# Notion of Complex Spherical Fuzzy Graph with Application

Authors: Muhammad Shoaib; Waqas Mahmood; Weded Albalawi; Faria Ahmad Shami
Journal: Journal of Function Spaces
 (2022)

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## Abstract

A complex spherical fuzzy set (CSFS) is a generalization of a spherical fuzzy set (CFS). CSFS handles vagueness more explicitly, and its range is expanded from the real subset to the complex with unit disc. The major goal of this research is to present the foundation of a complex spherical fuzzy graph (CSFG) due to the limitation of the complex neutral membership function in a complex Pythagorean fuzzy graph (CPFG). Complex spherical fuzzy models have more flexibility as compared to complex fuzzy models, complex intuitionistic fuzzy models, and complex Pythagorean fuzzy models due to their coverage in three directions: complex membership functions, neutral membership functions, and complex non-membership functions. Firstly, we present the motivation for CSFG. Furthermore, we define the order, degree of a vertex, size, and total degree of a vertex of CSFG. We elaborate on primary operations, including complement, join, and the union of CSFG. This research study introduces some operations, namely, strong product, composition, Cartesian product, and semi-strong product, on CSFG. Moreover, we present the application of CSFG, which ensures the ability to deal with problems in three directions.


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## Body

## 1. Introduction

Zadeh [1] proposed the fuzzy set (FS) as an extension of the crisp set, and it has numerous applications in networking, decision making, telecommunication, artificial intelligence, management sciences, computer science, social science, and the chemical industry. The FS theory was created in order to deal with problems involving a lack of information and impreciseness. FS can deal with uncertainty and vague problems whose membership functions lie in the interval [0,1]. Atanassov [2] extended the FS to the intuitionistic fuzzy set (IFS), which consists of truthness μ and falsity degree ν and fulfils the condition μ+ν≤1 involving hesitancy part.IFSs provide valuable scope in various fields of the world. Yager [3, 4] proposed the concept of the Pythagorean fuzzy set (PFS), which consists of truthness μ and falsity degree ν with the condition μ2+ν2≤1 due to a lack of both grades with the condition μ2+ν2≤1. For example, truthness grade 0.8 and falsity grade 0.3 fail to fulfil the condition μ+ν≤1, i.e., 0.8 + 0.3 = 1.1 is larger than 1. Both grades satisfy the μ2+ν2≤1 condition, i.e., 0.82+0.32=0.71. Therefore, a CPFS is much better than IFS. References [5–8] demonstrated decision making using the PFS. Later, the PFS was extended to include interval-valued PFSs. Cuong [9] proposed the picture fuzzy set (Pic-FS), which deals with neutrality, truth, and false membership grade.Graph models have many applications in networking, sciences, technology, and operational management. To analyze world problems, boundedness between the nodes exists due to a few relations. Many researchers [10–22] participated in improving fuzzy concepts in different fields. Thirunavukarasu et al. [23] initiated the concept of complex fuzzy graphs. A FS can be defined by a complex-valued truth membership function, which is a mix of a traditional truth membership function and an extra term known as the phase term. In this study, the motivation for a fuzzy graph is expanded to include a complex fuzzy graph. A complex fuzzy theory is important in mathematics because it provides greater flexibility, accuracy, and comparability to the system than a fuzzy model. The complexity of the FS originates in the range of values that its membership function may achieve. In contrast to a traditional fuzzy membership function, its range is expanded to the complex plane’s unit circle rather than [0, 1].Naveed et al. [24] studied a complex neutrosophic graph. Shoaib et al. [25] discussed the properties of CPFG. Akram et al. [26] described decision-making problems based on the spherical fuzzy graph (SFG). Akram et al. [27] described the properties of spherical fuzzy averaging operator. CSFS was proposed by Mahmood et al. [28]. Poulik and Ghori [29–31] worked on fuzzy graph theory. Poulik et al. [32] described the Randic index of bipolar fuzzy graphs and its application in network system. Poulik and Ghori [33] described well-known operations on bipolar fuzzy graphs. We apply these operations to a new type of graph, which is called CSFG. Shoaib et al. [34] defined new operations briefly on PFzG.Our main contributions are given as follows:The notion of CSFG is initiated.With the help of an example, the order and size of CSFG are defined. The complement of CSFG is determined.The join, union, and ring sum of CSFG are defined.CSFG’s degree and total degree are discussed.Strong product, composition, Cartesian product, and semi-strong product of CSFG are established.A three-dimensional problem is elaborated in the application of CSFG.The advantages of our concept are mentioned below:CSFG deals with a three-dimensional phenomenon for intuitive knowledge.The phase term of CSFG avoids any loss of knowledge. CSFG accommodates a large amount of data and is more applicable in different fields.CSFG has both the properties of a SFG and a complex fuzzy graph.The concept of CSFGs is elaborated in the CSFS environment. Different concepts and definitions, along with operations, are presented for CSFGs. The complex spherical fuzzy environment is more adaptive and generalized than the spherical fuzzy environment due to phase terms.


## 2. Preliminaries

Definition 1.
(see [35]). The score function of complex spherical fuzzy number Q=μQeiαQ,λQeiγQ,νQeiβQ is defined as follows.
SQ =  μQ2−λQ2−νQ2−1/4π2αQ2−γQ2−βQ2.Definition 2.
(see [36]). The accuracy function of complex spherical fuzzy number Q=μQeiαQ,λQeiγQ,νQeiβQ is defined as follows.
SQ =  μQ2+λQ2+νQ2 + 1/4π2αQ2+γQ2+βQ2.Definition 3.
(see [36]). For the comparison of two complex spherical fuzzy numbers Q1 =  μQ1eiαQ1,λQ1eiγQ1,νQ1eiβQ1 and Q2 =  μQ2eiαQ2,λQ2eiγQ2,νQ2eiβQ2,(1)
IfSQ1>SQ2, then Q1>Q2 (Q1 is superior to Q2).(2)
IfSQ1=SQ2, thenIfAQ1>AQ2, then Q1>Q2 (Q1 is superior to Q2).IfAQ1=AQ2, then Q1∼Q2 (Q1 is equivalent to Q2).Definition 4.
(see [36]). A CPFS Q on a universe Y is represented as Q =  x,μQxeiαQx,νQxeiβQx|x∈Y,where i2=−1, μQx, νQx are real-valued and both belong to [0,1] such that 0≤μQ2x+νQ2x≤1 and αQx, βQx∈0,2π with 0≤αQ2x/2π+βQ2x/2π≤1, for all x ∈Y. Note that μQx, νQx are called amplitude terms and αQx, βQx are called phase terms.
Degree of refusal is given bytQx =  1−μQ2x−νQ2x1/2ei2π1−αQ2x/2π2−βQ2x/2π21/2.Definition 5.
(see [36]). SFS Q on a universe Y is represented as Q =  x,μQx,λQx,νQx|x∈Y,where μQx, λQx, νQx are positive membership, neutral membership, and negative membership which are restricted to the unit interval [0,1].
0≤μQ2x+λQ2x+νQ2x≤1, for all x ∈Y. Degree of refusal is given by tQx =  1−μQ2x−λQ2x−νQ2x1/2.Definition 6.
(see [36]). CSFS Q on a universe Y is represented as Q =  x,μQxeiαQx,λQxeiγQx,νQxeiβQx|x∈Y,where i2=−1, μQx, λQx, νQx are real-valued and all belong to [0,1] such that 0≤μQ2x+λQ2x+νQ2x≤1 and αQx, γQx, βQx∈0,2π with 0≤αQ2x/2π+γQ2x/2π+βQ2x/2π≤1, for all x ∈Y. Note that μQx, λQx, νQx are known as amplitude terms and αQx, γQx, βQx are known as phase terms.
Degree of refusal is given bytQx =  1−μQ2x−λQ2x−νQ2x1/2ei2π1−αQ2x/2π2−γQ2x/2π2−βQ2x/2π21/2.Definition 7.
(see [25]). CPFG on a universe Y with underlying set A is an ordered pair G=Q,Z, Q is a CPFS on A, and Z is CSFS on B⊆A×A such that(1)μZxyeiαZxy≤minμQx,μQyeiminαQx,αQy,νZxyeiβZxy≤maxνQx,νQyeimaxβQx,βQy,where 0≤μQ2x+νQ2x≤1 and αQx, βQx∈0,2π∀x, y∈A.


## 3. Basic Concepts and Operation of CSFGs

This section defines and illustrates CSFG. We demonstrate with examples how to define the order, size, complement, union, join, and ring sum of CSFGs. On CSFGs, we define four distinct types of operations: Cartesian products, semi-strong products, compositions, and strong products.Definition 8.
