# Cubic Pythagorean Fuzzy Graphs

Authors: G. Muhiuddin; Saira Hameed; Afeefa Maryam; Uzma Ahmad
Journal: Journal of Mathematics
 (2022)

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## 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.


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## Body

## 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 λ and nonbelongingness μ with condition 0≤λ+μ≤1. 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 0≤λ2+μ2≤1. 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: Section2 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:(1)A=r,ℜALr,ℜAUr,ℵALr,ℵAUr,μr,νr|r∈R,where the functions ℜAL:R⟶D0,1, ℜAU:R⟶D0,1 and ℵAL:R⟶D0,1, ℵAU:R⟶D0,1 and μA:R⟶0,1, and νA:R⟶0,1 denote the degree of membership and nonmembership of the element r∈R, respectively, such that 0≤ℜAUr2+ℵAUr2≤1 and 0≤μr2+νr2≤1.Definition 2.
A CuPFSB on R×R is regarded as cubic Pythagorean fuzzy relation (CuPFR) in R indicated as(2)B=r,s,ℜALr,s,ℜAUr,s,ℵALr,s,ℵAUr,s,μr,s,νr,s|r,s∈R×R,where ℜBL:R⟶D0,1, ℜBU:R⟶D0,1 and ℵBL:R⟶D0,1, ℵBU:R⟶D0,1 and μB:R⟶0,1, and νB:R⟶0,1 are such that 0≤ℜBUr,s2+ℵBUr,s2≤1 and 0≤μr,s2+νr,s2≤1, for all r,s∈E.Definition 3.
A cubic Pythagorean fuzzy graph (CuPFG)G⌣ on a nonempty set R is a pair G⌣=A,B, where A is cubic Pythagorean fuzzy set and B is cubic Pythagorean fuzzy relation on R such thatℜBLr,s≤ℜALr∧ℜALs, ℜBUr,s≤ℜAUr∧ℜAUs, and ℵBLr,s≤ℵALr∨ℵALsℵBUr,s≤ℵAUr∨ℵAUs, μBr,s≤μAr∧μAs, and νBr,s≤νAr∨νAswhere 0≤ℜBUr,s2+ℵBUr,s2≤1 and 0≤μr,s2+νr,s2≤1, for all r,s∈E.Example 1.
Consider a graphG=R,E, where R=r1,r2,r3,r4 is the set of vertices and E=r1r2,r2r3,r3r4,r1r4 is the set of edges. The membership and nonmembership degrees of the vertices and edges are given in Figure 1.Figure 1
A cubic Pythagorean fuzzy graphG⌣.Definition 4.
The degree and total degree of a vertexr∈R in a CuPFG G⌣=A,B is described as(3)dG⌣r=dℜALr,dℜAUr,dℵALr,dℵAUr,dμAr,dνAr,(4)tdG⌣r=tdℜALr,tdℜAUr,tdℵALr,tdℵAUr,tdμAr,tdνAr,where dℜALr=∑r≠s,s∈EℜBLrs, dℜAUr=∑r≠s,s∈EℜBUrs,dℵALr=∑r≠s,s∈EℵBLrs, dℵAUr=∑r≠s,s∈EℵBUrs, dμAr=∑r≠s,s∈EμBrs, dνAr=∑r≠s,s∈EνBrs, and
tdℜALr=∑r≠s,sℜBLrs+ℜALr, tdℜAUr=∑r≠s,s∈EℜBUrs+ℜAUr,tdℵALr=∑r≠s,s∈EℵBLrs+ℵALr, tdℵAUr=∑r≠s,s∈EℵBUrs+ℵALr, and tdμAr=∑r≠s,s∈EμBrs+μAr, tdνAr=∑r≠s,s∈EνBrs+μAr.Definition 5.
The order of CuPFGG⌣=A,B is denoted as OG⌣ and defined as(5)OG⌣=∑r∈RℜALr,∑r∈RℜAUr,∑r∈RℵALr,∑r∈RℵAUr,∑r∈RμAr,∑r∈RνAr,and the size of CuPFG is denoted as SG⌣ and defined as(6)SG⌣=∑r≠s,s∈EℜALrs,∑r≠s,s∈EℜAUrs,∑r≠s,s∈EℵALrs,∑r≠s,s∈EℵAUrs,∑r≠s,s∈EμArs,∑r≠s,s∈EνArs.Definition 6.
A CuPFGG⌣=A,B of a graph G=R,E is said to be complete if it satisfies the following conditions:ℜBLr,s=ℜALr∧ℜALs, ℜBLr,s=ℜALr∧ℜALs, and ℵBLr,s=ℵALr∨ℵALs.ℵBUr,s=ℵAUr∨ℵAUs, μBr,s=μAr∧μAs, and νBr,s=νAr∨νAs.Now, we present certain operations on CuPFGs along with examples and some valuable results.Definition 7.
LetG⌣1=ℵ1,ℜ1 and G⌣2=ℵ2,ℜ2 be two CuPFGs of graphs G1=R1,E1 and G2=R2,E2, respectively. The semistrong product of G⌣1 and G⌣2 is denoted as G⌣1•G⌣2=ℵ1•ℵ2,ℜ1•ℜ2 and is defined as follows:(i)
ℜA1L•ℜA2Lr1,r2=ℜA1Lr1∧ℜA2Lr2ℜA1U•ℜA2Ur1,r2=ℜA1Ur1∧ℜA2Ur2ℵA1L•ℵA2Lr1,r2=ℵA1Lr1∨ℵA2Lr2ℵA1U•ℵA2Ur1,r2=ℵA1Ur1∨ℵA2Ur2μA1•μA2r1,r2=μA1r1∧μA2r2νA1•νA2r1,r2=νA1r1∨νA2r2,for allr1,r2∈R1×R2(ii)
ℜB1L•ℜB2Lr,r2,r,s2=ℜA1Lr∧ℜB2Lr2,s2ℜB1U•ℜB2Ur,r2,r,s2=ℜA1Ur∧ℜB2Ur2,s2ℵB1L•ℵB2Lr,r2,r,s2=ℵA1Lr1∧ℵB2Lr2,s2ℵB1U•ℵB2Ur,r2,r,s2=ℵA1Ur1∧ℵB2Ur2,s2μB1•μB2r,r2,r,s2=μA1r1∧μB2r2,s2νB1•νB2r,r2,r,s2=νA1r1∧νB2r2,s2for allr∈R1,for allr2s2∈E2(iii)
ℜB1L•ℜB2Lr1,r2,s1,s2=ℜB1Lr1,s1∧ℜB2Lr2,s2ℜB1U•ℜB2Ur1,r2,s1,s2=ℜB1Ur1,s1∧ℜB2Ur2,s2ℵB1L•ℵB2Lr1,r2,s1,s2=ℵB1Lr1,s1∧ℵB2Lr2,s2ℵB1U•ℵB2Ur1,r2,s1,s2=ℵB1Ur1,s1∧ℵB2Ur2,s2μB1•μB2r1,r2,s1,s2=μB1r1,s1∧μB2r2,s2νB1•νB2r1,r2,s1,s2=νB1r1,s1∧νB2r2,s2for allr1s1∈E1,for allr2s2∈E2Example 2.
Consider two CuPFGsG⌣1 and G⌣2 on R1=r1,s1 and R2=r2,s2, respectively, as shown in Figure 2. Then, their semistrong product G⌣1•G⌣2 is shown in Figure 3.Figure 2
CuPFGsG⌣1 and G⌣2.Figure 3
Semistrong product of two CuPFGsG⌣1 and G⌣2.Proposition 1.
The semistrong productG⌣1•G⌣2 of two CuPFGs G⌣1 and G⌣2 is also a CuPFG.Definition 8.
