# Graphical Structures of Cubic Intuitionistic Fuzzy Information

Authors: Sami Ullah Khan; Naeem Jan; Kifayat Ullah; Lazim Abdullah
Journal: Journal of Mathematics
 (2021)

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## 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.


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## Body

## 1. Introduction

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). The 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. The 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]. The concept of cubic soft sets was proposed by Muhiuddin and Al-roqi [7], which was further utilized by Muhiuddin et al. [8]. The 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. Some other works in this direction are given in [11–14].The theory of intuitionistic fuzzy set (IFS) was developed by Atanassov [15] as a generalization of FS by Rosenfeld [16]. An IFS described the membership and nonmembership degree of an element by two characteristic functions and can model phenomena of yes or no type easily. Garg and Kaur [17] initiated the concept of cubic intuitionistic fuzzy sets (CIFSs) and discussed their properties. Atanassov model of IFS provided a motivation for the concept of intuitionistic fuzzy graphs (IFGs) defined by Parvathi and Karunambigai [18]. The concept of IFG was a generalization of fuzzy graphs (FGs) proposed by Kauffman and Rosenfeld [19, 20] after Zadeh’s exemplary work in [16]. FG theory has a potential role in application point of view as described by Chan and Cheung [21] who studied an approach to clustering algorithm using the concepts of FGs. Some FG problems are solved by a novel technique in [22, 23] by discussing the domination of FGs in pattern recognitions. Mathew and Sunitha [24] worked on fuzzy attribute graphs applied to Chinese character recognitions, and Bhattacharya [25] used FGs in image classifications and so forth. For some other works on FG, one may refer to [26–31].The theory of IFG received great attention as Parvathi and Thamizhendhi [32] introduced the concept of strong IFGs; Akram and Dudek [33] discussed the order, degree, and size of IFGs; Akram and Alshehri [34] developed operations for IFGs; Karunambigai [35] worked on the domination of IFGs; Pasi et al. [36] developed the theory of intuitionistic fuzzy hypergraphs; Karunambigai et al. [37] studied the concepts of trees and cycles for IFGs; Parvathi [38] developed the idea of balanced IFGs, a multicriteria and multiperson decision-making based on IFGs was discussed by Chountas [39]; Akram and Dudek [40] studied constant IFGs; Mathew [41] discussed IF hypergraphs; and the authors of [42] discussed the matrix representation of IFGs. Interval-valued FGs have also been studied extensively after Akram [43] proposed interval-valued FGs, Rashmanlou and Pal [44] discussed the results proposed by [43], complete interval-valued FGs developed interval-valued fuzzy line graphs are discussed by Rashmanlou and Pal [45, 46], and Pramanik et al. [47] proposed balanced interval-valued FGs. Xiao et al. [48] worked on green supplier selection in steel industry with intuitionistic fuzzy Taxonomy method, Zhao et al. [49] proposed an extended CPT-TODIM method for IVIF MAGDM and applied it to urban ecological risk assessment, and Wu et al. [50] presented VIKOR method for financing risk assessment of rural tourism under IVIF environment. Further, for some works on interval-valued FGs, one may refer to [51–55]. Motivated by the existing theory, we proposed the framework of cubic intuitionistic fuzzy sets (CIFSs) and cubic intuitionistic fuzzy graphs (CIFGs). Several graphical and theoretical terms are illustrated with the help of examples and some results.The manuscript is organized as follows: In Section1, a brief introduction about existing concepts is presented. In Section 2, some basic definitions from the theories of FG, IFG, and IVFG are defined. The concept of CIFG is proposed in Section 3 along with some other related terms and results including the concepts of subgraphs, degrees, orders, and bridges in CIFGs. Section 4 is based on operations on CIFGs and their results. The applications of CIFG in decision-making problems are discussed in Section 5. Section 6 provides a comparison of CIFG with existing concepts, and Section 7 provides a brief discussion and concluding remarks.


## 2. Preliminaries

In this section, 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. Throughout 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.Definition 1.
(see [13]). A FS on Ẋ is defined as A=u,MAu/u∈X˙, where MA1/2 is a map on 0,1.Definition 2.
(see [20]). A pair Ğ∗=V,E is known as FG if(i)
V=Mi:i∈I and M1:V⟶0,1 is the association degree of Mi∈V(ii)
E=ui,uj:ui,uj∈V×V and M2:V×V⟶0,1 where M2ui,uj≤minM1ui,M1uj for all ui,uj∈E.Definition 3.
(see [15]). An IFS A on X is defined as A=u,MAu,ŊAu/u∈X˙, where MA and ŊA are mappings on 0,1 interval such that 0≤MA+ŊA≤1.Definition 4.
(see [18]). A Pair Ğ∗=(V, Ể) is known as IFG if(i)
V is the collection of nodes such thatM1 and Ŋ1are mappings on unit intervals from V with a condition 0≤M1ui + Ŋ1ui≤1 for all ui ∈ V, i ∈ I(ii)
E⊆V×V, where M2 and Ŋ2 are mappings that associate some grade to each ui,uj∈E from 0,1 interval such that M2ui,uj≤ min M1ui,M1uj and Ŋ2ui,uj≤ max Ŋ1ui,Ŋ1uj with a condition 0≤M2+Ŋ2≤1Example 1.
The graph in Figure1 is an IFG having four vertices and four edges.Figure 1
Intuitionistic fuzzy graph.Definition 5.
(see [33]). The complement of an IFG Ğ∗=V,E is Ğ∗c=Vc,Ec, where(i)
Vc=V(ii)
MA(ui)c =  MA(ui), ŊA(ui)c =  ŊA(ui),∀ ui ∈ V(iii)
MBui,ujc=minMBui,MBuj−MBui,uj,ŊBui,ujc=maxŊBui,ŊBuj−ŊBui,uj, for all ui,uj∈E
Hereui,MA,ŊA represent the vertices and eij,MB,ŊB represent the edges.Definition 6.
(see [32]). A Pair Ğ∗=V,E is known as strong IFG if(i)
V is the collection of nodes such that M1 and Ŋ1 are mappings on unit intervals from V with a condition 0≤M1ui+Ŋ1ui≤1 for all ui∈Vi∈I(ii)
E⊆V×V, where M2 and Ŋ2 are mappings that associate some grade to each ui,uj∈E from 0,1 interval such that M2ui,uj=minM1ui,M1uj and Ŋ2ui,uj=maxŊ1ui,Ŋ1uj with a condition 0≤M2+Ŋ2≤1Remark 1.
(see [32]). If Ğ∗=V,E is an IFG, then by the above definition Ğ∗cc=Ğ∗ and it is called self-complementary.Proposition 1.
(see [32]). If Ğ∗ is strong IFG, then it preserves self-complementary law.Example 2.
Figures2(a) and 2(b) provide a verification of Proposition 1.
ClearlyĞ∗cc=Ğ∗ is self-complementry.Figure 2
(a) Intuitionistic fuzzy graph. (b) Complement of intuitionistic fuzzy graph.
(a)(b)Definition 7.
(see [55]). A pair Ğ=A,ℬ of a graph Ğ∗=V,E is known as IVIFG, where A=MAL,MAU,ŊAL,ŊAU is IVFS on V, and ℬ=MℬL,MℬU,ŊℬL,ŊℬU is the IVF relation on E satisfying the following conditions: (i)
V=u1,u2,u3,…,unsuch that MAL:V⟶0,1,MAU:V⟶0,1 and ŊAL:V⟶0,1, ŊAU:V⟶0,1 represent the degrees of membership and nonmembership of the element u∈V, respectively, and 0≤MA+ŊA≤1 for all ui∈Vi=1,2,…,n(ii)
The functionsMℬL:V×V⟶0,1, MℬU:V×V⟶0,1,ŊℬL:V×V⟶0,1, and ŊℬU:V×V⟶0,1 are such that MℬLu,y≤minMALu,MALy, ŊℬLu,y≤maxŊALu,ŊALyMℬUu,y≤minMAUu,MAUy, and ŊℬUu,y≤maxŊAUu,ŊAUy; 0≤Mℬu,y+Ŋℬu,y≤1 for all ui,yj∈Ei,j=1,2,…,nExample 3.
LetĞ∗=V,E be a graph, where V=u1,u2,u3 is the set of vertices and E=u1u2,u2u3,u3u1 is the set of edges.