CSFG on a universe Y with underlying set A is an ordered pairG=Q,Z, Q is CSFS on A, and Z is CSFS on B⊆A×A such that(2)μZxyeiαZxy≤minμQx,μQyeiminαQx,αQy,λZxyeiγZxy≤minλQx,λQyeiminγQx,γQy,νZxyeiβZxy≤maxνQx,νQyeimaxβQx,βQy,where 0≤μQ2x+λQ2x+νQ2x≤1 and αQx, γQx, βQx∈0,2π∀x, y∈A.Definition 9.
LetQ=x,μQxeiαQx,λQxeiγQx,νQxeiβQx|x∈Y, Q1=x,μQ1xeiαQ1x,λQ1xeiγQ1x,νQ1xeiβQ1x|x∈Y, and Q2=x,μQ2xeiαQ2x,λQ2xeiγQ2x,νQ2xeiβQ2x|x∈Y be three CSFSs in Y; then,(i)
Q1⊆Q2 if and only if μQ1≤μQ2, λQ1≤λQ2,νQ1≤νQ2 for amplitude terms and αQ1≤αQ2, γQ1≤γQ2,βQ1≤βQ2 for phase terms, for all x ∈ Y.(ii)
Q1=Q2 if and only if μQ1=μQ2, λQ1=λQ2,νQ1=νQ2 for amplitude terms and αQ1=αQ2, γQ1=γQ2,βQ1=βQ2 for phase terms, for all x∈Y.(iii)
Q¯ =  x,νQxeiβQx,λQxeiγQx,μQxeiαQx|x∈Y.
For simplicity, the tripletμeiα,λeiγ,νeiβ is known as complex spherical fuzzy number.
μ, λ, ν∈ [0,1] such that μ2 + λ2 + ν2≤1 and α, γ,β∈0,2π such that 0≤μ2 +λ2 +ν2≤2π.Example 1.
LetY be a fixed set with just one element x, μQx = 0.6, αQx = 0.8 π, λQx = 0.4, γQx = 0.6 π, νQx = 0.4, βQx = 0.2 π. Then, Q=x,0.6ei2π0.4,0.4ei2π0.3,0.4ei2π0.1 is a complex spherical fuzzy number and denoted by Q=0.6ei2π0.4,0.4ei2π0.3,0.4ei2π0.1 for simplicity.Definition 10.
LetQ1=a,μQ1aeiαQ1a,λQ1aeiγQ1a,νQ1aeiβQ1a|x∈Yand Q2=a,μQ2aeiαQ2a,λQ2aeiγQ2a,νQ2aeiβQ2a|a∈Y be the two CSFSs in Y; then,(i)
Q1∪Q2  =  a,maxμQ1,μQ2aeimaxαQ1a,αQ2a,maxλQ1a,λQ2aeimaxγQ1a,γQ2a,minνQ1a,νQ2aeiminβQ1a,βQ2a|x∈Y.(ii)
Q1∩Q2  =  a,minμQ1,μQ2aeiminαQ1a,αQ2a,minλQ1a,λQ2aeiminγQ1a,γQ2a,maxνQ1a,νQ2aeimaxβQ1a,βQ2a|a∈Y.Example 2.
Suppose thatQ1=a,0.3ei0.2π,0.4ei0.2π,0.2ei0.4π and Q2=a,0.1ei0.3π,0.3ei0.3π,0.4ei0.2π are two CSFSs; then,Q1∪Q2 =  a,0.3ei0.3π,0.4ei0.3π,0.2ei0.2π.Q1∩Q2 =  a,0.1ei0.2π,0.3ei0.2π,0.4ei0.4π.Definition 11.
CSFSZ in Y × Y is called complex spherical fuzzy relation in Y, characterized by Q1=xy,μZxyeiαZxy,λZxyeiγZxy,νZxyeiβZxy|x∈Y×Y,
whereμZ: Y × Y ⟶ [0,1], λZ: Y × Y ⟶ [0,1], and νZ: Y × Y ⟶ [0,1] depict the membership, neutral, and non-membership functions of Z, respectively, such that 0≤μZ2x+λZ2x+νZ2x≤1 and 0≤αZ2x+γZ2x+βZ2x≤2π for all xy ∈ Y × Y.Example 3.
Consider a graphG = (A,B), Q=s1,s2,s3,s4,s5,s6 is the vertex set, and Z=s2s1,s1s3,s3s4,s4s5,s5s6 is the edge set of G. G=Q,Z be CSFG on A, as given in Figure 1, defined by(3)Q=<s10.3e0.4πi,s20.5e0.5πi,s30.4e0.6πi,s40.4e0.6πi,s50.4e0.3πi,s60.4e0.6πi,s10.4e0.3πi,s20.3e0.4πi,s30.3e0.1πi,s40.3e0.1πi,s50.2e0.1πi,s60.3e0.1πi,s10.5e0.2πi,s20.2e0.1πi,s30.4e0.4πi,s40.4e0.5πi,s50.3e0.3πi,s60.4e0.3πi>,Z=<s2s10.2e0.4πi,s1s30.3e0.4πi,s3s40.1e0.4πi,s4s50.3e0.3πi,s5s60.3e0.3πi,s2s10.2e0.2πi,s1s30.3e0.1πi,s3s40.3e0.1πi,s4s50.2e0.1πi,s5s60.2e0.1πi,s2s10.5e0.2πi,s1s30.5e0.4πi,s3s40.4e0.4πi,s4s50.4e0.5πi,s5s60.4e0.3πi
In order to compare CSFGs and SFGs, we must convert their vertices and edges from complex spherical fuzzy numbers to spherical fuzzy numbers by treating the phase terms of each complex spherical fuzzy value as zero, as shown in Figure2, which is the vertex set and edge set of the CSFG shown in Figure 1.
When employing the proposed extended fuzzy graph, known as the CSFG, it is more fair to incorporate facts into the decision-making process. There are three sorts of degrees available in an SFG: membership degrees, neutral degrees, and non-membership degrees. The amplitude term is the sole term that is investigated, resulting in information loss. In addition, CSFG is an extension of existing theories such as fuzzy graphs, complex fuzzy graphs, and SFGs in that it takes into account increasing quantities of information about vertices and relations and works with two-dimensional data in a single set.Figure 1
CSFG.Figure 2
SFG.Definition 12.
LetQ=a,μQaeiαQa,λQaeiγQa,νQaeiβQa|a∈A and Z=ab,μZabeiαZab,λZabeiγZab,νZabeiβZab|ab∈B be the vertex set and edge set of CSFG G; then, the order of CSFG is defined as(4)OG=∑ai∈AμQaiei∑ai∈AαQai,∑ai∈AλQaiei∑γQ∑ai∈AγQai,∑ai∈AνQaiei∑ai∈AβQai.
The size of CSFGG is defined as(5)SG=∑ai∈AμZaibjei∑aibj∈AαZaibj,∑aibj∈AλQaibjei∑aibj∈AγQaibj,∑aibj∈AνZaibjei∑aibj∈AβZaibj.Example 4.
The order and size of CSFG given in Figure1 are O G =  2.4e3πi,1.8e1.1πi,2.2e1.8πi and S G =  1.2e1.8πi,1.2e0.7πi,2.2e1.8πi, respectively.Definition 13.
The complement of CSFGG = (Q, Z) on the underlying graph G = (A, B) is CSFG G¯ =  Q¯,Q¯ defined by(1)
μQxeiαQx¯ =  μQxeiαQx, λQxeiγQx¯ =  λQxeiγQx and.νQxeiβQx¯ =  νQxeiβQx.(2)
μZxyeiαZxy¯=minμQx,μQyeminαQx,αQyiifμZxyeiαZxy=0,minμQx,μQyeminαQx,αQyi−μZxyeiαZxyif0<μZxyeiαZxy≤1.λZxyeiγZxy¯=minλQx,λQyeminγQx,γQyiifλZxyeiγZxy=0,minλQx,λQyeminγQx,γQyi−λZxyeiγZxyif0<λZxyeiγZxy≤1..νZxyeiβZxy¯=maxνQx,νQyemaxβQx,βQyiifνZxyeiβZxy=0,maxνQx,νQyemaxβQx,βQyi−νZxyiβZxyif0<νZxyeiβZxy≤1.Example 5.