LetG⌣1 and G⌣2 be two CuPFGs. Then, for any vertex r1,r2∈R1×R2,(7)dℜLG⌣1•G⌣2r1,r2=∑r1,r2s1,s2∈R1×R2ℜB1L•ℜB2Lr1,r2,s1,s2=∑r1=s1,r2s2∈E2ℜA1Lr1∧ℜB2Lr2s2+∑r1s1∈E1,r2s2∈E2ℜB1Lr1s1∧ℜB2Lr2s2,dℜUG⌣1•G⌣2r1,r2=∑r1,r2s1,s2∈R1×R2ℜB1U•ℜB2Ur1,r2,s1,s2=∑r1=s1,r2s2∈E2ℜA1Ur1∧ℜB2Ur2s2+∑r1s1∈E1,r2s2∈E2ℜB1Ur1s1∧ℜB2Ur2s2,(8)dℵLG⌣1•G⌣2r1,r2=∑r1,r2s1,s2∈R1×R2ℵB1L•ℵB2Lr1,r2,s1,s2=∑r1=s1,r2s2∈E2ℵA1Lr1∧ℵB2Lr2s2+∑r1s1∈E1,r2s2∈E2ℵB1Lr1s1∧ℵB2Lr2s2,(9)dℵUG⌣1•G⌣2r1,r2=∑r1,r2s1,s2∈R1×R2ℵB1U•ℵB2Ur1,r2,s1,s2=∑r1=s1,r2s2∈E2ℵA1Ur1∧ℵB2Ur2s2+∑r1s1∈E1,r2s2∈E2ℵB1Ur1s1∧ℵB2Ur2s2,(10)dμG⌣1•G⌣2r1,r2=∑r1,r2s1,s2∈R1×R2μB1L•μB2Lr1,r2,s1,s2=∑r1=s1,r2s2∈E2μA1r1∧μB2r2s2+∑r1s1∈E1,r2s2∈E2μB1r1s1∧μB2r2s2,(11)dνG⌣1•G⌣2r1,r2=∑r1,r2s1,s2∈R1×R2νB1•νB2r1,r2,s1,s2=∑r1=s1,r2s2∈E2νA1r1∧νB2r2s2+∑r1s1∈E1,r2s2∈E2νB1r1s1∧νB2r2s2.Theorem 1.
LetG⌣1 and G⌣2 be two CuPFGs. If ℜA1L≥ℜB2L,ℵA1L≤ℵB2L,ℜA1L≥ℜB2U,ℵA1U≤ℵB2U,μA1≥μB2,νA1≤νB2 and ℜA2L≥ℜB1L,ℵA2L≤ℵB1L,ℜA2L≥ℜB1U,ℵA2U≤ℵB1U,μA2≥μB1,νA2≤νB1, then dG⌣1•G⌣2r1,r2=dG⌣1r1+dG⌣2r2,for allr1,r2∈R1×R2.Definition 9.
LetG⌣1 and G⌣2 be two CuPFGs. Then, for any vertex r1,r2∈R1×R2,(12)tdℜLG⌣1•G⌣2r1,r2=∑r1,r2s1,s2∈R1×R2ℜB1L•ℜB2Lr1,r2,s1,s2+ℜA1L•ℜA2Lr1,r2=∑r1=s1,r2s2∈E2ℜA1Lr1∧ℜB2Lr2s2+∑r1s1∈E1,r2s2∈E2ℜB1Lr1s1∧ℜB2Lr2s2+ℜA1Lr1∧ℜA2Lr2,(13)tdℜUG⌣1•G⌣2r1,r2=∑r1,r2s1,s2∈R1×R2ℜB1U•ℜB2Ur1,r2,s1,s2+ℜA1U•ℜA2Ur1,r2=∑r1=s1,r2s2∈E2ℜA1Ur1∧ℜB2Ur2s2+∑r1s1∈E1,r2s2∈E2ℜB1Lr1s1∧ℜB2Ur2s2+ℜA1Ur1∧ℜA2Ur2,(14)tdℵLG⌣1•G⌣2r1,r2=∑r1,r2s1,s2∈R1×R2ℵB1L•ℵB2Lr1,r2,s1,s2+ℵA1L•ℵA2Lr1,r2=∑r1=s1,r2s2∈E2ℵA1Lr1∧ℵB2Lr2s2+∑r1s1∈E1,r2s2∈E2ℵB1Lr1s1∧ℵB2Lr2s2+ℵA1Lr1∧ℵA2Lr2,(15)tdℵUG⌣1•G⌣2r1,r2=∑r1,r2s1,s2∈R1×R2ℵB1U•ℵB2Ur1,r2,s1,s2+ℵA1U•ℵA2Ur1,r2=∑r1=s1,r2s2∈E2ℵA1Ur1∧ℵB2Ur2s2+∑r1s1∈E1,r2s2∈E2ℵB1Ur1s1∧ℵB2Ur2s2+ℵA1Ur1∧ℵA2Ur2,(16)tdμG⌣1•G⌣2r1,r2=∑r1,r2s1,s2∈R1×R2μB1•μB2r1,r2,s1,s2+μA1•μA2r1,r2=∑r1=s1,r2s2∈E2μA1r1∧μB2r2s2+∑r1s1∈E1,r2s2∈E2μB1r1s1∧μB2r2s2+μA1r1∧μA2r2,(17)tdνG⌣1•G⌣2r1,r2=∑r1,r2s1,s2∈R1×R2νB1•νB2r1,r2,s1,s2+νA1•νA2r1,r2=∑r1=s1,r2s2∈E2νA1r1∧νB2r2s2+∑r1s1∈E1,r2s2∈E2νB1r1s1∧νB2r2s2+νA1r1∧νA2r2.Theorem 2.
LetG⌣1 and G⌣2 be two CuPFGs. If(i)
ℜA1L≥ℜB2L and ℜA2L≥ℜB1L, then(18)tdℜALG⌣1•G⌣2r1,r2=tdℜALG⌣1r1+tdℜALG⌣2r2−ℜA1Lr1∨ℜA2Lr2.(ii)
ℜA1U≥ℜB2U and ℜA2U≥ℜB1U, then(19)tdℜAUG⌣1•G⌣2r1,r2=tdℜAUG⌣1r1+tdℜAUG⌣2r2−ℜA1Ur1∨ℜA2Ur2.(iii)
ℵA1L≥ℵB2L and ℵA2L≥ℵB1L, then(20)tdℵALG⌣1•G⌣2r1,r2=tdℵALG⌣1r1+tdℵALG⌣2r2−ℵA1Lr1∨ℵA2Lr2.(iv)
ℵA1U≥ℵB2U and ℵA2U≥ℵB1U, then(21)tdℵAUG⌣1•G⌣2r1,r2=tdℵAUG⌣1r1+tdℵAUG⌣2r2−ℵA1Ur1∨ℵA2Ur2.(v)
μA1≥μB2 and μA2≥μB1, then(22)tdμAG⌣1•G⌣2r1,r2=tdμAG⌣1r1+tdμAG⌣2r2−μA1r1∨μA2r2.(vi)
νA1≥νB2 and νA2≥νB1, then(23)tdνAG⌣1•G⌣2r1,r2=tdνAG⌣1r1+tdνAG⌣2r2−νA1r1∨νA2r2.proof.
It is easy to prove by using Definition9 and Theorem 2.Definition 10.
LetG⌣1=ℵ1,ℜ1 and G⌣2=ℵ2,ℜ2 be two CuPFGs of graphs G1=R1,E1 and G2=R2,E2, respectively. The lexicographical product of G⌣1 and G⌣2 is denoted as G⌣1°G⌣2=ℵ1∘ℵ2,ℜ1∘ℜ2 and is defined as follows:(i)
ℜA1L∘ℜA2Lr1,r2=ℜA1Lr1∧ℜA2Lr2ℜA1U∘ℜA2Ur1,r2=ℜA1Ur1∧ℜA2Ur2ℵA1L∘ℵA2Lr1,r2=ℵA1Lr1∧ℵA2Lr2ℵA1U∘ℵA2Ur1,r2=ℵA1Ur1∧ℵA2Ur2μA1∘μA2r1,r2=μA1r1∧μA2r2νA1∘νA2r1,r2=νA1r1∧νA2r2for allr1,r2∈R1×R2(ii)
ℜB1L∘ℜB2Lr,r2,r,s2=ℜA1Lr∧ℜB2Lr2,s2ℜB1U∘ℜB2Ur,r2,r,s2=ℜA1Ur∧ℜB2Ur2,s2ℵB1L∘ℵB2Lr,r2,r,s2=ℵA1Lr∧ℵB2Lr2,s2ℵB1U∘ℵB2Ur,r2,r,s2=ℵA1Ur∧ℵB2Ur2,s2μB1∘μB2r,r2,r,s2=μA1r∧μB2r2,s2νB1∘νB2r,r2,r,s2=νA1r∧νB2r2,s2for allr∈R1,for allr2s2∈E2(iii)
ℜA1L∘ℜB2Lr1,z,s1,z=ℜB1Lr1,s1∧ℜA2LzℜA1U∘ℜB2Ur1,z,s1,z=ℜB1Ur1,s1∧ℜA2UzℵA1L∘ℵB2Lr1,z,s1,z=ℵB1Lr1,s1∧ℵA2LzℵA1U∘ℵB2Ur1,z,s1,z=ℵB1Ur1,s1∧ℵA2UzμB1∘μB2r1,z,s1,z=μB1r1,s1∧μA2zνB1∘νB2r1,z,s1,z=νB1r1,s1∧νA2zfor allz∈R2,for allr1s1∈E1(iv)
ℜB1L∘ℜB2Lr1,r2,s1,s2=ℜB1Lr1,s1∧ℜA2Lr2∧ℜA2Ls2ℜB1U∘ℜB2Ur1,r2,s1,s2=ℜB1UUr1,s1∧ℜA2Ur2∧ℜA2Us2ℵB1L∘ℵB2Lr1,r2,s1,s2=ℵB1Lr1,s1∧ℵA2Lr2∧ℵA2Ls2ℵB1U∘ℵB2Ur1,r2,s1,s2=ℵB1Ur1,s1∧ℵA2Ur2∧ℵA2Us2μB1∘μB2r1,r2,s1,s2=μB1r1,s1∧μB2r2∧μB2s2μB1∘μB2r1,r2,s1,s2=μB1r1,s1∧μB2r2∧μB2s2for allr1s1∈E1,r2≠s2Example 3.