## 3. Cubic Intuitionistic Fuzzy Graphs

In this section, we discussed the basic concept of CIFG-like complement of CIFG, degree of CIFG, and bridge and cut vertex of CIFG with the help of examples and several results (Figures3 and 4).Figure 3
Interval-valued intuitionistic fuzzy graph.Figure 4
Cubic intuitionistic fuzzy graph.Definition 8.
A pairĞ=A,ℬ of a graph Ğ∗=V,E is known as cubic IFG, where A=MAL,MAU,ŊAL,ŊAU,MA,ŊA is a cubic IFS on V, and ℬ=MℬL,MℬU,ŊℬL,ŊℬU,Mℬ,Ŋℬ is the cubic IF relation on E satisfying the following conditions: (iii)
V=u1,u2,u3,…,un such that MAL:V⟶0,1,MAU:V⟶0,1 and ŊAL:V⟶0,1, ŊAU:V⟶0,1 and MA:V⟶0,1,ŊA:V⟶0,1⊸ represent the degrees of membership and nonmembership of the element u∈V, respectively, and 0≤MA+ŊA≤1 for all ui∈Vi=1,2,…,n(iv)
The functionsMℬL:V×V⟶0,1, MℬU:V×V⟶0,1,ŊℬL:V×V⟶0,1, ŊℬU:V×V⟶0,1 and Mℬ:V×V⟶0,1,Ŋℬ:V×V⟶0,1 are such that MℬLu,y≤minMALu,MALy, ŊℬLu,y≤maxŊALu,ŊALyMℬUu,y≤minMAUu,MAUy, and ŊℬUu,y≤maxŊAUu,ŊAUy; and Mℬu,y≤minMAu,MAy and Ŋℬu,y≤maxMAu,MAy such that 0≤Mℬu,y+Ŋℬu,y≤1 for all ui,yj∈Ei,j=1,2,…,nExample 4.
Consider a graphĞ∗=V,E, where V=u1,u2,u3 is the set of vertices and E=u1u2,u2u3,u3u1 is the set of edges.Definition 9.
A pairĞ=A,ℬ of a graph Ğ∗=V,E is known as strong cubic IFG, where A=MAL,MAU,ŊAL,ŊAU,MA,ŊA is a cubic IFS on V, and ℬ=MℬL,MℬU,ŊℬL,ŊℬU,Mℬ,Ŋℬ is a cubic IF relation on E satisfying the following conditions:(i)
V=u1,u2,u3,…,un such that MAL:V⟶0,1,MAU:V⟶0,1 and ŊAL:V⟶0,1, ŊAU:V⟶0,1 and MA:V⟶0,1,ŊA:V⟶0,1 represent the degrees of membership and nonmembership of the element u∈V, respectively, and 0≤MA+ŊA≤1 for all ui∈Vi=1,2,…,n(ii)
The functionsMℬL:V×V⟶0,1, MℬU:V×V⟶0,1,ŊℬL:V×V⟶0,1, ŊℬU:V×V⟶0,1, and Mℬ:V×V⟶0,1,Ŋℬ:V×V⟶0,1 are such that MℬLu,y=minMALu,MALy, ŊℬLu,y=maxŊALu,ŊALyMℬUu,y=minMAUu,MAUy, and ŊℬUu,y=maxŊAUu,ŊAUy; and Mℬu,y=minMAu,MAy and Ŋℬu,y=maxMAu,MAy such that 0≤Mℬu,y+Ŋℬu,y≤1 for all ui,yj∈Ei,j=1,2,…,nDefinition 10.
A cubic IFGH=V⋎, E⋎ is said to be cubic IFG subgraph of Ğ∗=V,E if V⋎⊆V and E⋎⊆E. In other words, MALi,MAUi⋎≤MALi,MAUi,ŊALi,ŊAUi⋎≤ŊALi,ŊAUi, and MAi,ŊAi⋎≤MAi,ŊAi and MℬLij,MℬUij⋎≤MℬLij,MℬUij,ŊℬLij,ŊℬUij⋎≤ŊℬLij,ŊℬUij, and Mℬij,Ŋℬij⋎≤Mℬij,Ŋℬij for i,j=1,2,…,n.Definition 11.
The order of cubic IFGĞ∗=V,Ể is denoted and defined by(1)OĞ∗=∑u∈VMAḼu,∑u∈VMAȖu,∑u∈VŊAḼu,∑u∈VŊAȖu,∑u∈VMAu,∑u∈VŊAu,and the size of cubic IFG is(2)SG=∑u≠yu,y∈VMℬLuy∑u≠yu,y∈VMℬAȖuy,∑u≠yu,y∈VŊℬLuy,∑u≠yu,y∈VŊℬAȖuy,∑u≠yu,y∈VMℬuy,∑u≠yu,y∈VŊℬuy.Definition 12.
The degree of a vertex in a cubic IFGĞ∗=V,E is denoted and defined by(3)du=dMALu,dMAUu,dŊAUu,dŊAUu,dMAu,dŊAu,where(4)dMALu=∑u≠yu∈VMℬLuy,dMAUu=∑u≠yu∈VMℬUuy,dŊALu=∑u≠yu,y∈VŊℬLuy,dŊAUu=∑u≠yu,y∈VŊℬUuy,dMAu=∑u≠yu,y∈VMℬuy,dŊAu=∑u≠yu,y∈VŊℬuy.Example 5.
Let Figure5 be a graph Ğ∗=V,E, where V=u1,u2,u3,u4 is the set of vertices and E=u1u2,u2u3,u3u4,u4u1 is the set of edges.
The degrees of vertices are(5)du1=0.3,0.6,0.5,0.8,0.3,0.8,du2=0.4,0.7,0.5,0.8,0.3,0.8,du3=0.3,0.7,0.4,0.8,0.2,0.8,du4=0.2,0.6,0.4,0.8,0.2,0.9.Figure 5
Cubic intuitionistic fuzzy graph.Definition 13.
The complement of a cubic IFGĞ=A,ℬ on Ğ∗=V,E is defined as follows:(i)
A¯=A(ii)