ConsiderCSFGG = (Q,Z) onA=s1,s2,s3, as shown in Figure 3, defined by(6)Q=<s10.3e0.4πi,s20.5e0.5πi,s30.4e0.6πi,s10.4e0.3πi,s20.3e0.4πi,s30.3e0.1πi,s10.5e0.2πi,s20.2e0.1πi,s30.4e0.4πi>Z=<s2s10.1e0.1πi,s1s30.3e0.4πi,s2s10.2e0.2πi,s1s30.3e0.1πi,s2s10.4e0.1πi,s1s30.5e0.4πi.
Utilizing Definition13, complement of CSFG can be obtained, as given in Figure 4, and defined as(7)Q¯=<s10.3e0.4πi,s20.5e0.5πi,s30.4e0.6πi,s10.4e0.3πi,s20.3e0.4πi,s30.3e0.1πi,s10.5e0.2πi,s20.2e0.1πi,s30.4e0.4πi>Z¯=<s2s10.2e0.3πi,s2s30.4e0.5πi,s2s10.1e0.1πi,s2s30.3e0.1πi,s2s10.1e0.2πi,s2s30.4e0.4πi.
It can be observed from Figure4 that G¯ =  Q¯,Z¯ is CSFG.Figure 3
CSFG.Figure 4
Complement of CSFG.Theorem 1.
The complement of a complement of CSFG is CSFG itself, that is,G¯¯ =  G.Proof.
LetG be a CSFG. Then, from Definition 13, we get μQx¯¯eiαQx¯¯ =  μQx¯eiαQx¯ =  μQxeiαQx, λQx¯¯eiγQx¯¯ =  λQx¯eiγQx¯ =  λQxeiγQx, νQx¯¯eiβQx¯¯ =  νQx¯eiβQx¯ =  νQxeiβQx, for all x ∈ A.
IfμZxyeiαZxy=0, λZxyeiγZxy=0, νZxyeiβZxy=0, then μZxy¯¯eiαZxy¯¯ =  minμQx¯,μQy¯eiminαQx¯,αQy¯, λZxy¯¯eiγZxy¯¯ =  minλQx¯,λQy¯eiminγQx¯,γQy¯, νZxy¯¯eiβZxy¯¯ =  maxνQx¯,νQy¯eimaxβQx¯,βQy¯.
If0<μZxyeiαZxy≤1, 0<λZxyeiγZxy≤1, 0<νZxyeiβZxy≤1, then μZxy¯¯eiαZxy¯¯ =  minμQx¯,μQy¯eiminαQx¯,αQy¯−μZxy¯eiαZxy¯, μZxy¯¯eiαZxy¯¯ =  minμQx¯,μQy¯eiminαQx¯,αQy¯ - minμQx,μQyeminαQx,αQyi−μZxyeiαZxy, μZxy¯¯eiαZxy¯¯ =  μZxyeiαZxy, λZxy¯¯eiγZxy¯¯ =  minλQx¯,λQy¯eiminγQx¯,γQy¯−λZxy¯eiγZxy¯, λZxy¯¯eiγZxy¯¯ =  minλQx¯,λQy¯eiminγQx¯,γQy¯ - minλQx,λQyeminγQx,γQyi−λZxyeiγZxy, νZxy¯¯eiβZxy¯¯ =  maxνQx¯,νQy¯eimaxβQx¯,βQy¯− νZxy¯eiβZxy¯, νZxy¯¯eiβZxy¯¯ =  maxνQx¯,νQy¯eimaxβQx¯,βQy¯ - maxνQx,νQyemaxβQx,βQyi−νZxyeiβZxy, νZxy¯¯eiβZxy¯¯ =  νZxyeiβZxy, for all x, y∈ A.
Hence,G¯¯ =  G.Definition 14.
The unionG1∪G2 =  Q1∪Q2,Z1∪Z2 of two CSFGs G1 =  Q1,Z1 and G2 =  Q2,Z2 of the graphs G1 =  A1,B1 and G2 =  A2,B2, respectively, is defined as follows.μQ1∪μQ2 (x)eiαQ1∪αQ2.(8)μQ1∪μQ2xeiαQ1∪αQ2x=μQ1xeiαQ1xifx∈A1−A2,μQ2xeiαQ1xifx∈A2−A1,maxμQ1x,μQ2xeimaxαQ1x,αQ2xifx∈A1∩A2,(9)λQ1∪λQ2xeiγQ1∪γQ2x=λQ1xeiγQ1x,ifx∈A1−A2,λQ2xeiγQ2x,ifx∈A2−A1,maxλQ1x,λQ2xeimaxγQ1x,γQ2x,ifx∈A1∩A2,(10)νQ1∪νQ2xeiβQ1∪βQ2x=νQ1xeiβQ1x,ifx∈A1−A2,νQ2xeiβQ2x,ifx∈A2−A1,minνQ1x,νQ2xeiminβQ1x,βQ2x,ifx∈A1∩A2,(11)μZ1∪μZ2xyeiαZ1∪αZ2xy=μZ1xyeiαZ1xy,ifxy∈B1−B2,μZ2xyeiαZ2xy,ifxy∈B2−B1,maxμZ1xy,μZ2xyeimaxαZ1xy,αZ2xy,ifxy∈B1∩B2,(12)λZ1∪λZ2xyeiγZ1∪γZ2xy=λZ1xyeiγZ1xyifxy∈B1−B2,λZ2xyeiγZ2xyifxy∈B2−B1,maxλZ1xy,λZ2xyeimaxγZ1xy,γZ2xyifxy∈B1∩B2,(13)νZ1∪νZ2xyeiβZ1∪βZ2xy=νZ1xyeiβZ1xy,ifxy∈B1−B2,νZ2xyeiβZ2xy,ifxy∈B2−B1,minνZ1xy,νZ2xyeiminβZ1xy,βZ2xy,ifxy∈B1∩B2.Definition 15.
The ring sumG1⊕G2 =  Q1⊕Q2,Z1⊕Z2 of two CSFGs G1 =  Q1,Z1 and G2 =  Q2,Z2 of the graphs G1 and G2, respectively, is defined as follows:μQ1⊕μQ2xeiαQ1⊕αQ2x =  μQ1∪μQ2xeiαQ1∪αQ2x,λQ1⊕λQ2xeiγQ1⊕γQ2x =  λQ1∪λQ2xeiγQ1∪γQ2x,νQ1⊕νQ2xeiβQ1⊕βQ2x =  νQ1∪νQ2xeiβQ1∪βQ2x, if x∈A1∪A2,(14)μZ1⊗μZ2xyeiαZ1∪αZ2xy=μZ1xyeiαZ1xy,ifxy∈B1−B2,μZ2xyeiαZ2xy,ifxy∈B2−B1,0,ifxy∈B1∩B2,(15)λZ1⊗λZ2xyeiγZ1⊗γZ2xy=λZ1xyeiγZ1xy,ifxy∈B1−B2,λZ2xyeiγZ2xy,ifxy∈B2−B1,0,ifxy∈B1∩B2,(16)νZ1⊗νZ2xyeiβZ1⊗βZ2xy=νZ1xyeiβZ1xy,ifxy∈B1−B2,νZ2xyeiβZ2xy,ifxy∈B2−B1,0,ifxy∈B1∩B2.Definition 16.
LetG1 =  Q1,Z1 and G2 =  Q2,Z2 be two CSFGs of G1 and G2, respectively. The join G1+G2 =  Q1+Q2,Z1+Z2 of G1 =  Q1,Z1 and G2 =  Q2,Z2 is defined as(i)
μQ1+μQ2xeiαQ1+αQ2x = μQ1∪μQ2xeiαQ1∪αQ2x,λQ1+λQ2xeiγQ1+γQ2x = λQ1∪λQ2xeiγQ1∪γQ2x,νQ1+νQ2xeiβQ1+βQ2x = νQ1∪νQ2xeiβQ1∪βQ2x, for all x∈A1∪A2,(ii)
μZ1+μZ2xyeiαZ1+αZ2xy =  μZ1∪μZ2xyeiαZ1∪αZ2xy,λZ1+λZ2xyeiγZ1+γZ2xy =  λZ1∪λZ2xyeiγZ1∪γZ2xy,νZ1+νZ2xyeiβZ1+βZ2xy =  νZ1∪νZ2xyeiβZ1∪βZ2xy, if xy∈B1∪B2,(iii)
μZ1+μZ2xyeiαZ1+αZ2xy =  minμQ1x,μQ2yeiminαQ1x,αQ2y,λZ1+λZ2xyeiγZ1+γZ2xy =  minλQ1x,λQ2yeiminγQ1x,γQ2y,νZ1+νZ2xyeiβZ1+βZ2xy =  maxνQ1x,νQ2yeimaxβQ1x,βQ2y, if xy. ∈B′,where B′ is the set of all edges joining the vertices of A1 and A2, A1∩A2=∅.Definition 17.