Consider two CuPFGsG⌣1 and G⌣2 on R1=r,s and R2=t,u,v, respectively, as shown in Figure 4. Their lexicographical product G⌣1°G⌣2 is shown in Figure 5.Figure 4
CuPFGsG⌣1 and G⌣2.Figure 5
Lexicographical product of two CuPFGsG⌣1 and G⌣2.Proposition 2.
The lexicographical productG⌣1°G⌣2 of two CuPFGs G⌣1 and G⌣2 is also a CuPFG.Definition 11.
LetG⌣1 and G⌣2 be two CuPFGs. Then, for any vertex r1,r2∈R1×R2,(24)dℜLG⌣1°G⌣2r1,r2=∑r1,r2s1,s2∈R1×R2ℜB1L∘ℜB2Lr1,r2,s1,s2,=∑r1=s1,r2s2∈E2ℜA1Lr1∧ℜB2Lr2s2+∑r2=s2,r1s1∈E1ℜA2Lr2∧ℜB1Lr1s1+∑r1s1∈E1,r2≠s2ℜB1Lr1s1∧ℜB2Lr2∧ℜB2Ls2,(25)dℜUG⌣1°G⌣2r1,r2=∑r1,r2s1,s2∈R1×R2ℜB1U∘ℜB2Ur1,r2,s1,s2,=∑r1=s1,r2s2∈E2ℜA1Ur1∧ℜB2Ur2s2+∑r2=s2,r1s1∈E1ℜA2Ur2∧ℜB1Ur1s1+∑r1s1∈E1,r2≠s2ℜB1Ur1s1∧ℜB2Ur2∧ℜB2Us2,(26)dℵG⌣1°G⌣2r1,r2=∑r1,r2s1,s2∈R1×R2ℵB1∘ℵB2r1,r2,s1,s2,=∑r1=s1,r2s2∈E2ℵA1Lr1∧ℵB2Lr2s2+∑r2=s2,r1s1∈E1ℵA2Lr2∧ℵB1Lr1s1+∑r1s1∈E1,r2≠s2ℵB1Lr1s1∧ℵB2Lr2∧ℵB2Ls2,(27)dℵUG⌣1°G⌣2r1,r2=∑r1,r2s1,s2∈R1×R2ℵB1U∘ℵB2Ur1,r2,s1,s2,=∑r1=s1,r2s2∈E2ℵA1Ur1∧ℵB2Ur2s2+∑r2=s2,r1s1∈E1ℵA2Ur2∧ℵB1Ur1s1+∑r1s1∈E1,r2≠s2ℵB1Ur1s1∧ℵB2Ur2∧ℵB2Us2,(28)dμLG⌣1°G⌣2r1,r2=∑r1,r2s1,s2∈R1×R2μB1L∘μB2Lr1,r2,s1,s2,=∑r1=s1,r2s2∈E2μA1r1∧μB2r2s2+∑r2=s2,r1s1∈E1μA2r2∧μB1r1s1+∑r1s1∈E1,r2≠s2μB1r1s1∧μB2r2∧μB2s2,(29)dνG⌣1°G⌣2r1,r2=∑r1,r2s1,s2∈R1×R2νB1∘νB2r1,r2,s1,s2,=∑r1=s1,r2s2∈E2νA1r1∧νB2r2s2+∑r2=s2,r1s1∈E1νA2r2∧νB1r1s1+∑r1s1∈E1,r2≠s2νB1r1s1∧νB2r2∧νB2s2.Theorem 3.
LetG⌣1 and G⌣2 be two CuPFGs. If ℜA1L≥ℜB2L,ℵA1L≤ℵB2L,ℜA1L≥ℜB2U,ℵA1U≤ℵB2U,μA1≥μB2,νA1≤νB2 and ℜA2L≥ℜB1L,ℵA2L≤ℵB1L,ℜA2L≥ℜB1U,ℵA2U≤ℵB1U,μA2≥μB1,νA2≤νB1, then dG⌣1°G⌣2r1,r2=a2dG⌣1r1+dG⌣2r2,where,a2=R2for allr1,r2∈R1×R2.proof.
From the Definition11 of vertex degree of G⌣1°G⌣2,(30)dℜLG⌣1°G⌣2r1,r2=∑r1,r2s1,s2∈R1×R2ℜB1L∘ℜB2Lr1,r2,s1,s2=∑r1=s1,r2s2∈E2ℜA1Lr1∧ℜB2Lr2s2+∑r2=s2,r1s1∈E1ℜA2Lr2∧ℜB1Lr1s1+∑r1s1∈E1,r2≠s2ℜB1Lr1∧ℜB1Ls1∧ℜB2Lr2s2=∑r2s2∈E2ℜA2Lr2s2+∑r1s1∈E1ℜA1Lr1s1+∑r1s1∈E1ℜA1Lr1s1by usingℜA1L≥ℜB2LandℜA2L≥ℜB1L=a2dℜLG⌣1r1+dℜLG⌣2r2,where a2=R2. By following the same steps, we have(31)dℜUG⌣1°G⌣2r1,r2=a2dℜUG⌣1r1+dℜUG⌣2r2,dℵLG⌣1°G⌣2r1,r2=a2dℵLG⌣1r1+dℵLG⌣2r2,dℵUG⌣1°G⌣2r1,r2=a2dℵUG⌣1r1+dℵUG⌣2r2,dμG⌣1°G⌣2r1,r2=a2dμG⌣1r1+dμG⌣2r2,dνG⌣1°G⌣2r1,r2=a2dνG⌣1r1+dνG⌣2r2.Definition 12.
Let us take two CuPFGsG⌣1 and G⌣2, then the total degree for any vertex r1,r2∈R1×R2 is defined as(32)tdℜLG⌣1°G⌣2r1,r2=∑r1,r2s1,s2∈R1×R2ℜB1L°ℜB2Lr1,r2,s1,s2+ℜA1L°ℜA2Lr1,r2∑r1=s1,r2s2∈E2ℜA1Lr1∧ℜB2Lr2s2+∑r2=s2,r1s1∈E1ℜA2Lr2∧ℜB1Lr1s1+∑r1s1∈E1,r2≠s2ℜB1Lr1∧ℜB1Ls1∧ℜB2Lr2s2+ℜA1Lr1∧ℜA2Lr2,(33)tdℜUG⌣1°G⌣2r1,r2=∑r1,r2s1,s2∈R1×R2ℜB1U°ℜB2Ur1,r2,s1,s2+ℜA1L°ℜA2Ur1,r2∑r1=s1,r2s2∈E2ℜA1Ur1∧ℜB2Lr2s2+∑r2=s2,r1s1∈E1ℜA2Ur2∧ℜB1Ur1s1+∑r1s1∈E1,r2≠s2ℜB1Ur1∧ℜB1Us1∧ℜB2Ur2s2+ℜA1Ur1∧ℜA2Ur2,(34)tdℵLG⌣1°G⌣2r1,r2=∑r1,r2s1,s2∈R1×R2ℵB1L°ℵB2Lr1,r2,s1,s2+ℵA1L°ℵA2Lr1,r2∑r1=s1,r2s2∈E2ℵA1Lr1∧ℵB2Lr2s2+∑r2=s2,r1s1∈E1ℵA2Lr2∧ℵB1Lr1s1+∑r1s1∈E1,r2≠s2ℵB1Lr1∧ℵB1Ls1∧ℵB2Lr2s2+ℵA1Lr1∧ℵA2Lr2,(35)tdℵUG⌣1°G⌣2r1,r2=∑r1,r2s1,s2∈R1×R2ℵB1U°ℵB2Ur1,r2,s1,s2+ℵA1U°ℵA2Ur1,r2∑r1=s1,r2s2∈E2ℵA1Ur1∧ℵB2Ur2s2+∑r2=s2,r1s1∈E1ℵA2Ur2∧ℵB1Ur1s1+∑r1s1∈E1,r2≠s2ℵB1Ur1∧ℵB1Us1∧ℵB2Ur2s2+ℵA1Ur1∧ℵA2Ur2,(36)tdμG⌣1°G⌣2r1,r2=∑r1,r2s1,s2∈R1×R2μB1°μB2r1,r2,s1,s2+μA1°μA2r1,r2∑r1=s1,r2s2∈E2μA1r1∧μB2r2s2+∑r2=s2,r1s1∈E1μA2r2∧μB1r1s1+∑r1s1∈E1,r2≠s2μB1r1∧μB1s1∧μB2r2s2+μA1r1∧μA2r2,(37)tdνG⌣1°G⌣2r1,r2=∑r1,r2s1,s2∈R1×R2νB1°νB2r1,r2,s1,s2+νA1°νA2r1,r2∑r1=s1,r2s2∈E2νA1r1∧νB2r2s2+∑r2=s2,r1s1∈E1νA2r2∧νB1r1s1+∑r1s1∈E1,r2≠s2νB1r1∧νB1s1∧νB2r2s2+νA1r1∧νA2r2.Theorem 4.