MAL¯ui=MALui, MAU¯ui=MAUui,ŊAL¯ui=ŊALui,ŊAU¯ui=ŊAUui and MA¯ui=MAui, ŊA¯ui=ŊAui for all ui∈V(iii)
MℬL¯ui,uj= minMALui, MALuj−MℬUui,uj, MℬU¯ui,uj= minMAUui, MAUuj−MℬUui,uj, ŊℬL¯ui,uj= maxŊALui, ŊALuj−ŊℬUui,uj, ŊℬU¯ui,uj=1/2max ŊALui, ŊALuj−ŊℬUui,uj for all ui,uj∈EProposition 2.
Ğ=Ğ¯¯ if and if Ğ is strong cubic IF graph.Proof.
The proof is straightforward.Definition 14.
A strong IFG is said to be self-complementary ifĞ≅Ğ¯¯, where Ğ¯ is the complement of IFG Ğ.Example 6.
Let Figures6 and 7 be two graphs of Ğ∗=V,E, where V=u1,u2,u3,u4 is the set of vertices and E=u1u2,u2u3,u3u4,u4u1 is the set of edges.
ClearlyĞ=Ğ¯¯; hence, Ğ is self-complementary.Figure 6
Cubic strong intuitionistic fuzzy graph.Figure 7
Complement of cubic strong intuitionistic fuzzy graph.Definition 15.
The power of edge relation in a cubic IFG is defined as(6)eij1=eij,MℬijL,MℬijU,ŊℬijL,ŊℬijU,Mℬij,Ŋℬijeij2=eij∗eij=eij,MℬijL,MℬijU2,ŊℬijL,ŊℬijU2,Mℬij2,Ŋℬij2eij3=eij∗eij∗eij=eij,MℬijL,MℬijU3,ŊℬijL,ŊℬijU3,Mℬij3,Ŋℬij3.
Also,(7)eij∞=eij,MℬijL,MℬijU∞,ŊℬijL,ŊℬijU∞,Mℬij∞,Ŋℬij∞.
Here,MℬijL,MℬijU∞=maxMℬijL,MℬijUk,Mℬij∞=maxMℬijk and ŊℬijL,ŊℬijU∞=minŊℬijL,ŊℬijUk, Ŋℬij∞=minŊℬijk are the M−strength and Ŋ−strength of the connectedness between the two vertices yi,yj.Definition 16.
An edge in a cubic IFGĞ∗=V,E is said to be bridge, if deleting that edge reduces the strength of connectedness between some pair of vertices.Example 7.
Let Figure8 be a graph Ğ∗=V,E, where V=u1,u2,u3,u4 is the set of vertices and E=u1u2,u2u3,u2u4,u4u1 is the set of edges.
The strength ofu1,u4 is 0.1,0.4,0.3,0.5,0.1,0.4, so u1,u4 is a bridge because when deleteing u1,u4 the strength of the connectedness between u1 and u4 is decreased.Figure 8
Cubic intuitionistic fuzzy graph.Theorem 1.
IfĞ∗=V,E is a cubic IFG, then, for any two vertices yi and yj, the following are equivalent:(i)
yi,yj is a bridge(ii)
MℬijL,MℬijU′∞<MℬijL,MℬijU,Mℬij′∞<Mℬij and ŊℬijL,ŊℬijU′∞>ŊℬijL,ŊℬijU,Ŋℬij′∞>Ŋℬij(iii)
yi,yj is not an edge of any cycleProof.
(ii)⟹ (i).
ConsiderMℬijL,MℬijU′∞<MℬijL,MℬijU,Mℬij′∞<Mℬij and ŊℬijL,ŊℬijU′∞>ŊℬijL,ŊℬijU,Ŋℬij′∞>Ŋℬij to show that yi,yj is a bridge; then MℬijL,MℬijU′∞=MℬijL,MℬijU∞≥MℬijL,MℬijU,Mℬij′∞=Mℬij∞≥Mℬij and ŊℬijL,ŊℬijU′∞=ŊℬijL,ŊℬijU∞≤ŊℬijL,ŊℬijU,Ŋℬij′∞=Ŋℬij∞≤Ŋℬij.
MℬijL,MℬijU′∞≥MℬijL,MℬijU,Mℬij′∞≥Mℬij and ŊℬijL,ŊℬijU′∞≤ŊℬijL,ŊℬijU,Ŋℬij′∞Ŋℬij′∞, which is a contradiction. Hence, yi,yj is a bridge.
(i)⟹ (iii).
Suppose thatyi,yj is a bridge to show that yi,yj is not an edge of any cycle. If yi,yj is an edge of cycle, then any path involving the edge yi,yj can be converted into a path not involving yi,yj by using the rest of the cycle as a path from yi to yj. This implies that yi,yj cannot be a bridge, which is a contradiction to our supposition. Hence, yi,yj is not an edge of any cycle.
(iii)⟹ (i).
The proof is straightforward.Definition 17.
A vertexui in a cubic IFG Ğ∗is said to be cut-vertex if deleting a vertex ui reduces the strength of connectedness between some pair of vertices.Example 8.
Consider a graphĞ∗=V,E, where V=u1,u2,u3,u4,u5 is the set of vertices and E=u1u2,u2u4,u4u3,u4u5,u4u1 is the set of edges.
In Figure9, u1 is a cut-vertex.Figure 9
Cubic intuitionistic fuzzy graph.