The degree of a vertexx∈A in CSFG G is denoted by dGx and is defined asdGx =  dμeiαx,dλeiγx,dνeiβx, wheredμeiαx =  ∑x,y≠x∈AμZxyei∑x,y≠x∈AαZxy,dλeiγx =  ∑x,y≠x∈AλZxyei∑x,y≠x∈AγZxy,dνeiβx =  ∑x,y≠x∈AνZxyei∑x,y≠x∈AβZxy.Definition 18.
The total degree of a vertex x∈ A in CSFG G is denoted by tdGx and is defined astdGx =  tdμeiαx,tdλeiγx,dνeiβx, wheretdμeiαx = ∑x,y≠x∈AμZxyei∑x,y≠x∈AαZxy + μQxeiαQx,tdλeiγx = ∑x,y≠x∈AλZxyei∑x,y≠x∈AγZxy + λQxeiγQx,tdνeiβx = ∑x,y≠x∈AνZxyei∑x,y≠x∈AβZx,y + νQxeiβQx.Definition 19.
LetG1 and G2 be two CSFGs. For any vertex x ∈A1∪A2, there are three cases to consider.Case1:Eitherx∈A1−A2 or x∈A2−A1. Then, there is no edge incident at x which lies in B1∩B2. Thus, for c∈C1−C2,dμeiαG1∪G2x = ∑xy∈B1μZ1xyei∑xy∈B1αZ1xy =  dμeiαG1x,dλeiγG1∪G2x = ∑xy∈B1λZ1xyei∑xy∈B1γZ1xy =  dλeiγG1x,dνeiβG1∪G2x = ∑xy∈B1νZ1xyei∑xy∈B1βZ1xy =  dνeiβG1x,tdμeiαG1∪G2x = tdμeiαG1x, tdλeiγG1∪G2x =  tdλeiγG1x, tdνeiβG1∪G2x = tdνeiβG1x.For x∈A2−A1,dμeiαG1∪G2x = ∑xy∈B2μZ2xyei∑xy∈B2αZ2xy =  dμeiαG2x,dλeiγG1∪G2x = ∑xy∈B2λZ2xyei∑xy∈B2γZ2xy =  dλeiγG2x,dνeiβG1∪G2x = ∑xy∈B2νZ2xyei∑xy∈B2βZ2xy =  dνeiβG2x,tdμeiαG1∪G2x = tdμeiαG2x, tdλeiγG1∪G2x =  tdλeiγG2x, tdνeiβG1∪G2x = tdνeiβG2x.Case2:X∈A1∩A2 but there is no edge incident at x which lies in B1∩B2. Then, any edge incident at x is either B1−B2 or B2−B1.dμeiαG1∪G2x = ∑xy∈B1∪B2μZ1∪μZ2xyei∑xy∈B1∪B2αZ1∪αZ2xy,dμeiαG1∪G2x =  ∑xy∈B1μZ1xyei∑xy∈B1αZ1xy + ∑xy∈B2μZ2xyei∑xy∈B2αZ2xy,dμeiαG1∪G2x =  dμeiαG1x + dμeiαG2x.Similarly,dλeiγG1∪G2x =  dλeiγG1x + dλeiγG2x and dνeiβG1∪G2x =  dνeiβG1x + dνeiβG2x,tdμeiαG1∪G2x = ∑xy∈B1∪B2μZ1∪μZ2xyei∑xy∈B1∪B2αZ1∪αZ2xy + maxμQ1x,μQ2xeimaxαQ1x,αQ2x,tdμeiαG1∪G2x = ∑xy∈B1μZ1xyei∑xy∈B1αZ1xy + ∑xy∈B2μZ2xyei∑xy∈B2αZ2xy + maxμQ1x,μQ2xeimaxαQ1x,αQ2x,tdμeiαG1∪G2x = dμeiαG1x + dνeiβG2x + maxμQ1x,μQ2xeimaxαQ1x,αQ2x,tdμeiαG1∪G2x = tdμeiαG1x + tdμeiαG2x − maxμQ1x,μQ2xeiminαQ1x,αQ2x,Similarly,tdλeiγG1∪G2x = tdλeiγG1x − tdλeiγG2x−minλQ1x,λQ2xeiminγQ1x,γQ2x, andtdνeiβG1∪G2x = tdνeiβG1x + tdνeiβG2x − maxνQ1x,νQ2xeimaxβQ1x,βQ2x.Case3:(17)dμeiαG1∪G2x=∑xy∈B1∪B2μZ1∪μZ2xy,=∑xy∈B1−B2μZ1xyei∑xy∈B1−B2αZ1xy+∑xy∈B2−B1μZ2xyei∑xy∈B2−B1αZ2xy,+∑xy∈B1∩B2maxμZ1xy,μZ2xyeimaxαZ1xy,αZ2xy,=∑xy∈B1−B2μZ1xyei∑xy∈B1−B2αZ1xy+∑xy∈B2−B1μZ2xyei∑xy∈B2−B1xy∈B2−B1xy,+∑xy∈B1∩B2maxμZ1xy,μZ2xyeimaxαZ1xy,αZ2xy,+∑xy∈B1∩B2minμZ1xy,μZ2xyeiminαZ1xy,αZ2xy,−∑xy∈B1∩B2minμZ1xy,μZ2xyeiminαZ1xy,αZ2xy,=∑xy∈B1μZ1xyei∑xy∈B1αZ1xy+∑xy∈B2μZ2xyei∑xy∈B2αZ2xy,−∑xy∈B1∩B2minμZ1xy,μZ2xyeiminαZ1xy,αZ2xy,=dμeiαG1x+dμeiαG2x−∑xy∈B1∩B2minμZ1xy,μZ2xyeiminαZ1xy,αZ2xy.
Similarly,(18)dλeiγG1∪G2x=dλeiγG1x+dλeiγG2x−∑xy∈B1∩B2minλZ1xy,λZ2xyeiminγZ1xy,γZ2xy,dνeiβG1∪G2x=dνeiβG1x+dνeiβG2x−∑xy∈B1∩B2maxνZ1xy,νZ2xyeimaxβZ1xy,βZ2xy.
In addition,(19)tdμeiαG1∪G2x=tdμeiαG1x+tdμeiαG2x−∑xy∈B1∩B2minμZ1xy,μZ2xyeiminαZ1xy,αZ2xy,−minμQ1x,μQ2xeiminαQ1x,αQ2x,tdλeiγG1∪G2x=tdλeiγG1x+tdλeiγG2x−∑xy∈B1∩B2minλZ1xy,λZ2xyeiminγZ1xy,γZ2xy,−minλQ1x,λQ2xeiminγQ1x,γQ2x,tdνeiβG1∪G2x=tdνeiβG1x+tdνeiβG2x−∑xy∈B1∩B2maxνZ1xy,νZ2xyeimaxβZ1xy,βZ2xy,−maxνQ1x,νQ2xeimaxβQ1x,βQ2x.Example 6.
Suppose thatG1=Q1,Z1 and G2=Q2,Z2 are two CSFGs on A1=s1,s2,s4 and A2=s1,s2,s3,s4, respectively, as shown in Figures 5 and 6.
Moreover,G1∪G2 is shown in Figure 7.
Sinces3∈A2A1, thendμeiαG1∪G2s3 =  dμeiαG2s3 =  0.5e0.7πidλeiγG1∪G2s3 =  dλeiγG2s3 =  0.2e0.4πidνeiβG1∪G2s3 =  dνeiβG2s3 =  0.8e0.5πiTherefore, dG1∪G2s3 =  dG2s3 =(0.5e0.7πi,0.2e0.4πi,0.8e0.5πi).tdμeiαG1∪G2s3 =  tdμeiαG2s3 =  1.1e1.4πitdλeiγG1∪G2s3 =  tdλeiγG2s3 =  0.3e0.8πitdνeiβG1∪G2s3 =  tdνeiβG2s3 =  0.9e0.9πiTherefore, tdG1∪G2s3 =  tdG2s3 = (1.1e1.4πi,0.3e0.8πi,0.9e0.9πi).Since s4∈A1∩A2, there is no edge incident at s4 that lies in B1∩B2.dμeiαG1∪G2s4 =  dμeiαG1s4 + dμeiαG2s4 =  0.5e1.1πidλeiγG1∪G2s4 =  dλeiγG1s4 + dλeiγG2s4 =  0.5e0.6πidμeiαG1∪G2s4 =  dμeiαG1s4 + dμeiαG2s4 =  1.1e0.9πiTherefore, dG1∪G2s4 =  dG1s4 +dG2s4 =(0.5e1.1πi,0.5e0.6πi,1.1e0.9πi).tdμeiαG1∪G2s4 = tdμeiαG1s4 +tdμeiαG2s4 + maxμQ1s4,μQ2s4emaxαQ1s4,αQ2s4i, =  0.9e1.9πitdλeiγG1∪G2s4 =  tdλeiγG1s4 +tdλeiγG2s4 +maxλQ1s4,λQ2s4emaxγQ1s4,γQ2s4i =  0.9e1.1πitdνeiβG1∪G2s4 = tdνeiβG1s4 +tdνeiβG2s4 =  0.8e0.7πi
Therefore,tdνeiβG1∪G2s4 =  tdG1s4 +tdG2s4 =  maxνQ1s4,νQ2s4emaxβQ1s4,βQ2s4i =(0.9e1.9πi,0.9e1.1πi,0.8e0.7πi).