Let us take two CuPFGsG⌣1 and G⌣2. If(i)
ℜA1L≥ℜB2L and ℜA2L≥ℜB1L, then(38)tdℜALG⌣1°G⌣2r1,r2=a2tdℜALG⌣1r1+tdℜALG⌣2r2−a2−1ℜA1Lr1−ℜA1Lr1∨ℜA2Lr2.(ii)
ℜA1U≥ℜB2U and ℜA2U≥ℜB1U, then(39)tdℜAUG⌣1°G⌣2r1,r2=a2tdℜAUG⌣1r1+tdℜAUG⌣2r2−a2−1ℜA1Ur1−ℜA1Ur1∨ℜA2Ur2.(iii)
ℵA1L≥ℵB2L and ℵA2L≥ℵB1L, then(40)tdℵALG⌣1°G⌣2r1,r2=a2tdℵALG⌣1r1+tdℵALG⌣2r2−a2−1ℵA1Lr1−ℵA1Lr1∨ℵA2Lr2.(iv)
ℵA1U≥ℵB2U and ℵA2U≥ℵB1U, then(41)tdℵAUG⌣1°G⌣2r1,r2=a2tdℵAUG⌣1r1+tdℵAUG⌣2r2−a2−1ℵA1Ur1−ℵA1Ur1∨ℵA2Ur2.(v)
μA1≥μB2 and μA2≥μB1, then(42)tdμAG⌣1°G⌣2r1,r2=a2tdμAG⌣1r1+tdμAG⌣2r2−a2−1μA1r1−μA1r1∨μA2r2.(vi)
νA1≥νB2 and νA2≥νB1, then(43)tdνAG⌣1°G⌣2r1,r2=a2tdνAG⌣1r1+tdνAG⌣2r2−a2−1μA1r1−νA1r1∨νA2r2.where a2=R2.proof.
It is easy to prove by using Definition12 and Theorem 4.Definition 13.
LetG⌣1=ℵ1,ℜ1 and G⌣2=ℵ2,ℜ2 be two CuPFGs of graphs G1=R1,E1 and G2=R2,E2, respectively. The symmetric difference of G⌣1 and G⌣2 is denoted by G⌣1⊕G⌣2=ℵ1⊕ℵ2,ℜ1⊕ℜ2 and is defined as follows:(i)
ℜA1L⊕ℜA2Lr1,r2=ℜA1Lr1∧ℜA2Lr2ℜA1U⊕ℜA2Ur1,r2=ℜA1Ur1∧ℜA2Ur2ℵA1L⊕ℵA2Lr1,r2=ℵA1Lr1∧ℵA2Lr2ℵA1U⊕ℵA2Ur1,r2=ℵA1Ur1∧ℵA2Ur2μA1⊕μA2r1,r2=μA1r1∧μA2r2νA1⊕νA2r1,r2=νA1r1∧νA2r2for allr1,r2∈R(ii)
ℜB1L⊕ℜB2Lr,r2,r,s2=ℜA1Lr∧ℜB2Lr2,s2ℜB1U⊕ℜB2Ur,r2,r,s2=ℜA1Ur∧ℜB2Ur2,s2ℵB1L⊕ℵB2Lr,r2,r,s2=ℵA1Lr∧ℵB2Lr2,s2ℵB1U⊕ℵB2Ur,r2,r,s2=ℵA1Ur∧ℵB2Ur2,s2μB1U⊕ℵB2Ur,r2,r,s2=ℵA1Ur∧ℵB2Ur2,s2νB1⊕νB2r,r2,r,s2=νA1r1∨νB2r2,s2,for allr∈R1,for allr2s2∈E2(iii)
ℜA1L⊕ℜB2Lr1,z,s1,z=ℜB1Lr1,s1∧ℜA2LzℜA1U⊕ℜB2Ur1,z,s1,z=ℜB1Ur1,s1∧ℜA2UzℵA1L⊕ℵB2Lr1,z,s1,z=ℵB1Lr1,s1∧ℵA2LzℵA1U⊕ℵB2Ur1,z,s1,z=ℵB1Ur1,s1∧ℵA2UzμB1⊕μB2r1,z,s1,z=μB1r1,s1∧μA2zνB1⊕νB2r1,z,s1,z=νB1r1,s1∧νA2zfor allz∈R2,for allr1s1∈E1(iv)
ℜB1L⊕ℜB2Lr1,r2,s1,s2=ℜB1Lr1,s1∧ℜA2Lr2∧ℜA2Ls2∀r1s1∉E1,r2s2∈E2or=ℜB2Lr2,s2∧ℜA1Lr1∧ℜA1Ls1∀r2s2∉E2,r1s1∈E1ℜB1U⊕ℜB2Ur1,r2,s1,s2=ℜB1Ur1,s1∧ℜA2Ur2∧ℜA2Us2∀r1s1∉E1,r2s2∈E2or=ℜB2Ur2,s2∧ℜA1Ur1∧ℜA1Us1∀r2s2∉E2,r1s1∈E1ℵB1L⊕ℵB2Lr1,r2,s1,s2=ℵB1Lr1,s1∧ℵA2Lr2∧ℵA2Ls2∀r1s1∉E1,r2s2∈E2or=ℵB2Lr2,s2∧ℵA1Lr1∧ℵA1Ls1∀r2s2∉E2,r1s1∈E1ℵB1U⊕ℵB2Ur1,r2,s1,s2=ℵB1Ur1,s1∧ℵA2Ur2∧ℵA2Us2∀r1s1∉E1,r2s2∈E2or=ℵB2Ur2,s2∧ℵA1Ur1∧ℵA1Us1∀r2s2∉E2,r1s1∈E1μB1⊕μB2r1,r2,s1,s2=μB1r1,s1∧μA2r2∧μA2s2∀r1s1∉E1,r2s2∈E2or=μB2r2,s2∧μA1r1∧μA1s1∀r2s2∉E2,r1s1∈E1νB1⊕νB2r1,r2,s1,s2=νB1r1,s1∧νA2r2∧νA2s2∀r1s1∉E1,r2s2∈E2or=νB2r2,s2∧νA1r1∧νA1s1∀r2s2∉E2,r1s1∈E1Example 4.
Consider two CuPFGsG⌣1 and G⌣2 on R1=r,s and R2=t,u,v, respectively, as shown in Figure 4. Their symmetric difference G⌣1⊕G⌣2 is shown in Figure 6.Figure 6
Symmetric difference of two CuPFGsG⌣1 and G⌣2.Proposition 3.
The symmetric differenceG⌣1⊕G⌣2 of two CuPFGs G⌣1 and G⌣2 of G1 and G2 is also a CuPFG.Definition 14.