## 4. Operations on Cubic IFG

In this section, the operations of CIFG-like Cartesian product of CIFG, union of CIFG, joint operation of CIFG, and so forth with the help of examples are discussed and some interesting results related to these operations are proved.Definition 18.
The Cartesian productĞ=Ğ1×Ğ2=A1×A2,ℬ1×ℬ2of two cubic IFGs Ğ1=A1,ℬ1 and Ğ2=A2,ℬ2 of the graphs Ğ1∗=V1,E1 and Ğ2∗=V2,E2 is defined as follows:(i)
(8)MA1L×MA2Lu1,u2=minMA1Lu1,MA2Lu2,MA1U×MA2Uu1,u2=minMA1Uu1,MA2UMu2,ŊA1L×ŊA2Lu1,u2=maxŊA1Lu1,ŊA2Lu2,ŊA1U×ŊA2Uu1,u2=maxŊA1Uu1,ŊA2Uu2,MA1×MA2u1,u2=minMA1u1,MA2u2,ŊA1×ŊA2u1,u2=maxŊA1u1,ŊA2u2,for allu1,u2∈V.(ii)
(9)Mℬ1L×Mℬ2Lu,u2u,y2=minMA1Lu,Mℬ2Lu2y2,Mℬ1U×Mℬ2Uu,u2u,y2=minMA1Uu,Mℬ2Uu2y2,Ŋℬ1L×Ŋℬ2Lu,u2u,y2=maxŊA1Lu,Ŋℬ2Lu2y2,Ŋℬ1U×Ŋℬ2Uu,u2u,y2=maxŊA1Uu,Ŋℬ2Uu2y2,Mℬ1×Mℬ2u,u2u,y2=minMA1u,Mℬ2u2y2,Ŋℬ1×Ŋℬ2u,u2u,y2=maxŊA1u,Ŋℬ2u2y2,for allu∈V1andu2y2∈E2.(iii)
(10)Mℬ1L×Mℬ2Lu1,zy1,z=minMℬ1Lu1y1,MA2Lz,Mℬ1U×Mℬ2Uu1,zy1,z=minℬA1Uu1y1,MA2Uz,Ŋℬ1L×Ŋℬ2Lu1,zy1,z=maxŊℬ1Lu1y1,ŊA2Lz,Ŋℬ1U×Ŋℬ2Uu1,zy1,z=maxŊℬ1Uu1y1,ŊA2Uz,Mℬ1×Mℬ2u1,zy1,z=minMℬ1u1y1,MA2z,Ŋℬ1×Ŋℬ2u1,zy1,z=maxŊℬ1u1y1,ŊA2z,for allz∈V2andu1y1∈E1.Example 9.
LetĞ∗=V,E be a graph, where V is the set of vertices and E is the set of edges; then the product of two cubic IFGs in Figures 10–12 is given below.Figure 10
Cubic intuitionistic fuzzy graph.Figure 11
Cubic intuitionistic fuzzy graph.Figure 12
Cartesian product of cubic intuitionistic fuzzy graph.Proposition 3.
IfĞ1 and Ğ2 are strong cubic IFGs, then the Cartesian product Ğ1×Ğ2 is also strong cubic IFG.Proof.
Suppose thatĞ1 and Ğ2 are strong cubic IFGs; then there exist ui,yi∈Ei such that(11)MℬLui,yi=minMALui,MALyi,ŊℬLui,yi=maxŊALui,ŊALyi,MℬUui,yi=minMAUui,MAUyi,ŊℬUui,yi=maxŊAUui,ŊAUyi,Mℬui,yi=minMAui,MAy,Ŋℬui,yi=maxMAui,MAyi.
ConsiderE=u,u2u,y2/u2∈V1,u2y2∈E2∪u1,zy1,z/z∈V2,u1y1∈E1.
Letu,u2u,y2∈E; then(12)Mℬ1L×Mℬ2Lu,u2u,y2=minMA1Lu,Mℬ2Lu2y2=minMA1Lu,MA2Lu2,MA2Ly2.
Similarly,(13)Mℬ1L×Mℬ2Lu,u2u,y2=minMA1Lu,Mℬ2Lu2y2=minMA1Uu,MA2Uu2,MA2Uy2,MA1L×MA2Lu1,u2=minMA1Lu1,MA2Lu2,MA1L×MA2Lu1,u2=minMA1Lu1,MA2LMu2,MA1U×MA2Uu1,y2=minMA1Uu1,MA2Uy2,MA1U×MA2Uu1,y2=minMA1Uu1,MA2Uy2,=minMA1U×MA2Uu,u2,MA1U×MA2Uu,y2=minminMA1Uu,MA2Uu2,minMA1Uu,MA2Uy2=minMA1Uu,MA2Uu2,MA2Uy2.
Hence,(14)Mℬ1L×Mℬ2Lu,u2u,y2=minMA1L×MA2Lu,u2,MA1L×MA2Lu,y2,Mℬ1U×Mℬ2Uu,u2u,y2=minMA1U×MA2Uu,u2,MA1U×MA2Uu,y2.
Similarly, we can show that(15)Ŋℬ1L×Ŋℬ2Lu,u2u,y2=maxŊA1L×ŊA2Lu,u2,ŊA1L×ŊA2Lu,y2,Ŋℬ1U×Ŋℬ2Uu,u2u,y2=maxŊA1U×ŊA2Uu,u2,ŊA1U×ŊA2Uu,y2,Mℬ1×Mℬ2u,u2u,y2=minMA1×MA2u,u2,MA1×MA2u,y2,Ŋℬ1×Ŋℬ2u,u2u,y2=maxŊA1×ŊA2u,u2,ŊA1×ŊA2u,y2.Proposition 4.