Sinces2∈A1∩A2 and s1s2∈B1∩B2,dμeiαG1∪G2s2 =  dμeiαG1s2 +dμeiαG2s2− minμZ1s1s2,μZ2s1s2eminαZ1s1s2,αZ2s1s2i  =  0.4e0.9πi,dλeiγG1∪G2s2 =  dλeiγG1s2 +dλeiγG2s2− minλZ1s1s2,λZ2s1s2eminγZ1s1s2,γZ2s1s2i  =  0.4e1.0πi,dνeiβG1∪G2s2 =  dνeiβG1s2 +dνeiβG2s2− minνZ1s1s2,νZ2s1s2eminβZ1s1s2,βZ2s1s2i  =  1.5e0.9πi.
Therefore,dG1∪G2s2 =  0.4e0.9πi,0.4e1.0πi,1.5e0.9πi.
tdμeiαG1∪G2s2 =  tdμeiαG1s2 +tdμeiαG2s2− minμZ1s1s2,μZ2s1s2eminαZ1s1s2,αZ2s1s2 + maxμQ1s2,μQ2s2emaxαQ1s2,αQ2s1i, =  1.0e1.5πi,
tdλeiγG1∪G2s2 =  tdλeiγG1s2 + tdλeiγG2s2− minλZ1s1s2,λZ2s1s2eminγZ1s1s2,γZ2s1s2 + maxλQ1s2,λQ2s2emaxγQ1s2,γQ2s1i =  0.6e1.5πi,
tdνeiβG1∪G2s2 =  tdνeiβG1s2 +tdνeiβG2s2 -minνZ1s1s2,νZ2s1s2eminβZ1s1s2,βZ2s1s2 + maxνQ1s2,νQ2s2emaxβQ1s2,βQ2s1i =(2.0e1.2πi.
Therefore,tdG1∪G2s2 =  1.0e1.5πi,0.6e1.5πi,2.0e1.2πi).Figure 5
G1.Figure 6
G2.Figure 7
G1∪G2.Definition 20.
Suppose thatG1=Q1,Z1 and G2=Q2,Z2 are two CSFGs defined on G1=A1,B1 and G2=A2,B2, respectively. The Cartesian product of G1 and G2 is represented by G1×G2=Q1×Q2,Z1×Z2 and defined as follows:(i)
(20)μQ1×μQ2x1,x2eiαQ1×αQ2x1,x2=minμQ1x1,μQ2x2eiminαQ1x1,αQ2x2,λQ1×λQ2x1,x2eiγQ1×αQ2x1,x2=minλQ1x1,λQ2x2eiminγQ1x1,αQ2x2,νQ1×νQ2x1,x2eiβQ1×αQ2x1,x2=minνQ1x1,νQ2x2eiminβQ1x1,αQ2x2,∀x1,x2∈A1×A2.(ii)
(21)μZ1×μZ2m,x2m,y2eiαZ1×αZ2m,x2m,y2=minμQ1m,μZ2x2y2eiminαQ1m,αZ2x2y2,λZ1×λZ2m,x2m,y2eiγZ1×γZ2m,x2m,y2=minλQ1m,λZ2x2y2eiminγQ1m,γZ2x2y2,νZ1×νZ2m,x2m,y2eiβZ1×βZ2m,x2m,y2=minνQ1m,νZ2x2y2eiminβQ1m,βZ2x2y2,∀m∈A1andx2y2∈B2.(iii)
(22)μZ1×μZ2x1,zy1,zeiαZ1×αZ2x1,zy1,z=minμZ1x1y1,μQ2zeiminαZ1x1y1,αQ2z,λZ1×λZ2x1,zy1,zeiγZ1×γZ2x1,zy1,z=minλZ1x1y1,λQ2zeiminγZ1x1y1,γQ2z,νZ1×νZ2x1,zy1,zeiβZ1×βZ2x1,zy1,z=minνZ1x1y1,νQ2zeiminβZ1x1y1,βQ2z,∀z∈A2andx1y1∈B1.Example 7.
Suppose thatG1=Q1,Z1 and G2=Q2,Z2 are two CSFGs on A1=a,bandA2=c,d, respectively, which are shown in Figures 8 and 9. Also, the Cartesian product is shown in Figure 10.Figure 8
G1.Figure 9
G2.Figure 10
G1×G2.Proposition 1.
Suppose thatG1=Q1,Z1 and G2=Q2,Z2 are two CSFGs of graph G1=A1,B1 and G2=A2,B2, respectively. Then Cartesian product G1×G2 of G1=A1,B1 and G2=A2,B2 is CSFG.Proof.
Suppose thatG1=Q1,Z1 and G2=Q2,Z2 are two CSFGs of graph G1=A1,B1 and G2=A2,B2, respectively. Then, the Cartesian product G1×G2 of G1=A1,B1 and G2=A2,B2 can be proved. Let x1,x2y1,y2∈B1×B2.(i)
It is trivial.(ii)
Ifx1=y1=m,(23)μZ1×μZ2m,x2m,y2eiαZ1×αZ2m,x2m,y2=minμQ1m,μZ2x2y2eiminαQ1m,αZ2x2y2,≤minμQ1m,minμQ2x2,μQ2y2eiminαQ1m,minαQ2x2αQ2y2,=minminμQ1m,μQ2x2,minμQ1m,μQ2y2eiminminαQ1m,αQ2x2,minαQ1m,αQ2y2,=minμQ1×μQ2m,x2,μQ1×μQ2m,y2eiminαQ1×αQ2m,x2,αQ1×αQ2m,y2,(24)λZ1×λZ2m,x2m,y2eiγZ1×γZ2m,x2m,y2=minλQ1m,λZ2x2y2eiminγQ1m,γZ2x2y2,≤minλQ1m,minλQ2x2,λQ2y2eiminγQ1m,minγQ2x2,γQ2y2,=minminλQ1m,λQ2x2,minλQ1m,λQ2y2eiminminγQ1m,γQ2x2,minγQ1m,γQ2y2,=minλQ1×λQ2m,x2,λQ1×λQ2m,y2eiminγQ1×γQ2m,x2,γQ1×γQ2m,y2,(25)νZ1×νZ2m,x2m,y2eiβZ1×βZ2m,x2m,y2=maxνQ1m,νZ2x2y2eimaxβQ1m,βZ2x2y2≤maxνQ1m,maxνQ2x2,νQ2y2eimaxβQ1m,maxβQ2x2,βQ2y2=maxmaxνQ1m,νQ2x2,maxνQ1m,νQ2y2eimaxmaxβQ1m,βQ2x2,maxβQ1m,βQ2y2=maxνQ1×νQ2m,x2,νQ1×νQ2m,y2eimaxβQ1×βQ2m,x2,βQ1×βQ2m,y2.(iii)
Ifx2=y2=z,(26)μZ1×μZ2x1,zy1,zeiαZ1×αZ2x1,zy1,z=minμZ1x1y1,μQ2zeiminαZ1x1y1,αQ2z≤minminμZ1x1y1,μQ2zeiminminαZ1x1y1,αQ2z,=minminμZ1x1,μQ2z,minμQ1y1,μQ2zeiminminαZ1x1,αQ2z,minαQ1y1,αQ2z,=minμQ1×μQ2x1,z,μQ1×μQ2y1,zeiminαQ1×αQ2x1,z,αQ1×αQ2y1,z,(27)λZ1×λZ2x1,zy1,zeiγZ1×γZ2x1,zy1,z=minλZ1x1y1,λQ2zeiminγZ1x1y1,γQ2z,≤minminλZ1x1y1,λQ2zeiminminγZ1x1y1,γQ2z,=minminλZ1x1,λQ2z,minλQ1y1,λQ2zeiminminγZ1x1,γQ2z,minγQ1y1,γQ2z=minλQ1×λQ2x1,z,λQ1×λQ2y1,zeiminγQ1×γQ2x1,z,γQ1×γQ2y1,z,(28)νZ1×νZ2x1,zy1,zeiβZ1×βZ2x1,zy1,z=minνZ1x1y1,νQ2zeiminβZ1x1y1,βQ2z,≤minminνZ1x1y1,νQ2zeiminminβZ1x1y1,βQ2z,=minminνZ1x1,νQ2z,minνQ1y1,νQ2zeiminminβZ1x1,βQ2z,minβQ1y1,βQ2z,=minνQ1×νQ2x1,z,νQ1×νQ2y1,zeiminβQ1×βQ2x1,z,βQ1×βQ2y1,z.
