Cubic Pythagorean Fuzzy Graphs

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.


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 eld 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 (λ) and nonbelongingness (μ) with condition 0 ≤ λ + μ ≤ 1. After successful implementation of IFS in numerous elds, 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 exible constraint 0 ≤ λ 2 + μ 2 ≤ 1. PFSs have more capacity than IFS and IVFS to model the ambiguity and to handle complex coarsens. e 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 decisionmaking.
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. us, 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. us, 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 exible environment to describe those situations where degree of belongingness or nonbelongingness uctuates 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, manmade 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][25][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 intervalvalued 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][33][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][40][41][42]. e objective of our proposed work is to apply graph terminology on cubic Pythagorean fuzzy sets. e 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. e 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.

Definition 4.
e degree and total degree of a vertex r ∈ R in a CuPFG G ⌣ � (A, B) is described as is said to be complete if it satisfies the following conditions: Definition 7. Let G ⌣ 1 � (ℵ 1 , R 1 ) and G ⌣ 2 � (ℵ 2 , R 2 ) be two CuPFGs of graphs G 1 � (R 1 , E 1 ) and G 2 � (R 2 , E 2 ), respectively. e semistrong product of G ⌣ 1 and G ⌣ 2 is denoted as and is defined as follows:
en, their semistrong product G ⌣ 1 •G ⌣ 2 is shown in Figure 3.   Journal of Mathematics (d])  Journal of Mathematics 5 Definition 9. Let G ⌣ 1 and G ⌣ 2 be two CuPFGs. en, for any vertex (r 1 , 6 Journal of Mathematics proof. It is easy to prove by using Definition 9 and eorem 2.
and is defined as follows: respectively, as shown in Figure 4. eir lexicographical product G ⌣ 1°G ⌣ 2 is shown in Figure 5.

Journal of Mathematics 7
Definition 11. Let G ⌣ 1 and G ⌣ 2 be two CuPFGs. en, for any vertex (r 1 , (r, t) proof. From the Definition 11 of vertex degree of G Journal of Mathematics 9 where a 2 � |R 2 |. By following the same steps, we have □ Definition 12. Let us take two CuPFGs G ⌣ 1 and G ⌣ 2 , then the total degree for any vertex (r 1 , r 2 ) ∈ R 1 × R 2 is defined as 10 Journal of Mathematics
proof. It is easy to prove by using Definition 12 and eorem 4.

Journal of Mathematics
respectively, as shown in Figure 4. eir symmetric difference G ⌣ 1 ⊕G ⌣ 2 is shown in Figure 6.
12 Journal of Mathematics (49) proof. It is easy to prove by using Definition 15 and eorem 5.

Application
e purpose of decision-making activity is to solve an issue ended up with satisfactory results. In current section, we investigate MCDM problem in cubic PF environment, demonstrating the validity of our proposed work in realistic scenarios. To express the experts' preference among the given alternatives, preference relation is the most beneficial technique by which the ranking of alternatives can be obtained. For set of alternatives E � E 1 , E 2 , E 3 , . . . , E n , the decision makers compare each one alternative to other and obtained preference relations, respectively. If every element in the preference relations belongs to a CuPFSs then the concept of the cubic Pythagorean fuzzy preference relation (CuPFPR) can be taken as follows.

Definition 16. A CuPFPR on the set of alternatives
for all (j, k � 1, 2, 3, . . . , n). Such that, R L jk and R U jk represent the lower and upper limits of interval-valued preference degree to which the object E j is preferred to E k , ℵ L jk and ℵ U jk represent the lower and upper limits of interval-valued degree to which the object E j is not preferred to E k , μ jk is the simple fuzzy valued degree of preference, and ] jk is the simple fuzzy valued degree of nonpreference of the alternative E j over E k , respectively, with the following conditions:

Selection of the Best Captain for a Cricket Team.
Teams often consist of multitalented individuals who have the right skills and experience to deliver results. But a team leader (captain) is always needed to direct the energy of the members of team toward more effectiveness. e captain of a team provides a road map outlining the steps the team needs to achieve the preferred destination. He helps everybody to identify the roles that best fit skills and experiences. Successful captains are committed to the success of their team and its players. Captain should not just be the best players of the team or the player that the coach likes the most, but it is about having certain qualities that make that player a good leader. e selection of a captain is one of the most important and critical decisions that can be made as its influence is deep across all players of the team. A poor process and little thought invite risk and disaster, while a well-considered and inviting process can establish a foundation for progress and success. Keeping in view the role of captain, the experts aim is to choose the best among four candidates I 1 , I 2 , I 3 , I 4 on the basis of certain criterion: C 1 : personal traits: a courageous leader is someone who is not afraid to honestly and openly seek out feedback from those around them and always motivate his team. He is self-confident and a good instructor. C 2 : previous performance and experience: the past performance of a player defines his expertise and experience helps him to guide others. C 3 : professional skills: the best captain cares passionately about team's success and about their teammates with his leadership qualities. He is a good planner, courageous, and consistent. e experts of the game assigned the weight vector, W � 0.4, 0.3, 0.3 { } T , where 3 m�1 w m � 1, demonstrating the significance of that particular criterion C l (l � 1, 2, 3). e decision-making experts of the game pairwise compare the four players I h (h � 1, 2, 3, 4) under consideration with respect to the three criteria C l (l � 1, 2, 3) and provides information according to their preference in the form of cubic Pythagorean fuzzy preference relation (CuPFPR) is cubic Pythagorean fuzzy element assigned by the experts, such that interval-valued terms present the preference and rejection grade of each attribute in past and simple fuzzy terms represents the preference and rejection grade of each attribute in present. e CuPFPRs, jk4×4 , are shown in Tables 1-3 as follows. e cubic Pythagorean fuzzy influenced networks corresponding to CuPFPRs Z (l) (l � 1, 2, 3) given in Tables 1-3, are shown in Figure 7. Now, we find aggregated CuPFPR by applying CuPFWA operator.
e Pythagorean fuzzy influenced graph, corresponding to the aggregated CuPFPR given in Table 4, is displayed in Figure 8. Under the constraint on degree of truthness η jk ≥ 0.5 (i, j � 1, 2, 3, 4) on both interval value and simple value, a partial influenced graph is constructed and depicted in Figure 9.
e out-degrees out-d (I h ) (h � 1, 2, 3, 4) of all considered players in the partial graph are calculated as follows: According to membership value of out-d (I h )(h � 1, 2, 3, 4), we obtain the optimal ranking order of the four players I h (h � 1, 2, 3, 4) as follows: (64) us, I 4 is decided as the best for captaincy of the team. Now, we find aggregated CuPFPR by applying CuPFWG operator.
e Pythagorean fuzzy influenced graph, corresponding to the aggregated CuPFPR given in Table 5 is displayed in Figure 10. Within the constraint on degree of truthness η jk ≥ 0.5 (j, k � 1, 2, 3, 4) on both interval value and simple value, a partial influenced graph is constructed as depicted in Figure 11. e out-degrees out-d (I h ) (h � 1, 2, 3, 4) of all considered players in the partial graph are calculated as out  1, 2, 3, 4), we obtain ranking order of the four players I h (h � 1, 2, 3, 4) as Hence, in both cases I 4 is decided as the best for captaincy of the team.
Our proposed method in MCDM of CuPFGs is outlined in Algorithm 1.
Furthermore, for better understanding the organization of the proposed method is presented in Figure 12.

Comparative Analysis.
In order to demonstrate the effectiveness and validity of the propose method, a comparative analysis with already existing techniques is carried Table 1: CuPFPR with respect to the criteria "personal traits."  Table 2: CuPFPR with respect to the criteria "previous performance and experience."  Table 3: CuPFPR with respect to the criteria "professional skills."       corresponding to these approaches are summarized in Table 6. Hence, the final result for the ranking of the alternatives of the existing technique (PFWG operator and PFWA operator) is the same as the proposed method which indicates the rationality and effectiveness of the proposed technique.
Obviously, the method presented in this paper is more accurate, flexible, and generalized. e existing method in Pythagorean fuzzy environment is a special case of the proposed method. e originality of this decision-making approach is that we have described cubic Pythagorean fuzzy preference relations among the alternatives and developed MADM model with the interrelated alternatives by operating the corresponding graphical structures with cubic Pythagorean fuzzy information.

Conclusion
Fuzzy graph theory has effectively operated to solve a variety of issues, where information is uncertain or vague, such as producing, social networking, man-made reasoning, image processing, and network security. Several types of graphs were introduced in the past few years to model the vagueness and uncertainties in information: cubic fuzzy graphs is one of them where the data are in the form of intervals as well as simple fuzzy numbers. is research paper proposes the novel concept of CuPFGs, where the flexibility of information is increased within the fuzzy environment. Furthermore, we present several terms related to CuPFGs and develop certain operations including semistrong product, lexicographical product, and symmetric difference of two CuPFGs. In addition, some essential and mandatory constraints are illustrated. Lastly, the term CuPFPR is defined, and to show the validity of our proposed work, an application in a cubic Pythagorean fuzzy environment is solved. e presented work is more suitable to handle different issues, but it has certain limitations. In coming years, one can further extend the concept of CuPFGs to cubic Fermatean fuzzy graphs.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare no conflicts of interest.