IfĞ1×Ğ2 is a strong cubic IFG, then at least Ğ1 or Ğ2 must be strong.Proof.
Suppose thatĞ1 and Ğ2 are not strong cubic IFGs, then there exist ui,yi∈Ei such that(16)MℬLui,yi<minMALui,MALyi,ŊℬLui,yi>maxŊALui,ŊALyi,MℬUui,yi<minMAUui,MAUyi,ŊℬUui,yi>maxŊAUui,ŊAUyi,Mℬui,yi<minMAui,MAy,Ŋℬui,yi>maxMAui,MAyi.
ConsiderE=u,u2u,y2/u2∈V1,u2y2∈E2∪u1,zy1,z/z∈V2,u1y1∈E1.
Letu,u2u,y2∈E, then(17)Mℬ1L×Mℬ2Lu,u2u,y2=minMA1Lu,Mℬ2Lu2y2<minMA1Lu,MA2Lu2,MA2Ly2.
Similarly,(18)Mℬ1L×Mℬ2lu,u2u,y2=minMA1Lu,Mℬ2Lu2y2<minMA1Uu,MA2Uu2,MA2Uy2,MA1L×MA2Lu1,u2=minMA1Lu1,MA2Lu2,MA1U×MA2UMu1,u2=minMA1Uu1,MA2Uu2,MA1L×MA2Lu1,y2=minMA1Lu1,MA2Ly2,MA1U×MA2Uu1,y2=minMA1Uu1,MA2Uy2=minMA1U×MA2Uu,u2,MA1U×MA2Uu,y2=minminMA1Uu,MA2Uu2,minMA1Uu,MA2Uy2=minMA1Uu,MA2Uu2,MA2Uy2.
Hence,(19)Mℬ1L×Mℬ2Lu,u2u,y2<minMA1L×MA2Lu,u2,MA1L×MA2Lu,y2,Mℬ1U×Mℬ2Uu,u2u,y2<minMA1U×MA2Uu,u2,MA1U×MA2Uu,y2.
Similarly, we can show that(20)Ŋℬ1L×Ŋℬ2Lu,u2u,y2>maxŊA1L×ŊA2Lu,u2,ŊA1L×ŊA2Lu,y2,Ŋℬ1U×Ŋℬ2Uu,u2u,y2>maxŊA1U×ŊA2Uu,u2,ŊA1U×ŊA2Uu,y2,Mℬ1×Mℬ2u,u2u,y2<minMA1×MA2u,u2,MA1×MA2u,y2,Ŋℬ1×Ŋℬ2u,u2u,y2>maxŊA1×ŊA2u,u2,ŊA1×ŊA2u,y2.
Therefore,Ğ1×Ğ2 is not a strong cubic IFG, which is a contradiction. This completes the proof.Definition 19.
The compositionĞ1Ğ2=Ğ1°Ğ2=A1°A2,ℬ1°ℬ2 of two cubic IFGs Ğ1=A1,ℬ1 and Ğ2=A2,ℬ2 of the graphs Ğ1∗=V1,E1 and Ğ2∗=V2,E2 is defined as follows:(i)
(21)MA1L°MA2Lu1,u2=minMA1Lu1,MA2Lu2,MA1U°MA2Uu1,u2=minMA1Uu1,MA2Uu2,ŊA1L°ŊA2Lu1,u2=maxŊA1Lu1,ŊA2Lu2,ŊA1U°ŊA2Uu1,u2=maxŊA1Uu1,ŊA2Uu2,MA1°MA2u1,u2=minMA1u1,MA2u2,ŊA1°ŊA2u1,u2=maxŊA1u1,ŊA2u2,for allu1,u2∈V(ii)
(22)Mℬ1L°Mℬ2Lu,u2u,y2=minMA1Lu,Mℬ2Lu2y2,Mℬ1U°Mℬ2Uu,u2u,y2=minMA1Uu,Mℬ2Uu2y2,Ŋℬ1L°Ŋℬ2Lu,u2u,y2=maxŊA1Lu,Ŋℬ2Lu2y2,Ŋℬ1U°Ŋℬ2Uu,u2u,y2=maxŊA1Uu,Ŋℬ2Uu2y2,Mℬ1°Mℬ2u,u2u,y2=minMA1u,Mℬ2u2y2,Ŋℬ1°Ŋℬ2u,u2u,y2=maxŊA1u,Ŋℬ2u2y2,for allu∈V1andu2y2∈E2.(iii)
(23)Mℬ1L°Mℬ2Lu1,zy1,z=minMℬ1Lu1y1,MA2Lz,Mℬ1U°Mℬ2Uu1,zy1,z=minℬA1Uu1y1,MA2Uz,Ŋℬ1L°Ŋℬ2Lu1,zy1,z=maxŊℬ1Lu1y1,ŊA2Lz,Ŋℬ1U°Ŋℬ2Uu1,zy1,z=maxŊℬ1Uu1y1,ŊA2Uz,Mℬ1°Mℬ2u1,zy1,z=minMℬ1u1y1,MA2z,Ŋℬ1°Ŋℬ2u1,zy1,z=maxŊℬ1u1y1,ŊA2z,for allz∈V2andu1y1∈E1.(iv)(24)Mℬ1L°Mℬ2Lu1,u2y1,y2=minMA2Lu2,MA2Ly2,Mℬ1Lu1y1,Mℬ1U°Mℬ2Uu1,u2y1,y2=minMA2Uu2,MA2Uy2,Mℬ1Uu1y1,Ŋℬ1L°Ŋℬ2Lu1,u2y1,y2=maxŊA2Lu2,ŊA2Ly2,Ŋℬ1Lu1y1,Ŋℬ1U°Ŋℬ2Uu1,u2y1,y2=maxŊA2Uu2,ŊA2Uy2,Ŋℬ1Uu1y1,Mℬ1°Mℬ2u1,u2y1,y2=minMA2u2,MA2y2,Mℬ1u1y1,Ŋℬ1°Ŋℬ2u1,u2y1,y2=maxŊA2u2,ŊA2y2,Ŋℬ1u1y1,for allu1,u2y1,y2∈E°−E.Example 10.
LetĞ∗=V,E be a graph; then the compositions of two cubic IFGs in Figures 13–15 are given as follows.Figure 13
