied Pythagorean cubic fuzzy aggregation operators to aggregate cubic information and studied their application in MCDM. Ashraf et al. [12] presented the concept of cubic picture fuzzy sets. Furthermore, Mehmood et al. [13] proposed cubic hesitant fuzzy sets and its applications to MCDM. Muhiuddin and Al-roqi [14] introduced cubic soft sets with applications in BCK/BCI-algebras.
Graphs communicate information visually, but if the information is ambiguous and uncertain, then it can be recognized as fuzzy graphs (FGs). FGs have been shown to be a powerful tool for modeling complex problems such as communications, social networking, data hypothesis, man-made reasoning, system analysis, operations research, and economics. Kaufmann [15] initiated the idea of FG theory. Rosenfeld [16] gave the theoretical concept of FG by introducing relations between fuzzy sets and establishing the structure of FGs. Mordeson and Chang-Shyh [17] illustrated certain operations on FGs. Bhattacharya [18] added some remarks on FGs. Yeh and Bang [19] defined fuzzy relations, fuzzy graphs, and their applications to clustering analysis, fuzzy sets, and their applications in cognitive and decision process. Dogra [20] presented different types of products of FGs, and Al-Hawary [21] introduced complete FGs.
Afterward, Parvathi and Karunambigai [22] initiated the concept of intuitionistic fuzzy graphs (IFGs). Gani and Begum [23] illustrated degree, order, and size in IFGs. Akram et al. [24–26] introduced the idea of PFGs and presented specific types of PFGs, their direct sum, and their application in decision-making. Garg [23] presented Pythagorean fuzzy geometric aggregation operators for multiple attribute group decision analysis. Mandal and Ranadive [27] gave the idea of Pythagorean fuzzy preference relations and their applications in group decision-making systems. Wang and Garg [28] presented the algorithm for MADM with interactive Archimedean norm operations under Pythagorean fuzzy uncertainty. Furthermore, Akram and Naz [29] studied energy of Pythagorean fuzzy graphs (PFGs) with applications.
Moreover, Akram et al. [24] extended FGs to interval-valued fuzzy graphs (IVFGs), in which the ambiguous information is expressed in the form of interval. Naz et al. [30] gave the concept of simplified IVFGs with application. Akram et al. [31] presented certain types of interval-valued fuzzy graphs. Mohamed and Ali [32–34] developed some products on interval-valued Pythagorean fuzzy graphs and defined strong interval-valued Pythagorean fuzzy graphs. Rashid et al. [35] put forward the idea of graphical structure of cubic sets and discuss some operations on them. Moreover, Muhiuddin et al. [36,37] extended the concept cubic graphs and presented their application.
Recently, Khan et al. [38] introduced graphical structures of cubic intuitionistic fuzzy information and its application in multiattribute decision-making (MADM). For other concepts, the readers are suggested to [39–42].
The objective of our proposed work is to apply graph terminology on cubic Pythagorean fuzzy sets. The presented work is devoted to elaborate the degree, order, and size of CuPFGs. Furthermore, cubic Pythagorean fuzzy preference relation (CuPFPR) is defined and to examine the rationality of the purposed work an application is presented. Aggregation operators are commonly used to compose all the inputted individual information into a single value. So, we use cubic Pythagorean fuzzy weighted averaging (CuPFWA) operator and cubic Pythagorean fuzzy weighted geometric (CuPFWG) operator to aggregate all cubic Pythagorean fuzzy preference relation matrices. Finally, we develop a CuPFG based MADM approach to handle situations in which the attributes’ graphic structure is uncertain.
The proposed work is organized as follows: Section 2 presents the basic notions and definitions of CuPFGs, degree, total degree, order, size, and complete CuPFGs. In addition, certain operations including semistrong product, lexicographical product, and symmetric difference of two cubic Pythagorean fuzzy graphs are elaborated. Section 3 defines CuPFPR to compare the preference of the experts. Furthermore, the information of CuPFGs in MCDM is applied, and the proposed method is presented in a frame diagram. Section 4 concludes the entire work with certain remarks and further directions for future work.
2. Cubic Pythagorean Fuzzy Graphs
Definition 1. (see [10]). Let R be the universe of discourse. A cubic Pythagorean fuzzy set (CuPFS) A, defined on R is given as follows:where the functions , and , and , and denote the degree of membership and nonmembership of the element , respectively, such that and .
Definition 2. A CuPFS on is regarded as cubic Pythagorean fuzzy relation (CuPFR) in indicated aswhere , and , and , and are such that and , for all .
Definition 3. A cubic Pythagorean fuzzy graph (CuPFG) on a nonempty set is a pair , where is cubic Pythagorean fuzzy set and is cubic Pythagorean fuzzy relation on such that , , and , , and where and , for all .
Example 1. Consider a graph , where is the set of vertices and is the set of edges. The membership and nonmembership degrees of the vertices and edges are given in Figure 1.
Definition 4. The degree and total degree of a vertex in a CuPFG is described aswhere , , , , , and
, , , and , .
Definition 5. The order of CuPFG is denoted as and defined asand the size of CuPFG is denoted as and defined as
Definition 6. A CuPFG of a graph is said to be complete if it satisfies the following conditions: , , and . , , and .
Now, we present certain operations on CuPFGs along with examples and some valuable results.
Definition 7. Let and be two CuPFGs of graphs and , respectively. The semistrong product of and is denoted as and is defined as follows:(i) (ii) (iii)
Example 2. Consider two CuPFGs and on and , respectively, as shown in Figure 2. Then, their semistrong product is shown in Figure 3.
Proposition 1. The semistrong product of two CuPFGs and is also a CuPFG.
Definition 8. Let and be two CuPFGs. Then, for any vertex ,
Theorem 1. Let and be two CuPFGs. If and , then .
Definition 9. Let and be two CuPFGs. Then, for any vertex ,
Theorem 2. Let and be two CuPFGs. If(i) and , then(ii) and , then(iii) and , then(iv) and , then(v) and , then(vi) and , then
proof. It is easy to prove by using Definition 9 and Theorem 2.
Definition 10. Let and be two CuPFGs of graphs and , respectively. The lexicographical product of and is denoted as and is defined as follows:(i) (ii) (iii) (iv)