Hence,G1×G2 is a CSFG.Definition 21.
Suppose thatG1=Q1,Z1 and G2=Q2,Z2 are CSFGs defined on G1=A1,B1 and G2=A2,B2, respectively. The composition of G1 and G2 is represented by G1°G2=Q1°Q2,Z1°Z2. The composition of G1andG2 is defined as follows:(i)
μQ1°μQ2x1,x2eiαQ1°αQ2x1,x2=minμQ1x1,μQ2x2eiminαQ1x1,αQ2x2,λQ1°λQ2x1,x2eiγQ1°γQ2x1,x2=minλQ1x1,λQ2x2eiminγQ1x1,γQ2x2,νQ1°νQ2x1,x2eiβQ1°βQ2x1,x2=maxνQ1x1,νQ2x2eimaxβQ1x1,βQ2x2,∀x1,x2∈A1×A2.(ii)
μZ1°μZ2m,x2m,y2eiαZ1°αZ2m,x2m,y2=minμQ1m,μZ2x2y2eiminαQ1m,αZ2x2y2,λZ1°λZ2m,x2m,y2eiγZ1°γZ2m,x2m,y2=minλQ1m,λZ2x2y2eiminγQ1m,γZ2x2y2,νZ1°νZ2m,x2m,y2eiβZ1°βZ2m,x2m,y2=maxνQ1m,νZ2x2y2eimaxβQ1m,βZ2x2y2,∀m∈A1andx2y2∈B2.(iii)
μZ1°μZ2x1,zy1,zeiαZ1°αZ2x1,zy1,z=minμZ1x1y1,μQ2zeiminαZ1x1y1,αQ2z,λZ1°λZ2x1,zy1,zeiγZ1°γZ2x1,zy1,z=minλZ1x1y1,λQ2zeiminγZ1x1y1,γQ2z,νZ1°νZ2x1,zy1,zeiβZ1°βZ2x1,zy1,z=maxνZ1x1y1,νQ2zeimaxβZ1x1y1,βQ2z,∀z∈A2andx1y1∈B1.(iv)
μZ1∘μZ2x1,x2y1,y2eiαZ1∘αZ2x1,x2y1,y2=minμQ2x2,μQ2y2,μZ1x1y1eiminαQ2x2,αQ2y2,αZ1x1y1,for allx2andy2∈A2,x2≠y2andx1y1∈B1,λZ1∘λZ2x1,x2y1,y2eiγZ1∘γZ2x1,x2y1,y2=minλQ2x2,λQ2y2,λZ1x1y1eiminγQ2x2,γQ2y2,γZ1x1y1,for allx2andy2∈A2,x2≠y2andx1y1∈B1,νZ1∘νZ2x1,x2y1,y2eiβZ1∘βZ2x1,x2y1,y2=maxνQ2x2,νQ2y2,νZ1x1y1eimaxβQ2x2,βQ2y2,βZ1x1y1,for allx2andy2∈A2,x2≠y2andx1y1∈B1.Example 8.
Suppose thatG1=Q1,Z1 and G2=Q2,Z2 are two CSFGs on A1=a,bandA2=c,d, respectively, which are shown in Figures 11 and 12. Also, composition is shown in Figure 13.Figure 11
G1.Figure 12
G2.Figure 13
G1°G2.Proof.
Suppose thatG1=Q1,Z1 and G2=Q2,Z2 are two CSFGs of graph G1=A1,B1 and G2=A2,B2, respectively. Then, the composition G1°G2 of G1=A1,B1 and G2=A2,B2 can be proved. Let x1,x2y1,y2∈B1×B2.(i)
It is trivial.(ii)
Ifx1=y1=m,(29)μZ1∘μZ2m,x2m,y2eiαZ1∘αZ2m,x2m,y2=minμQ1m,μZ2x2y2eiminαQ1m,αZ2x2y2,≤minμQ1m,minμQ2x2,μQ2y2eiminαQ1m,minαQ2x2,αQ2y2,=minminμQ1m,μQ2x2,minμQ1m,μQ2y2eiminminαQ1m,αQ2x2,minαQ1m,αQ2y2,=minμQ1∘μQ2m,x2,μQ1∘μQ2m,y2eiminαQ1∘αQ2m,x2,αQ1∘αQ2m,y2,(30)λZ1∘λZ2m,x2m,y2eiγZ1∘γZ2m,x2m,y2=minλQ1m,λZ2x2y2eiminγQ1m,γZ2x2y2,≤minλQ1m,minλQ2x2,λQ2y2eiminγQ1m,minγQ2x2,γQ2y2,=minminλQ1m,λQ2x2,minλQ1m,λQ2y2eiminminγQ1m,γQ2x2,minγQ1m,γQ2y2,=minλQ1∘λQ2m,x2,λQ1∘λQ2m,y2eiminγQ1∘γQ2m,x2,γQ1∘γQ2m,y2,(31)νZ1∘νZ2m,x2m,y2eiβZ1∘βZ2m,x2m,y2=minνQ1m,νZ2x2y2eiminβQ1m,βZ2x2y2,≤minνQ1m,minνQ2x2,νQ2y2eiminβQ1m,minβQ2x2,βQ2y2=minminνQ1m,νQ2x2,minνQ1m,νQ2y2eiminminβQ1m,βQ2x2,minβQ1m,βQ2y2,=minνQ1∘νm,x2,νQ1∘νQ2m,y2eiminβQ1∘βQ2m,x2,βQ1∘βQ2m,y2.(iii)
Ifx2=y2=z,(32)μZ1∘μZ2x1,zy1,zeiαZ1∘αZ2x1,zy1,z=minμZ1x1y1,μQ2zeiminαZ1x1y1,αQ2z,≤minminμZ1x1y1,μQ2zeiminminαZ1x1y1,αQ2z,=minminμZ1x1,μQ2z,minμQ1y1,μQ2zeiminminαZ1x1,αQ2z,minαQ1y1,αQ2z,=minμQ1∘μQ2x1,z,μQ1∘μQ2y1,zeiminαQ1∘αQ2x1,z,αQ1∘αQ2y1,z,(33)λZ1∘λZ2x1,zy1,zeiγZ1∘γZ2x1,zy1,z,=minλZ1x1y1,λQ2zeiminγZ1x1y1,γQ2z,≤minminλZ1x1y1,λQ2zeiminminγZ1x1y1,γQ2z,=minminλZ1x1,λQ2z,minλQ1y1,λQ2zeiminminγZ1x1,γQ2z,minγQ1y1,γQ2z,=minλQ1∘λQ2x1,z,λQ1∘λQ2y1,zeiminγQ1∘γQ2x1,z,γQ1∘γQ2y1,z,(34)νZ1∘νZ2x1,zy1,zeiβZ1∘γZ2x1,zy1,z=minνZ1x1y1,νQ2zeiminβZ1x1y1,βQ2z,≤minminνZ1x1y1,νQ2zeiminminβZ1x1y1,βQ2z,=minminνZ1x1,νQ2z,minνQ1y1,νQ2zeiminminβZ1x1,βQ2z,minβQ1y1,βQ2z,=minνQ1∘νQ2x1,z,νQ1∘νQ2y1,zeiminβQ1∘γQ2x1,z,βQ1∘βQ2y1,z.(iv)
For allx2,y2∈A2,x2≠y2 and x1y1∈B1,(35)μZ1∘μZ2x1,x2y1,y2eiαZ1∘αZ2x1,x2y1,y2=minμQ2x2,μQ2y2,μZ1x1y1eiminαQ2x2,αQ2y2,αZ1x1y1,≤minμQ2x2,μQ2y2,minμQ1x1μQ1y1eiminαQ2x2,αQ2y2,minαQ1x1αQ1y1,=minminμQ1x1,μQ2x2,μQ1y1,μQ2y2eiminminαQ1x1,αQ2x2,αQ1y1,αQ2y2,=minμQ1∘μQ2x1,x2,μQ1∘μQ2y1,y2eiminαQ1∘αQ2x1,x2,αQ1∘αQ2y1,y2,(36)λZ1∘λZ2x1,x2y1,y2eiγZ1∘γZ2x1,x2y1,y2=minλQ2x2,λQ2y2,λZ1x1y1eiminγQ2x2,γQ2y2,γZ1x1y1,≤minλQ2x2,λQ2y2,minλQ1x1λQ1y1eiminγQ2x2,γQ2y2,minγQ1x1γQ1y1,=minminλQ1x1,λQ2x2,λQ1y1,λQ2y2eiminminγQ1x1,γQ2x2,γQ1y1,γQ2y2,=minλQ1∘λQ2x1,x2,λQ1∘λQ2y1,y2eiminγQ1∘γQ2x1,x2,γQ1γQ2y1,y2,(37)νZ1∘νZ2x1,x2y1,y2eiβZ1∘βZ2x1,x2y1,y2=minνQ2x2,νQ2y2,νZ1x1y1eiminβQ2x2,βQ2y2,βZ1x1y1,≤minνQ2x2,νQ2y2,minνQ1x1νQ1y1eiminβQ2x2,βQ2y2,minβQ1x1βQ1y1,=minminνQ1x1,νQ2x2,νQ1y1,νQ2y2eiminminβQ1x1,βQ2x2,βQ1y1,βQ2y2,=minνQ1∘νQ2x1,x2,νQ1∘νQ2y1,y2eiminβQ1∘βQ2x1,x2,βQ1βQ2y1,y2.