Cubic intuitionistic fuzzy graph.Figure 14
Cubic intuitionistic fuzzy graph.Figure 15
Composition of cubic intuitionistic fuzzy graph.Proposition 5.
The compositionĞ1Ğ2 of cubic IFG for the graphs Ğ1 and Ğ2 of the graphs Ğ1∗ and Ğ2∗ is a cubic IFG of Ğ1∗Ğ2∗.Proof.
The proof is straightforward.Definition 20.
The unionĞ1∪Ğ2=A1∪A2,ℬ1∪ℬ2of two cubic IFGs Ğ1=A1,ℬ1 and Ğ2=A2,ℬ2 of the graphs Ğ1∗=V1,E1 and Ğ2∗=V2,E2 is defined as follows:(i)
(25)MA1L∪MA2Lu=MA1Lu,ifu∈V1−V2,MA1L∪MA2Lu=MA2Lu,ifu∈V2−V1,MA1L∪MA2Lu=maxMA1Lu,MA2Lu,ifu∈V1∩V2.(ii)
(26)MA1U∪MA2Uu=MA1Uu,ifu∈V1−V2,MA1U∪MA2Uu=MA2Uu,ifu∈V2−V1,MA1U∪MA2Uu=maxMA1Uu,MA2Uu,ifu∈V1∩V2.(iii)
(27)ŊA1L∩ŊA2Lu=ŊA1Lu,ifu∈V1−V2,ŊA1L∩ŊA2Lu=ŊA2Lu,ifu∈V2−V1,ŊA1L∩ŊA2Lu=minŊA1Lu,ŊA2Lu,ifu∈V1∩V2.(iv)
(28)ŊA1U∩ŊA2Uu=ŊA1Uu,ifu∈V1−V2,ŊA1U∩ŊA2Uu=ŊA2Uuifu∈V2−V1,ŊA1U∩ŊA2Uu=minŊA1Uu,ŊA2Uu,ifu∈V1∩V2.(v)
(29)MA1∪MA2u=MA1u,ifu∈V1−V2,MA1∪MA2u=MA2u,ifu∈V2−V1,MA1∪MA2u=maxMA1u,MA2u,ifu∈V1∩V2.(vi)
(30)ŊA1∩ŊA2u=ŊA1u,ifu∈V1−V2,ŊA1∩ŊA2u=ŊA2u,ifu∈V2−V1,ŊA1∩ŊA2u=minŊA1u,ŊA2u,ifu∈V1∩V2.(vii)
(31)Mℬ1L∪Mℬ2Luy=Mℬ1Luy,ifuy∈E1−E2,Mℬ1L∪Mℬ2Luy=Mℬ2Luy,ifuy∈E2−E1,Mℬ1L∪Mℬ2Luy=maxMℬ1Luy,Mℬ2Luy,ify∈E1∩E2.(viii)
(32)Mℬ1U∪Mℬ2Uuy=Mℬ1Uuy,ifuy∈E1−E2,Mℬ1U∪Mℬ2Uuy=Mℬ2Uuy,ifuy∈E2−E1,Mℬ1U∪Mℬ2Uuy=maxMℬ1Uuy,Mℬ2Uuy,ifuy∈E1∩E2.(ix)
(33)Ŋℬ1L∩Ŋℬ2Luy=Ŋℬ1Luy,ifuy∈E1−E2,Ŋℬ1L∩Ŋℬ2Luy=Ŋℬ2Luy,ifuy∈E2−E1,Ŋℬ1L∩Ŋℬ2Luy=minŊℬ1Luy,Ŋℬ2Luy,ifuy∈E1∩E2.(x)
(34)Ŋℬ1U∩Ŋℬ2Uuy=Ŋℬ1Uuy,ifuy∈E1−E2,Ŋℬ1U∩Ŋℬ2Uuy=Ŋℬ2Uuy,ifuy∈E2−E1,Ŋℬ1U∩Ŋℬ2Uuy=minŊℬ1Uuy,Ŋℬ2Uuy,ifuy∈E1∩E2.(xi)
(35)Mℬ1∪Mℬ2uy=Mℬ1uy,ifuy∈E1−E2,Mℬ1∪Mℬ2uy=Mℬ2uy,ifuy∈E2−E1,Mℬ1∪Mℬ2uy=maxMℬ1uy,Mℬ2uy,ifuy∈E1∩E2.(xii)
(36)Ŋℬ1∩Ŋℬ2uy=Ŋℬ1uy,ifuy∈E1−E2,Ŋℬ1∩Ŋℬ2uy=Ŋℬ2uy,ifuy∈E2−E1,Ŋℬ1∩Ŋℬ2uy=minŊℬ1uy,Ŋℬ2uy,ifuy∈E1∩E2.Example 11.
LetĞ∗=V,E be a graph; then the union of two cubic IFGs is given below.
In Figures16–18 the union of two CIFGs is defined.Figure 16
Cubic intuitionistic fuzzy graph.Figure 17
Cubic intuitionistic fuzzy graph.Figure 18
Union of cubic intuitionistic fuzzy graphs.Proposition 6.
The union of two cubic IFGs is a cubic IFG.Proof.
LetĞ1=A1,ℬ1 and Ğ2=A2,ℬ2 be the cubic IFGs Ğ1∗ and Ğ2∗, respectively. Then, we have to prove Ğ1∪Ğ2=A1∪A2,ℬ1∪ℬ2is a cubic IFG and of the graphs Ğ1∗∪Ğ2∗. As all the conditions of A1∪A2 are satisfied, we only have to verify the conditions of ℬ1∪ℬ2.