LetG⌣1 and G⌣2 be two CuPFGs. Then, for any vertex r1,r2∈V1×V2,(44)dℜLG⌣1⊕G⌣2r1,r2=∑r1,r2s1,s2∈R1×R2ℜB1L⊕ℜB2Lr1,r2,s1,s2,=∑r1=s1,r2s2∈E2ℜA1Lr1∧ℜB2Lr2s2+∑r2=s2,r1s1∈E1ℜA2Lr2∧ℜB1Lr1s1+∑r1s1∈E1,r2s2∉E2ℜB1Lr1s1∧ℜB2Lr2∧ℜB2Ls2+∑r2s2∈E2,r1s1∉E1ℜB2Lr2s2∧ℜB1Lr1∧ℜB1Ls1,(45)dℜUG⌣1⊕G⌣2r1,r2=∑r1,r2s1,s2∈R1×R2ℜB1U⊕ℜB2Ur1,r2,s1,s2,=∑r1=s1,r2s2∈E2ℜA1Ur1∧ℜB2Ur2s2+∑r2=s2,r1s1∈E1ℜA2Ur2∧ℜB1Ur1s1+∑r1s1∈E1,r2s2∉E2ℜB1Ur1s1∧ℜB2Ur2∧ℜB2Us2+∑r2s2∈E2,r1s1∉E1ℜB2Ur2s2∧ℜB1Ur1∧ℜB1Us1,(46)dℵLG⌣1⊕G⌣2r1,r2=∑r1,r2s1,s2∈R1×R2ℵB1L⊕ℵB2Lr1,r2,s1,s2,=∑r1=s1,r2s2∈E2ℵA1Lr1∧ℵB2Lr2s2+∑r2=s2,r1s1∈E1ℵA2Lr2∧ℵB1Lr1s1+∑r1s1∈E1,r2s2∉E2ℵB1Lr1s1∧ℵB2Lr2∧ℵB2Ls2+∑r2s2∈E2,r1s1∉E1ℵB2Lr2s2∧ℵB1Lr1∧ℵB1Ls1,(47)dℵUG⌣1⊕G⌣2r1,r2=∑r1,r2s1,s2∈R1×R2ℵB1U⊕ℵB2Ur1,r2,s1,s2,=∑r1=s1,r2s2∈E2ℵA1Ur1∧ℵB2Ur2s2+∑r2=s2,r1s1∈E1ℵA2Ur2∧ℵB1Ur1s1+∑r1s1∈E1,r2s2∉E2ℵB1Ur1s1∧ℵB2Ur2∧ℵB2Us2+∑r2s2∈E2,r1s1∉E1ℵB2Ur2s2∧ℵB1Ur1∧ℵB1Us1,(48)dμLG⌣1⊕G⌣2r1,r2=∑r1,r2s1,s2∈R1×R2μB1L⊕μB2Lr1,r2,s1,s2,=∑r1=s1,r2s2∈E2μA1r1∧μB2r2s2+∑r2=s2,r1s1∈E1μA2r2∧μB1r1s1+∑r1s1∈E1,r2s2∉E2μB1r1s1∧μB2r2∧μB2s2+∑r2s2∈E2,r1s1∉E1μB2r2s2∧μB1r1∧μB1s1,(49)dνG⌣1⊕G⌣2r1,r2=∑r1,r2s1,s2∈R1×R2νB1⊕νB2r1,r2,s1,s2,=∑r1=s1,r2s2∈E2νA1r1∧νB2r2s2+∑r2=s2,r1s1∈E1νA2r2∧νB1r1s1+∑r1s1∈E1,r2s2∉E2νB1r1s1∧νB2r2∧νB2s2+∑r2s2∈E2,r1s1∉E1νB2r2s2∧νB1r1∧νB1s1.Theorem 5.
LetG⌣1 and G⌣2 be two CuPFGs. If ℜA1L≥ℜB2L,ℵA1L≤ℵB2L,ℜA1L≥ℜB2U,ℵA1U≤ℵB2U,μA1≥μB2,νA1≤νB2 and ℜA2L≥ℜB1L,ℵA2L≤ℵB1L,ℜA2L≥ℜB1L,ℵA2U≤ℵB1U,μA2≥μB1,νA2≤νB1, then dG⌣1⊕G⌣2r1,r2=b2dG⌣1r1+b1dG⌣2r2, where b2=R2−dG⌣2r2 and b1=R1−dG⌣1r1, for all r1,r2∈R1×R2.Definition 15.
LetG⌣1 and G⌣2 be two CuPFGs. Then, for any vertex r1,r2∈R1×R2,(50)tdℜLG⌣1⊕G⌣2r1,r2=∑r1,r2s1,s2∈R1×R2ℜB1L⊕ℜB2Lr1,r2,s1,s2+ℜA1L⊕ℜA2Lr1,r2=∑r1=s1,r2s2∈E2ℜA1Lr1∧ℜB2Lr2s2+∑r2=s2,r1s1∈E1ℜA2Lr2∧ℜB1Lr1s1+∑r1s1∈E1,r2s2∉E2ℜA2Lr2∧ℜA2Ls2∧ℜB1Lr1s1+∑r2s2∈E2,r1s1∉E1ℜA1Lr1∧ℜA1Ls1∧ℜB2Lr2s2+ℜA1Lr1∧ℜA2Lr2,(51)tdℜUG⌣1⊕G⌣2r1,r2=∑r1,r2s1,s2∈R1×R2ℜB1U⊕ℜB2Ur1,r2,s1,s2+ℜA1U⊕ℜA2Ur1,r2=∑r1=s1,r2s2∈E2ℜA1Ur1∧ℜB2Ur2s2+∑r2=s2,r1s1∈E1ℜA2Ur2∧ℜB1Ur1s1+∑r1s1∈E1,r2s2∉E2ℜA2Ur2∧ℜA2Us2∧ℜB1Ur1s1+∑r2s2∈E2,r1s1∉E1ℜA1Ur1∧ℜA1Us1∧ℜB2Ur2s2+ℜA1Ur1∧ℜA2Ur2,(52)tdℵLG⌣1⊕G⌣2r1,r2=∑r1,r2s1,s2∈R1×R2ℵB1L⊕ℵB2Lr1,r2,s1,s2+ℵA1L⊕ℵA2Lr1,r2=∑r1=s1,r2s2∈E2ℵA1Lr1∧ℵB2Lr2s2+∑r2=s2,r1s1∈E1ℵA2Lr2∧ℵB1Lr1s1+∑r1s1∈E1,r2s2∉E2ℵA2Lr2∧ℵA2Ls2∧ℵB1Lr1s1+∑r2s2∈E2,r1s1∉E1ℵA1Lr1∧ℵA1Ls1∧ℵB2Lr2s2+ℵA1Lr1∧ℵA2Lr2,(53)tdℵUG⌣1⊕G⌣2r1,r2=∑r1,r2s1,s2∈R1×R2ℵB1U⊕ℵB2Ur1,r2,s1,s2+ℵA1U⊕ℵA2Ur1,r2=∑r1=s1,r2s2∈E2ℵA1Ur1∧ℵB2Ur2s2+∑r2=s2,r1s1∈E1ℵA2Ur2∧ℵB1Ur1s1+∑r1s1∈E1,r2s2∉E2ℵA2Ur2∧ℵA2Us2∧ℵB1Ur1s1+∑r2s2∈E2,r1s1∉E1ℵA1Ur1∧ℵA1Us1∧ℵB2Ur2s2+ℵA1Ur1∧ℵA2Ur2,(54)tdμG⌣1⊕G⌣2r1,r2=∑r1,r2s1,s2∈R1×R2μB1⊕μB2r1,r2,s1,s2+μA1⊕μA2r1,r2=∑r1=s1,r2s2∈E2μA1r1∧μB2r2s2+∑r2=s2,r1s1∈E1μA2r2∧μB1r1s1+∑r1s1∈E1,r2s2∉E2μA2r2∧μA2s2∧μB1r1s1+∑r2s2∈E2,r1s1∉E1μA1r1∧μA1s1∧μB2r2s2+μA1r1∧μA2r2,(55)tdνG⌣1⊕G⌣2r1,r2=∑r1,r2s1,s2∈R1×R2μB1⊕μB2r1,r2,s1,s2+νA1⊕νA2r1,r2=∑r1=s1,r2s2∈E2νA1r1∧νB2r2s2+∑r2=s2,r1s1∈E1μA2r2∧μB1r1s1+∑r1s1∈E1,r2s2∉E2νA2r2∧νA2s2∧μB1r1s1+∑r2s2∈E2,r1s1∉E1νA1r1∧νA1s1∧μB2r2s2+μA1r1∧μA2r2.Theorem 6.
LetG⌣1 and G⌣2 be two CuPFGs. If(i)
ℜA1L≥ℜB2L and ℜA2L≥ℜB1L, then(56)tdℜALG⌣1⊕G⌣2r1,r2=b2tdℜALG⌣1r1+b1tdℜALG⌣2r2−b2−1ℜA1Lr1−b1−1ℜA2Lr2−ℜA1Lr1∨ℜA2Lr2.(ii)
ℜA1U≥ℜB2U and ℜA2U≥ℜB1U, then(57)tdℜAUG⌣1⊕G⌣2r1,r2=b2tdℜAUG⌣1r1+b1tdℜAUG⌣2r2−b2−1ℜA1Ur1−b1−1ℜA2Ur2−ℜA1Ur1∨ℜA2Ur2.(iii)
ℵA1L≥ℵB2L and ℵA2L≥ℵB1L, then(58)tdℵALG⌣1⊕G⌣2r1,r2=b2tdℵALG⌣1r1+b1tdℵALG⌣2r2−b2−1ℵA1Lr1−b1−1ℵA2Lr2−ℵA1Lr1∨ℵA2Lr2.(iv)
ℵA1U≥ℵB2U and ℵA2U≥ℵB1U, then(59)tdℵAUG⌣1⊕G⌣2r1,r2=b2tdℵAUG⌣1r1+b1tdℵAUG⌣2r2−b2−1ℵA1Ur1−b1−1ℵA2Ur2−ℵA1Ur1∨ℵA2Ur2.(v)
μA1≥μB2 and μA2≥μB1, then(60)tdμAG⌣1⊕G⌣2r1,r2=b2tdμAG⌣1r1+b1tdμAG⌣2r2−b2−1μA1r1−b1−1μA2r2−μA1r1∨μA2r2.(vi)
νA1≥νB2 and νA2≥νB1, then(61)tdνAG⌣1⊕G⌣2r1,r2=b2tdνAG⌣1r1+b1tdνAG⌣2r2−b2−1νA1r1−b1−1νA2r2−νA1r1∨νA2r2,where b2=R2−dG⌣2r2 and b1=R1−dG⌣1r1, for all r1,r2∈R1×R2.proof.