Hence,G1°G2 is a CSFG.Definition 22.
Suppose thatG1=Q1,Z1 and G2=Q2,Z2 are CSFGs defined on G1=A1,B1 and G2=A2,B2, respectively. The strong product of G1 and G2 is represented by G1⊗G2=Q1⊗Q2,Z1⊗Z2. The strong product of G1andG2 is defined as follows:(i)
(38)μQ1⊗μQ2x1,x2eiαQ1⊗αQ2x1,x2=minμQ1x1,μQ2x2eiminαQ1x1,αQ2x2,λQ1⊗λQ2x1,x2eiγQ1⊗γQ2x1,x2=minλQ1x1,λQ2x2eiminγQ1x1,γQ2x2,νQ1⊗νQ2x1,x2eiβQ1⊗βQ2x1,x2=minνQ1x1,νQ2x2eiminβQ1x1,βQ2x2,∀x1,x2∈A1×A2.(ii)
(39)μZ1⊗μZ2m,x2m,y2eiαZ1⊗αZ2m,x2m,y2=minμQ10m,μZ2x2y2eiminαQ1m,αZ2x2y2,λZ1⊗λZ2m,x2m,y2eiγZ1⊗αZ2m,x2m,y2=minλQ10m,λZ2x2y2eiminγQ1m,αZ2x2y2,νZ1⊗νZ2m,x2m,y2eiβZ1⊗αZ2m,x2m,y2=minνQ10m,νZ2x2y2eiminβQ1m,αZ2x2y2,∀m∈A1andx2y2∈B2.(iii)
(40)μZ1⊗μZ2x1,zy1,zeiαZ1⊗αZ2x1,zy1,z=minμZ1x1y1,μQ2zeiminαZ1x1y1,αQ2z,λZ1⊗λZ2x1,zy1,zeiγZ1⊗γZ2x1,zy1,z=minλZ1x1y1,λQ2zeiminγZ1x1y1,γQ2z,νZ1⊗νZ2x1,zy1,zeiβZ1⊗βZ2x1,zy1,z=minνZ1x1y1,νQ2zeiminβZ1x1y1,βQ2z,∀z∈A2andx1y1∈B1.(4)
(41)μZ1⊗μZ2x1,x2y1,y2eiαZ1⊗αZ2x1,x2y1,y2=minμZ1x1y1,μZ2x2y2eiminαZ1x1y1,αZ2x2y2,λZ1⊗λZ2x1,x2y1,y2eiγZ1⊗γZ2x1,x2y1,y2=minλZ1x1y1,λZ2x2y2eiminγZ1x1y1,γZ2x2y2,νZ1⊗νZ2x1,x2y1,y2eiβZ1⊗αZ2x1,x2y1,y2=minνZ1x1y1,νZ2x2y2eiminβZ1x1y1,αZ2x2y2,for allx1y1∈B1andforallx2y2∈B2.Example 9.
Suppose thatG1=Q1,Z1 and G2=Q2,Z2 are two CSFGs on A1=a,bandA2=c,d, respectively, which are shown in Figures 14 and 15. Also, strong product is shown in Figure 16.Figure 14
G1.Figure 15
G2.Figure 16
G1⊗G2.Proposition 2.
Suppose thatG1=Q1,Z1 and G2=Q2,Z2 are two CSFGs of graph G1=A1,B1 and G2=A2,B2, respectively. Then, strong product G1⊗G2 of G1=A1,B1 and G2=A2,B2 is CSFG.Proof.
Suppose thatG1=Q1,Z1 and G2=Q2,Z2 are two CSFG of graph G1=A1,B1 and G2=A2,B2, respectively. Then, the strong product G1⊗G2 of G1=A1,B1 and G2=A2,B2 can be proved. Let x1,x2y1,y2∈B1×B2.(i)
It is trivial.(ii)
Ifx1=y1=m,(42)μZ1⊗μZ2m,x2m,y2eiαZ1⊗αZ2m,x2m,y2,=minμQ1m,μZ2x2y2eiminαQ1m,αZ2x2y2,≤minμQ1m,minμQ2x2,μQ2y2eiminαQ1m,minαQ2x2,αQ2y2,=minminμQ1m,μQ2x2,minμQ1m,μQ2y2eiminminαQ1m,αQ2x2,minαQ1m,αQ2y2,=minμQ1⊗μQ2m,x2,μQ1⊗μQ2m,y2eiminαQ1⊗αQ2m,x2,αQ1⊗αQ2m,y2,(43)λZ1⊗λZ2m,x2m,y2eiγZ1⊗γZ2m,x2m,y2=minλQ1m,λZ2x2y2eiminγQ1m,γZ2x2y2,≤minλQ1m,minλQ2x2,λQ2y2eiminγQ1m,minγQ2x2,γQ2y2,=minminλQ1m,λQ2x2,minλQ1m,λQ2y2eiminminγQ1m,γQ2x2,minγQ1m,γQ2y2,=minλQ1⊗λQ2m,x2,λQ1⊗λQ2m,y2eiminγQ1⊗γQ2m,x2,γQ1⊗γQ2m,y2,(44)νZ1⊗νZ2m,x2m,y2eiβZ1⊗βZ2m,x2m,y2=minνQ1m,νZ2x2y2eiminβQ1m,βZ2x2y2,≤minνQ1m,minνQ2x2,νQ2y2eiminβQ1m,minβQ2x2,βQ2y2,=minminνQ1m,νQ2x2,minνQ1m,νQ2y2eiminminβQ1m,βQ2x2,minβQ1m,βQ2y2,=minνQ1⊗νQ2m,x2,νQ1⊗νQ2m,y2eiminβQ1⊗βQ2m,x2,βQ1⊗βQ2m,y2.(iii)
Ifx2=y2=z,(45)μZ1⊗μZ2x1,zy1,zeiαZ1⊗αZ2x1,zy1,z=minμZ1x1y1,μQ2zeiminαZ1x1y1,αQ2z,≤minminμZ1x1y1,μQ2zeiminminαZ1x1y1,αQ2z,=minminμZ1x1,μQ2z,minμQ1y1,μQ2zeiminminαZ1x1,αQ2z,minαQ1y1,αQ2z,=minμQ1⊗μQ2x1,z,μQ1⊗μQ2y1,zeiminαQ1⊗αQ2x1,z,αQ1⊗αQ2y1,z,(46)λZ1⊗λZ2x1,zy1,zeiγZ1⊗γZ2x1,zy1,z=minλZ1x1y1,λQ2zeiminγZ1x1y1,γQ2z,≤minminλZ1x1y1,λQ2zeiminminγZ1x1y1,γQ2z,=minminλZ1x1,λQ2z,minλQ1y1,λQ2zeiminminγZ1x1,γQ2z,minγQ1y1,γQ2z,=minλQ1⊗λQ2x1,z,λQ1⊗λQ2y1,zeiminγQ1⊗γQ2x1,z,γQ1⊗γQ2y1,z,(47)νZ1⊗νZ2x1,zy1,zeiβZ1⊗βZ2x1,zy1,z=maxνZ1x1y1,νQ2zeimaxβZ1x1y1,βQ2z,≤maxmaxνZ1x1y1,νQ2zeimaxmaxβZ1x1y1,βQ2z,=maxmaxνQ1x1,νQ2z,maxνQ1y1,νQ2zeimaxmaxβQ1x1,βQ2z,maxβQ1y1,βQ2z,=maxνQ1⊗νQ2x1,z,νQ1⊗νQ2y1,zeimaxβQ1⊗βQ2x1,z,βQ1⊗βQ2y1,z.(iv)