First assume thatuy∈E1∩E2. Then,(37)Mℬ1L∪Mℬ2Luy=maxMℬ1Luy,Mℬ2Luy≤maxminMA1Lu,MA1Ly,minMA2Lu,MA2Ly,=minmaxMA1Lu,MA2Lu,maxMA1Ly,MA2Ly,=minMA1L∪MA2Lu,MA1L∪MA2Ly,Mℬ1U∪Mℬ2Uuy=maxMℬ1Uuy,Mℬ2Uuy≤maxminMA1Uu,MA1Uy,minMA2Uu,MA2Ly=minmaxMA1Uu,MA2Uu,maxMA1UMy,MA2Uy=minMA1U∪MA2Uu,MA1U∪MA2Uy,Ŋℬ1L∪Ŋℬ2Luy=minŊℬ1Luy,Ŋℬ2Luy≤minmaxŊA1Lu,ŊA1Ly,maxŊA2Lu,ŊA2Ly,=maxminŊA1Lu,ŊA2Lu,minŊA1Ly,ŊA2Ly,=maxŊA1L∪ŊA2Lu,ŊA1L∪ŊA2Ly,Ŋℬ1U∪Ŋℬ2Uuy=minŊℬ1Uuy,Ŋℬ2Uuy≤minmaxŊA1Uu,ŊA1Uy,maxŊA2Uu,ŊA2Uy,=maxminŊA1Uu,ŊA2Uu,minŊA1Uy,ŊA2Uy,=maxŊA1U∪ŊA2Uu,ŊA1U∪ŊA2Uy,Mℬ1∪Mℬ2uy=maxMℬ1uy,Mℬ2uy≤maxminMA1u,MA1y,minMA2u,MA2y,=minmaxMA1u,MA2u,maxMA1y,MA2y,=minMA1∪MA2u,MA1∪MA2y,Ŋℬ1∪Ŋℬ2uy=minŊℬ1uy,Ŋℬ2uy≤minmaxŊA1u,ŊA1y,maxŊA2u,ŊA2y=maxminŊA1u,ŊA2u,minŊA1y,ŊA2y=maxŊA1∪ŊA2u,ŊA1∪ŊA2y.
Ifuy∈E1 and uy∉E2, then(38)Mℬ1L∪Mℬ2Luy≤minMA1L∪MA2Lu,MA1L∪MA2Ly,Mℬ1U∪Mℬ2Uuy≤minMA1U∪MA2Uu,MA1U∪MA2Uy,Ŋℬ1L∪Ŋℬ2Luy≤maxŊA1L∪ŊA2Lu,ŊA1L∪ŊA2Ly,Ŋℬ1U∪Ŋℬ2Uuy≤maxŊA1U∪ŊA2Uu,ŊA1U∪ŊA2Uy,Mℬ1∪Mℬ2uy≤minMA1∪MA2u,MA1∪MA2y,Ŋℬ1∪Ŋℬ2uy≤maxŊA1∪ŊA2u,ŊA1∪ŊA2y.
Ifuy∉E1 and uy∈E2, then(39)Mℬ1L∪Mℬ2Luy≤minMA1L∪MA2Lu,MA1L∪MA2Ly,Mℬ1U∪Mℬ2Uuy≤minMA1U∪MA2Uu,MA1U∪MA2Uy,Ŋℬ1L∪Ŋℬ2Luy≤maxŊA1L∪ŊA2Lu,ŊA1L∪ŊA2Ly,Ŋℬ1U∪Ŋℬ2Uuy≤maxŊA1U∪ŊA2Uu,ŊA1U∪ŊA2Uy,Mℬ1∪Mℬ2uy≤minMA1∪MA2u,MA1∪MA2y,Ŋℬ1∪Ŋℬ2uy≤maxŊA1∪ŊA2u,ŊA1∪ŊA2y.
This completes the proof.Definition 21.
The jointĞ1+Ğ2=A1+A2,ℬ1+ℬ2 of two cubic IFGs Ğ1=A1,ℬ1 and Ğ2=A2,ℬ2 of the graphs Ğ1∗=V1,E1 and Ğ2∗=V2,E2 is defined as follows:(i)
(40)MA1L+MA2Lu=MA1L∪MA2Lu,MA1U+MA2Uu=MA1U∪MA2UuŊA1L+ŊA2Lu=ŊA1L∪ŊA2Lu,ŊA1U+ŊA2Uu=MA1U∪MA2Uu,MA1+MA2u=MA1∪MA2uŊA1+ŊA2u=ŊA1∪ŊA2u.Ifu∈V1∪V2,(ii)
(41)Mℬ1L+Mℬ2Luy=Mℬ1L∪Mℬ2Luy,Mℬ1U+Mℬ2Uuy=Mℬ1U∪Mℬ2Uuy,Ŋℬ1L+Ŋℬ2Luy=Ŋℬ1L∪Ŋℬ2Luy,Ŋℬ1U+Ŋℬ2Uuy=Mℬ1U∪Mℬ2Uuy,Mℬ1+Mℬ2uy=Mℬ1∪Mℬ2uy,Ŋℬ1+Ŋℬ2uy=Ŋℬ1∪Ŋℬ2uyuy∈E1∩E2, and then(42)Mℬ1L+Mℬ2Luy=minMA1Lu,MA2Ly,Mℬ1U+Mℬ2Uuy=minMA1Uu,MA2Uy,Ŋℬ1L+Ŋℬ2Luy=maxŊA1Lu,ŊA2Ly,Ŋℬ1U+Ŋℬ2Uuy=maxŊA1Uu,ŊA2Uy,Mℬ1+Mℬ2uy=minMA1u,MA2y,Ŋℬ1+Ŋℬ2uy=maxŊA1u,ŊA2y,uy∈E′, where E' is the set of all edges joining the nodes of V1 and V2.Proposition 7.
The joint of two cubic IFGs is a cubic IFG.Proof.
Assume thatĞ1=A1,ℬ1 and Ğ2=A2,ℬ2 are two cubic IFGs of the graphs Ğ1∗=V1,E1 and Ğ2∗=V2,E2. Then, we have to prove Ğ1+Ğ2=A1+A2,ℬ1+ℬ2 is a cubic IFG. In view of proposition 6 is sufficient to verify the case when uy∈E′. In this case, we have(43)Mℬ1L∪Mℬ2Luy=minMA1Lu,MA2UyMℬ1L∪Mℬ2Luy=minMA1Lu,MA2Uy=minMA1L+MA2Lu,MA1L+MA2Ly,Mℬ1U∪Mℬ2Uuy=minMA1Uu,MA2Uy≤minMA1U∪MA2Uu,MA1U∪MA2Uy=minMA1U+MA2Uu,MA1U+MA2Uy,Ŋℬ1L∪Ŋℬ2Luy=maxŊA1Lu,ŊA2Ly≤maxŊA1L∪ŊA2Lu,ŊA1L∪ŊA2Ly=maxŊA1L+ŊA2Lu,ŊA1L+ŊA2Ly,Ŋℬ1U∪Ŋℬ2Uuy=maxŊA1Uu,ŊA2Uy≤maxŊA1U∪ŊA2Uu,ŊA1U∪ŊA2Uy=maxŊA1U+ŊA2Uu,ŊA1U+ŊA2Uy,Mℬ1∪Mℬ2uy=minMA1u,MA2y≤minMA1∪MA2u,MA1∪MA2y=minMA1+MA2u,MA1+MA2y,Ŋℬ1∪Ŋℬ2uy=maxŊA1u,ŊA2y≤maxŊA1∪ŊA2u,ŊA1∪ŊA2y=maxŊA1+ŊA2u,ŊA1+ŊA2y.
This completes the proof.