It is easy to prove by using Definition15 and Theorem 5.


## 3. Application

The 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 alternativesE=E1,E2,E3,…,En, 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 alternativesE=E1,E2,E3,…,En is represented by a matrix Z=ejkn×n, for j≠k, where(62)ejkn×n=ℜjkL,ℜjkU,ℵjkL,ℵjkU,μjk,νjk,for all j,k=1,2,3,…,n. Such that, ℜjkL and ℜjkU represent the lower and upper limits of interval-valued preference degree to which the object Ej is preferred to Ek, ℵjkL and ℵjkU represent the lower and upper limits of interval-valued degree to which the object Ej is not preferred to Ek, μjk is the simple fuzzy valued degree of preference, and νjk is the simple fuzzy valued degree of nonpreference of the alternative Ej over Ek, respectively, with the following conditions:ℜjkL,ℜjkU,ℵjkL,ℵjkU,μjk,νjk∈0,1,ℜjkU2+ℵjkU2≤1 and μjk2+νjk2≤1ℜjkL=ℵkjL, ℜjkU=ℵkjU, μjk=νkj.


### 3.1. 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. The 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.The 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 candidatesℑ1,ℑ2,ℑ3,ℑ4 on the basis of certain criterion:C1: 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.C2: previous performance and experience: the past performance of a player defines his expertise and experience helps him to guide others.C3: 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.The experts of the game assigned the weight vector,W=0.4,0.3,0.3T, where ∑m=13wm=1, demonstrating the significance of that particular criterion Cll=1,2,3. The decision-making experts of the game pairwise compare the four players ℑhh=1,2,3,4 under consideration with respect to the three criteria Cll=1,2,3 and provides information according to their preference in the form of cubic Pythagorean fuzzy preference relation (CuPFPR) Zl=ejk4×4l, where ejkl4×4=〈ℜjkL,ℜjkU,ℵjkL,ℵUjk,μjk,νjk 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. The CuPFPRs, Zl=ejk4×4l, are shown in Tables 1–3 as follows.Table 1
CuPFPR with respect to the criteria “personal traits.”
Z1ℑ1ℑ2ℑ3ℑ4ℑ1—0.6,0.7,0.5,0.7,0.6,0.70.8,0.9,0.3,0.4,0.7,0.50.4,0.8,0.5,0.6,0.9,0.4ℑ20.8,0.9,0.3,0.4,0.8,0.6—0.6,0.8,0.4,0.5,0.9,0.30.7,0.8,0.5,0.6,0.7,0.5ℑ30.3,0.5,0.7,0.8,0.3,0.90.5,0.6,0.7,0.8,0.8,0.4—0.6,0.7,0.5,0.7,0.8,0.6ℑ40.7,0.8,0.5,0.6,0.7,0.60.8,0.9,0.3,0.4,0.5,0.80.3,0.4,0.8,0.9,0.8,0.5—Table 2
CuPFPR with respect to the criteria “previous performance and experience.”
Z2ℑ1ℑ2ℑ3ℑ4ℑ1—0.3,0.4,0.8,0.9,0.3,0.80.4,0.5,0.7,0.8,0.7,0.60.2,0.3,0.8,0.9,0.7,0.5ℑ20.7,0.9,0.3,0.4,0.5,0.8—0.6,0.7,0.5,0.6,0.5,0.70.5,0.7,0.6,0.7,0.8,0.6ℑ30.6,0.7,0.5,0.6,0.6,0.70.3,0.4,0.4,0.5,0.7,0.6—0.6,0.8,0.4,0.6,0.9,0.4ℑ40.3,0.4,0.8,0.9,0.8,0.50.7,0.8,0.4,0.5,0.7,0.40.8,0.9,0.3,0.4,0.4,0.9—Table 3
CuPFPR with respect to the criteria “professional skills.”
Z3ℑ1ℑ2ℑ3ℑ4ℑ1—0.8,0.9,0.3,0.4,0.8,0.30.5,0.8,0.5,0.6,0.8,0.50.3,0.4,0.7,0.8,0.4,0.9ℑ20.6,0.7,0.5,0.7,0.8,0.5—0.6,0.7,0.5,0.6,0.7,0.60.4,0.6,0.6,0.7,0.5,0.8ℑ30.5,0.6,0.6,0.7,0.7,0.60.6,0.7,0.5,0.6,0.6,0.4—0.6,0.7,0.5,0.6,0.6,0.5ℑ40.4,0.5,0.5,0.7,0.5,0.80.5,0.8,0.4,0.6,0.4,0.80.4,0.5,0.6,0.7,0.5,0.4—The cubic Pythagorean fuzzy influenced networks corresponding to CuPFPRsZll=1,2,3 given in Tables 1–3, are shown in Figure 7.Figure 7
Cubic Pythagorean fuzzy influenced networks corresponding to CuPFPRs.
(a)(b)(c)Now, we find aggregated CuPFPR by applying CuPFWA operator.The Pythagorean fuzzy influenced graph, corresponding to the aggregated CuPFPR given in Table4, is displayed in Figure 8. Under the constraint on degree of truthness ηjk≥0.5i,j=1,2,3,4 on both interval value and simple value, a partial influenced graph is constructed and depicted in Figure 9. The out-degrees out-d ℑh (h = 1, 2, 3, 4) of all considered players in the partial graph are calculated as follows:(63)out−dℑ1=1.0956,1.3665,1.1294,1.3801,1.2598,1.2071,out−dℑ2=1.8400,2.2780,1.4037,1.7545,2.0531,1.8709,out−dℑ3=0.6,0.7286,0.4733,0.6446,0.7602,0.5213,out−dℑ4=1.1264,1.4029,0.9953,1.2663,1.1759,1.3917.Table 4
Aggregated cubic Pythagorean fuzzy preference relation by CuPFWA operator.
Zℑ1ℑ2ℑ3ℑ4ℑ1—0.53124,0.6382,0.6019,0.7480,0.5312,0.67340.5643,0.7283,0.5275,0.6321,0.7286,0.53370.2980,0.4842,0.6812,0.7911,0.6544,0.6929ℑ20.7050,0.8346,0.3759,0.5261,0.6947,0.6597—0.6,0.7384,0.4639,0.5639,0.6997,0.55840.5349,0.7050,0.5639,0.6645,0.6586,0.6528ℑ30.4305,0.5842,0.6222,0.7246,0.4762,0.79510.4531,0.5564,0.6492,0.7662,0.7050,0.4752—0.6,0.7286,0.4734,0.6645,0.7602,0.5213ℑ40.4590,0.5643,0.6311,0.7646,0.6586,0.65970.6675,0.8386,0.3642,0.5016,0.5172,0.73190.4389,0.5454,0.6595,0.7752,0.5643,0.6972—Figure 8
Influenced graph of aggregated CuPFPR.Figure 9
Partial influenced graph of aggregated CuPFPR.According to membership value of out-dℑhh=1,2,3,4, we obtain the optimal ranking order of the four players ℑhh=1,2,3,4 as follows:(64)ℑ2>ℑ4>ℑ1>ℑ3.Thus,ℑ4 is decided as the best for captaincy of the team.Now, we find aggregated CuPFPR by applying CuPFWG operator.The Pythagorean fuzzy influenced graph, corresponding to the aggregated CuPFPR given in Table5 is displayed in Figure 10. Within the constraint on degree of truthness ηjk≥0.5j,k=1,2,3,4 on both interval value and simple value, a partial influenced graph is constructed as depicted in Figure 11. The out-degrees out-d ℑh (h = 1, 2, 3, 4) of all considered players in the partial graph are calculated as(65)out−dℑ1=1.2828,1.5558,0.9448,1.1943,1.3689,1.0932,out−dℑ2=1.9033,2.3333,1.3648,1.6890,2.2221,1.7036,out−dℑ3=0.6,0.7352,0.4676,0.6382,0.8042,0.5030,out−dℑ4=1.8239,2.1913,1.4790,1.8470,1.9008,1.8269.Table 5
Aggregated cubic Pythagorean fuzzy preference relation by CuPFWG operator.