## 5. Application

In this section, we apply the concept of CIFGs in multiattribute decision-making problem, where the selection of suitable subjects has been carried out.There are many career options for the students of present times. Moreover, some of the courses are usually chosen where all the available choices remain superior and best choices until a single student has to choose a field of his interest by keeping in view his preferences. At the finishing of college level education requires selecting their first choice of career planning. During this time, pupils must be given enough information about choosing career according to their interest. According to the survey of random sample of 100 pupils of classX carried out in this part, pupils with favour of interests and no favouring of choices of a specific subject up to class X are measured and given below. Based on the data, cubic nonrational fuzzy graph is used as a tool as it makes the level of membership (interval-valued membership) (percentage of students who favour a subject or a pair of subjects) and level of nonmembership (interval-valued nonmembership) (percentage of students who disfavour a subject or a pair of subjects). Employing CIFS, the best subject’s combination may be evaluated that are the class having subjects that could be productive to most students and have best academic performance of most of the students.LetS=EnglishE,LanguageL,MathsM,ScienceS,Social SciencesSS be the set of vertices. Tables 1 and 2 illustrate the percentages of students with interest/disinterest towards a subject or a pair of subjects.Table 1
Subject combination.
Subject combinationInterest percentageDisinterest percentageE0.3,0.4,0.30.4,0.5,0.7L0.2,0.4,0.40.55,0.6,0.6M0.2,0.3,0.30.6,0.7,0.5S0.1,0.4,0.50.5,0.6,0.4SS0.2,0.3,0.70.3,0.6,0.3Table 2
Subjects combinations.
Subjects combinationInterest percentageDisinterest percentageE−M0.2,0.3,0.30.6,0.7,0.7E−L0.2,0.4,0.30.55,0.6,0.7E−S0.1,0.4,0.30.5,0.6,0.7E−SS0.2,0.3,0.30.4,0.6,0.7L−M0.2,0.3,0.30.6,0.7,0.6L−S0.1,0.4,0.40.55,0.6,0.6L−SS0.2,0.3,0.40.55,0.6,0.6M−S0.1,0.3,0.30.6,0.7,0.5M−SS0.2,0.3,0.30.6,0.7,0.5S−SS0.1,0.3,0.50.5,0.6,0.4Based on the above information, we generate an CIFG as follows (Figure19).Figure 19
Cubic intuitionistic fuzzy graph.In every vertex of the graph, the degree of membership shows the percentage of students with zeal for a specific subject and the degree of nonmembership is the percentage of students with no zeal in subject from a random sample of 100 students of classX chosen for survey. Also, the corners of graph of both membership and nonmembership show the favour and disfavour of students to study the combined two subjects at higher secondary corner. From the given graph, the corner (L−SS) possesses high degree of nonmembership, which shows that majority of pupils do not like to study the combined subjects Language and Social Science, and the corner (M−S) possesses high degree of membership, which shows that majority of pupils have zeal for studying the combined subjects of Math and Science. There is disfavour to study the combined subjects of Tamil and Math, which indicates that these subjects do not require to be combined. Therefore, a high (low) level of membership of any corner shows the high (low) weightage of combined subjects at higher studies.


## 6. Comparison

Proposition 8.
A cubic IFG is a generalization of cubic FG.Proof.
LetĞ∗=V,E be a cubic IFG. Then if we put the value of nonmembership of the vertex set and edge set as zero in the IVFS and FS, then the cubic IFG reduces to cubic FG.Proposition 9.
An IVIFG is a generalization of IVFG.Proof.
LetĞ∗=V,E be an IVIFG. If we put the value of nonmembership of the vertex set and edge set as zero, then the IVIFG reduces to IVFG.Proposition 10.
An IFG is a generalization of FG.Proof.
LetĞ∗=V,E be an IFG. If we put the value of nonmembership of the vertex set and edge set as zero, then the IFG reduces to FG.


## 7. Conclusion

In this article, we developed a novel concept of CIFG as a generalization of IFGs. The graph theoretic terms like subgraphs, complements, degree of vertices, strength of graphs, paths, and cycle are briefly presented with the help of examples. Some related results and properties of the defined concepts are discussed. The generalization of CIFG is proved by some examples and remarks. A comparison of CIFG with IFG and other related concepts is given. The theory of CIFG is a generalization of IFG and can be applied to many real-life problems such as shortest path problem, communication problem, cluster analysis, and traffic signal problems. In the future, the graphs of the cubic Pythagorean fuzzy sets, cubic q-rung orthopair fuzzy sets, and cubic spherical fuzzy sets can be developed and different aggregation operators are defined for better decision-making.



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