Zℑ1ℑ2ℑ3ℑ4ℑ1—0.6336,0.7480,0.4939,0.6382,0.6336,0.56500.6492,0.8078,0.4509,0.5562,0.7353,0.52810.3231,0.6220,0.6369,0.7387,0.7752,0.5455ℑ20.7246,0.8628,0.3497,0.4731,0.7425,0.6193—0.6,0.7459,0.4573,0.5578,0.7838,0.47620.5788,0.7246,0.5578,0.6581,0.6959,0.6081ℑ30.4768,0.6025,0.6042,0.7050,0.5584,0.73900.4920,0.5928,0.5643,0.6681,0.7246,0.4517—0.6,0.7458,0.4676,0.6382,0.8042,0.5030ℑ40.5433,0.6492,0.5757,0.7097,0.6959,0.56190.7085,0.8492,0.3565,0.4830,0.5573,0.64980.5721,0.6929,0.5468,0.6544,0.6492,0.5578—Figure 10
Influenced graph of aggregated CuPFPR.Figure 11
Partial influenced graph of aggregated CuPFPR.According to membership value of out-dℑhh=1,2,3,4, we obtain ranking order of the four players ℑhh=1,2,3,4 as(66)ℑ2>ℑ4>ℑ1>ℑ3.Hence, in both casesℑ4 is decided as the best for captaincy of the team.Our proposed method in MCDM of CuPFGs is outlined in Algorithm 1.Algorithm 1:
(1)
Input: first of all, take discrete set of alternativesE=E1,E2,E3,…,En and attributes C=C1,C2,C3,…,Cn under some parameters in order to achieve the target(2)
Let the cubic Pythagorean fuzzy preference relation(CuPFPR) is of the form:Z=ejk4×4=〈ℜjkL,ℜjkU,ℵjkL,ℵjkU,μjk,νjkj,k=1,2,3,…,n of each alternative Ej over the Ek with respect to the all considered criteria Cll=1,2,3…,n(3)
Apply cubic Pythagorean fuzzy weighted averaging(CuPFWA) operator [10]ej=CuPFWAej1,ej2,ej3,…,ejn=Πj=1nℜejLwi,ℜj=1nμejUwj,1−Πj=1n1−ℵejL2wj,1−Πj=1n1−ℵejU2wj,Πj=1nμejwj,1−Πj=1n1−νej2wjand cubic Pythagorean fuzzy weighted geometric(CuPFWG) operator [10]ej=CuPFWGej1,ej2,ej3,…,ejn1−Πj=1n1−ℵejL2wj,1−Πj=1n1−ℵejU2wj,Πj=1nℜejLwi,ℜj=1nμejUwj,1−Πj=1n1−νej2wj,Πj=1nμejwjto aggregate the individual CuPFPR(4)
Construct the collective averaged CuPFRejk4×4=ℜjkL,ℜjkU,ℵjkL,ℵjkU,μjk,νjkj,k=1,2,3,4,computed by CuPFWA and CuPFWG operator(5)
Draw an influenced model relating the collective CuPFPR(6)
Draw a partially influenced model relating the collective CuPFPR by using constraint on degree of truthnessηjk≥0.5(7)
Investigate the out-degrees of all alternativesEj in partial influenced network(8)
Rank each of the alternative on the basis of out-degreesηjOutput: the alternative having maximum out-degree is selected as the best oneFurthermore, for better understanding the organization of the proposed method is presented in Figure12.Figure 12
Flowchart of the purposed work.


### 3.2. Comparative Analysis

In order to demonstrate the effectiveness and validity of the propose method, a comparative analysis with already existing techniques is carried out in this section. The Pythagorean fuzzy weighted average (PFWA) operator and the Pythagorean fuzzy weighted geometric (PFWG) operator introduced by Yager [6] are applied for solving the above decision-making problem. For comparison with previous techniques, we have to take information in Pythagorean fuzzy environment. The results corresponding to these approaches are summarized in Table 6.Table 6
Comparative analysis with existing techniques.
TechniquesOut-degrees of the alternativesRankings of the alternativesExisting PFWA [6] technique(1.2598,1.2071) (2.0531,1.8709) (0.7602,0.5213) (1.1759,1.3917)ℑ2>ℑ4>ℑ1>ℑ3Existing PFWG [6] technique(1.3689,1.0932) (2.2221,1.7036),(0.8042,0.5030) (1.9008,1.8269)ℑ2>ℑ4>ℑ1>ℑ3Our purposed technique using CuPFWA [10]1.0956,1.3665,1.1294,1.3801,1.2598,1.20711.8400,2.2780,1.4037,1.7545,2.0531,1.8709ℑ2>ℑ4>ℑ1>ℑ30.6,0.7286,0.4733,0.6446,0.7602,0.52131.1264,1.4029,0.9953,1.2663,1.1759,1.3917Our purposed technique using CuPFWG [10]1.2828,1.5558,0.9448,1.1943,1.3689,1.09321.9033,2.3333,1.3648,1.6890,2.2221,1.7036ℑ2>ℑ4>ℑ1>ℑ30.6,0.7352,0.4676,0.6382,0.8042,0.50301.8239,2.1913,1.4790,1.8470,1.9008,1.8269Hence, 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. The existing method in Pythagorean fuzzy environment is a special case of the proposed method. The 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.


## 3.1. 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. The 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.The 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 candidatesℑ1,ℑ2,ℑ3,ℑ4 on the basis of certain criterion:C1: 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.C2: previous performance and experience: the past performance of a player defines his expertise and experience helps him to guide others.C3: 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.The experts of the game assigned the weight vector,W=0.4,0.3,0.3T, where ∑m=13wm=1, demonstrating the significance of that particular criterion Cll=1,2,3. The decision-making experts of the game pairwise compare the four players ℑhh=1,2,3,4 under consideration with respect to the three criteria Cll=1,2,3 and provides information according to their preference in the form of cubic Pythagorean fuzzy preference relation (CuPFPR) Zl=ejk4×4l, where ejkl4×4=〈ℜjkL,ℜjkU,ℵjkL,ℵUjk,μjk,νjk 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. The CuPFPRs, Zl=ejk4×4l, are shown in Tables 1–3 as follows.Table 1
CuPFPR with respect to the criteria “personal traits.”
Z1ℑ1ℑ2ℑ3ℑ4ℑ1—0.6,0.7,0.5,0.7,0.6,0.70.8,0.9,0.3,0.4,0.7,0.50.4,0.8,0.5,0.6,0.9,0.4ℑ20.8,0.9,0.3,0.4,0.8,0.6—0.6,0.8,0.4,0.5,0.9,0.30.7,0.8,0.5,0.6,0.7,0.5ℑ30.3,0.5,0.7,0.8,0.3,0.90.5,0.6,0.7,0.8,0.8,0.4—0.6,0.7,0.5,0.7,0.8,0.6ℑ40.7,0.8,0.5,0.6,0.7,0.60.8,0.9,0.3,0.4,0.5,0.80.3,0.4,0.8,0.9,0.8,0.5—Table 2
CuPFPR with respect to the criteria “previous performance and experience.”
Z2ℑ1ℑ2ℑ3ℑ4ℑ1—0.3,0.4,0.8,0.9,0.3,0.80.4,0.5,0.7,0.8,0.7,0.60.2,0.3,0.8,0.9,0.7,0.5ℑ20.7,0.9,0.3,0.4,0.5,0.8—0.6,0.7,0.5,0.6,0.5,0.70.5,0.7,0.6,0.7,0.8,0.6ℑ30.6,0.7,0.5,0.6,0.6,0.70.3,0.4,0.4,0.5,0.7,0.6—0.6,0.8,0.4,0.6,0.9,0.4ℑ40.3,0.4,0.8,0.9,0.8,0.50.7,0.8,0.4,0.5,0.7,0.40.8,0.9,0.3,0.4,0.4,0.9—Table 3
CuPFPR with respect to the criteria “professional skills.”
Z3ℑ1ℑ2ℑ3ℑ4ℑ1—0.8,0.9,0.3,0.4,0.8,0.30.5,0.8,0.5,0.6,0.8,0.50.3,0.4,0.7,0.8,0.4,0.9ℑ20.6,0.7,0.5,0.7,0.8,0.5—0.6,0.7,0.5,0.6,0.7,0.60.4,0.6,0.6,0.7,0.5,0.8ℑ30.5,0.6,0.6,0.7,0.7,0.60.6,0.7,0.5,0.6,0.6,0.4—0.6,0.7,0.5,0.6,0.6,0.5ℑ40.4,0.5,0.5,0.7,0.5,0.80.5,0.8,0.4,0.6,0.4,0.80.4,0.5,0.6,0.7,0.5,0.4—The cubic Pythagorean fuzzy influenced networks corresponding to CuPFPRsZll=1,2,3 given in Tables 1–3, are shown in Figure 7.Figure 7
Cubic Pythagorean fuzzy influenced networks corresponding to CuPFPRs.
(a)(b)(c)Now, we find aggregated CuPFPR by applying CuPFWA operator.The Pythagorean fuzzy influenced graph, corresponding to the aggregated CuPFPR given in Table4, is displayed in Figure 8. Under the constraint on degree of truthness ηjk≥0.5i,j=1,2,3,4 on both interval value and simple value, a partial influenced graph is constructed and depicted in Figure 9. The out-degrees out-d ℑh (h = 1, 2, 3, 4) of all considered players in the partial graph are calculated as follows:(63)out−dℑ1=1.0956,1.3665,1.1294,1.3801,1.2598,1.2071,out−dℑ2=1.8400,2.2780,1.4037,1.7545,2.0531,1.8709,out−dℑ3=0.6,0.7286,0.4733,0.6446,0.7602,0.5213,out−dℑ4=1.1264,1.4029,0.9953,1.2663,1.1759,1.3917.Table 4
Aggregated cubic Pythagorean fuzzy preference relation by CuPFWA operator.
Zℑ1ℑ2ℑ3ℑ4ℑ1—0.53124,0.6382,0.6019,0.7480,0.5312,0.67340.5643,0.7283,0.5275,0.6321,0.7286,0.53370.2980,0.4842,0.6812,0.7911,0.6544,0.6929ℑ20.7050,0.8346,0.3759,0.5261,0.6947,0.6597—0.6,0.7384,0.4639,0.5639,0.6997,0.55840.5349,0.7050,0.5639,0.6645,0.6586,0.6528ℑ30.4305,0.5842,0.6222,0.7246,0.4762,0.79510.4531,0.5564,0.6492,0.7662,0.7050,0.4752—0.6,0.7286,0.4734,0.6645,0.7602,0.5213ℑ40.4590,0.5643,0.6311,0.7646,0.6586,0.65970.6675,0.8386,0.3642,0.5016,0.5172,0.73190.4389,0.5454,0.6595,0.7752,0.5643,0.6972—Figure 8
Influenced graph of aggregated CuPFPR.Figure 9
Partial influenced graph of aggregated CuPFPR.According to membership value of out-dℑhh=1,2,3,4, we obtain the optimal ranking order of the four players ℑhh=1,2,3,4 as follows:(64)ℑ2>ℑ4>ℑ1>ℑ3.Thus,ℑ4 is decided as the best for captaincy of the team.Now, we find aggregated CuPFPR by applying CuPFWG operator.The Pythagorean fuzzy influenced graph, corresponding to the aggregated CuPFPR given in Table5 is displayed in Figure 10. Within the constraint on degree of truthness ηjk≥0.5j,k=1,2,3,4 on both interval value and simple value, a partial influenced graph is constructed as depicted in Figure 11. The out-degrees out-d ℑh (h = 1, 2, 3, 4) of all considered players in the partial graph are calculated as(65)out−dℑ1=1.2828,1.5558,0.9448,1.1943,1.3689,1.0932,out−dℑ2=1.9033,2.3333,1.3648,1.6890,2.2221,1.7036,out−dℑ3=0.6,0.7352,0.4676,0.6382,0.8042,0.5030,out−dℑ4=1.8239,2.1913,1.4790,1.8470,1.9008,1.8269.Table 5
Aggregated cubic Pythagorean fuzzy preference relation by CuPFWG operator.
Zℑ1ℑ2ℑ3ℑ4ℑ1—0.6336,0.7480,0.4939,0.6382,0.6336,0.56500.6492,0.8078,0.4509,0.5562,0.7353,0.52810.3231,0.6220,0.6369,0.7387,0.7752,0.5455ℑ20.7246,0.8628,0.3497,0.4731,0.7425,0.6193—0.6,0.7459,0.4573,0.5578,0.7838,0.47620.5788,0.7246,0.5578,0.6581,0.6959,0.6081ℑ30.4768,0.6025,0.6042,0.7050,0.5584,0.73900.4920,0.5928,0.5643,0.6681,0.7246,0.4517—0.6,0.7458,0.4676,0.6382,0.8042,0.5030ℑ40.5433,0.6492,0.5757,0.7097,0.6959,0.56190.7085,0.8492,0.3565,0.4830,0.5573,0.64980.5721,0.6929,0.5468,0.6544,0.6492,0.5578—Figure 10
Influenced graph of aggregated CuPFPR.Figure 11
Partial influenced graph of aggregated CuPFPR.According to membership value of out-dℑhh=1,2,3,4, we obtain ranking order of the four players ℑhh=1,2,3,4 as(66)ℑ2>ℑ4>ℑ1>ℑ3.Hence, in both casesℑ4 is decided as the best for captaincy of the team.Our proposed method in MCDM of CuPFGs is outlined in Algorithm 1.Algorithm 1:
(1)
Input: first of all, take discrete set of alternativesE=E1,E2,E3,…,En and attributes C=C1,C2,C3,…,Cn under some parameters in order to achieve the target(2)
Let the cubic Pythagorean fuzzy preference relation(CuPFPR) is of the form:Z=ejk4×4=〈ℜjkL,ℜjkU,ℵjkL,ℵjkU,μjk,νjkj,k=1,2,3,…,n of each alternative Ej over the Ek with respect to the all considered criteria Cll=1,2,3…,n(3)
Apply cubic Pythagorean fuzzy weighted averaging(CuPFWA) operator [10]ej=CuPFWAej1,ej2,ej3,…,ejn=Πj=1nℜejLwi,ℜj=1nμejUwj,1−Πj=1n1−ℵejL2wj,1−Πj=1n1−ℵejU2wj,Πj=1nμejwj,1−Πj=1n1−νej2wjand cubic Pythagorean fuzzy weighted geometric(CuPFWG) operator [10]ej=CuPFWGej1,ej2,ej3,…,ejn1−Πj=1n1−ℵejL2wj,1−Πj=1n1−ℵejU2wj,Πj=1nℜejLwi,ℜj=1nμejUwj,1−Πj=1n1−νej2wj,Πj=1nμejwjto aggregate the individual CuPFPR(4)
Construct the collective averaged CuPFRejk4×4=ℜjkL,ℜjkU,ℵjkL,ℵjkU,μjk,νjkj,k=1,2,3,4,computed by CuPFWA and CuPFWG operator(5)
Draw an influenced model relating the collective CuPFPR(6)
Draw a partially influenced model relating the collective CuPFPR by using constraint on degree of truthnessηjk≥0.5(7)
Investigate the out-degrees of all alternativesEj in partial influenced network(8)
Rank each of the alternative on the basis of out-degreesηjOutput: the alternative having maximum out-degree is selected as the best oneFurthermore, for better understanding the organization of the proposed method is presented in Figure12.Figure 12
Flowchart of the purposed work.


## 3.2. Comparative Analysis

In order to demonstrate the effectiveness and validity of the propose method, a comparative analysis with already existing techniques is carried out in this section. The Pythagorean fuzzy weighted average (PFWA) operator and the Pythagorean fuzzy weighted geometric (PFWG) operator introduced by Yager [6] are applied for solving the above decision-making problem. For comparison with previous techniques, we have to take information in Pythagorean fuzzy environment. The results corresponding to these approaches are summarized in Table 6.Table 6
Comparative analysis with existing techniques.
TechniquesOut-degrees of the alternativesRankings of the alternativesExisting PFWA [6] technique(1.2598,1.2071) (2.0531,1.8709) (0.7602,0.5213) (1.1759,1.3917)ℑ2>ℑ4>ℑ1>ℑ3Existing PFWG [6] technique(1.3689,1.0932) (2.2221,1.7036),(0.8042,0.5030) (1.9008,1.8269)ℑ2>ℑ4>ℑ1>ℑ3Our purposed technique using CuPFWA [10]1.0956,1.3665,1.1294,1.3801,1.2598,1.20711.8400,2.2780,1.4037,1.7545,2.0531,1.8709ℑ2>ℑ4>ℑ1>ℑ30.6,0.7286,0.4733,0.6446,0.7602,0.52131.1264,1.4029,0.9953,1.2663,1.1759,1.3917Our purposed technique using CuPFWG [10]1.2828,1.5558,0.9448,1.1943,1.3689,1.09321.9033,2.3333,1.3648,1.6890,2.2221,1.7036ℑ2>ℑ4>ℑ1>ℑ30.6,0.7352,0.4676,0.6382,0.8042,0.50301.8239,2.1913,1.4790,1.8470,1.9008,1.8269Hence, 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. The existing method in Pythagorean fuzzy environment is a special case of the proposed method. The 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.


## 4. 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. This 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. The 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.



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*Source: 1144666-2022-06-28.xml*