(48)μZ1⊗μZ2x1,x2y1,y2eiαZ1⊗αZ2x1,x2y1,y2=minμZ1x1y1,μZ2x2y2eiminαZ1x1y1,αZ2x2y2,≤minminμQ1x1,μQ1y1,μQ2x2,μQ2y2eiminαQ1x1,αQ1y1,αQ2x2,αQ2y2,=minminμQ1m,μQ2x2,minμQ1m,μQ2y2eiminminαQ1m,αQ2x2,minαQ1m,αQ2y2,=minμQ1⊗μQ2x1,x2,μQ1⊗μQ2y1,y2eiminαQ1⊗αQ2x1,x2,αQ1⊗αQ2y1,y2,for allx1y1∈B1,and for allx2y2∈B2.(49)λZ1⊗λZ2x1,x2y1,y2eiγZ1⊗γZ2x1,x2y1,y2=minλZ1x1y1,λZ2x2y2eiminγZ1x1y1,γZ2x2y2≤minminλQ1x1,λQ1y1,λQ2x2,λQ2y2eiminγQ1x1,γQ1y1,γQ2x2,γQ2y2=minminλQ1m,λQ2x2,minλQ1m,λQ2y2eiminminγQ1m,γQ2x2,minγQ1m,γQ2y2=minλQ1⊗λQ2x1,x2,λQ1⊗λQ2y1,y2eiminγQ1⊗γQ2x1,x2,γQ1⊗γQ2y1,y2,for allx1y1∈B1,and for allx2y2∈B2.(50)νZ1⊗νZ2x1,x2y1,y2eiβZ1⊗βZ2x1,x2y1,y2=maxνZ1x1y1,νZ2x2y2eimaxβZ1x1y1,βZ2x2y2≤maxmaxνQ1x1,νQ1y1,νQ2x2,νQ2y2eimaxβQ1x1,βQ1y1,βQ2x2,βQ2y2=maxmaxνQ1m,νQ2x2,maxνQ1m,νQ2y2eimaxmaxβQ1m,βQ2x2,maxβQ1m,βQ2y2=maxνQ1⊗νQ2x1,x2,νQ1⊗νQ2y1,y2eimaxβQ1⊗βQ2x1,x2,βQ1⊗βQ2y1,y2,for allx1y1∈B1,andforallx2y2∈B2.
Hence,G1⊗G2.Definition 23.
Suppose thatG1=Q1,Z1 and G2=Q2,Z2 are CSFGs defined on G1=A1,B1 and G2=A2,B2, respectively. The semi-strong product of G1 and G2 is represented by G1⊙G2=Q1⊙Q2,Z1⊙Z2. The semi-strong product of G1andG2 is defined as follows:(i)
(51)μQ1⊙μQ2x1,x2eiαQ1⊙αQ2x1,x2=minμQ1x1,μQ2x2eiminαQ1x1,αQ2x2,λQ1⊙λQ2x1,x2eiγQ1⊙γQ2x1,x2=minλQ1x1,λQ2x2eiminγQ1x1,γQ2x2,νQ1⊙νQ2x1,x2eiβQ1⊙βQ2x1,x2=maxνQ1x1,νQ2x2emaxβQ1x1,βQ2x2,∀x1,x2∈A1×A2.(ii)
(52)μZ1⊙μZ2m,x2m,y2eiαZ1⊙αZ2m,x2m,y2=minμQ1m,μZ2x2y2eiminαQ1m,αZ2x2y2,λZ1⊙λZ2m,x2m,y2eiγZ1⊙γZ2m,x2m,y2=minλQ1m,λZ2x2y2eiminγQ1m,γZ2x2y2,νZ1⊙νZ2m,x2m,y2eiβZ1⊙βZ2m,x2m,y2=maxνQ1m,νZ2x2y2eimaxβQ1m,βZ2x2y2,∀m∈A1andx2y2∈B2.(iii)
(53)μZ1μ˙Z2x1,x2y1,y2eiαZ1⊙αZ2x1,x2y1,y2=minμZ1x1y1,μZ2x2y2eiminαZ1x1y1,αZ2x2y2,λZ1λ˙Z2x1,x2y1,y2eiγZ1⊙γZ2x1,x2y1,y2=minλZ1x1y1,λZ2x2y2eiminγZ1x1y1,γZ2x2y2,νZ1⊙νZ2x1,x2y1,y2eiβZ1⊙βZ2x1,x2y1,y2=maxνQ1x1y1,νQ2x2y2eimaxβQ1x1y1,βQ2x2y2,for allx1y1∈B1and for allx2y2∈B2.Example 10.
Suppose thatG1=Q1,Z1 and G2=Q2,Z2 are two CSFGs on A1=a,bandA2=c,d, respectively, which are shown in Figures 17 and 18. Also, semi-strong product is shown in Figure 19.Figure 17
G1.Figure 18
G2.Figure 19
G1⊙G2.Proposition 3.
Suppose thatG1=Q1,Z1 and G2=Q2,Z2 are two CSFGs of graph G1=A1,B1 and G2=A2,B2, respectively. Then, semi-strong product G1⊙G2 of G1=A1,B1 and G2=A2,B2 is CSFG.Proof.
Suppose thatG1=Q1,Z1 and G2=Q2,Z2 are two CSFG of graph G1=A1,B1 and G2=A2,B2, respectively. Then, the semi-strong product G1⊙G2 of G1=A1,B1 and G2=A2,B2 can be proved. Let x1,x2y1,y2∈B1×B2.(i)
It is trivial.(ii)
Ifx1=y1=m,(54)μZ1⊙μZ2m,x2m,y2eiαZ1⊙αZ2m,x2m,y2=minμQ1m,μZ2x2y2eiminαQ1m,αZ2x2y2,≤minμQ1m,minμQ2x2,μQ2y2eiminαQ1m,minαQ2x2,αQ2y2,=minminμQ1m,μQ2x2,minμQ1m,μQ2y2eiminminαQ1m,αQ2x2,minαQ1m,αQ2y2,=minμQ1⊙μQ2m,x2,μQ1⊙μQ2m,y2eiminαQ1⊙αQ2m,x2,αQ1⊙αQ2m,y2,(55)λZ1⊙λZ2m,x2m,y2eiγZ1⊙γZ2m,x2m,y2=minλQ1m,λZ2x2y2eiminγQ1m,γZ2x2y2,≤minλQ1m,minλQ2x2,λQ2y2eiminγQ1m,minγQ2x2,γQ2y2,=minminλQ1m,λQ2x2,minλQ1m,λQ2y2eiminminγQ1m,γQ2x2,minγQ1m,γQ2y2,=minλQ1⊙λQ2m,x2,λQ1⊙λQ2m,y2eiminγQ1⊙γQ2m,x2,γQ1⊙γQ2m,y2,(56)νZ1⊙νZ2m,x2m,y2eiβZ1⊙βZ2m,x2m,y2=maxνQ1m,νZ2x2y2eimaxβQ1m,βZ2x2y2,≤maxνQ1m,minνQ2x2,νQ2y2eimaxβQ1m,maxβQ2x2,βQ2y2,=maxmaxνQ1m,νQ2x2,maxνQ1m,νQ2y2eimaxmaxβQ1m,βQ2x2,maxβQ1m,βQ2y2,=maxνQ1⊙νQ2m,x2,νQ1⊙νQ2m,y2eimaxβQ1⊙βQ2m,x2,βQ1⊙βQ2m,y2.(iii)
(57)μZ1eμZ2x1,x2y1,y2eiαZ1⊙αZ2x1,x2y1,y2=minμZ1x1y1,μZ2x2y2eiminαZ1x1y1,αZ2x2y2,≤minminμQ1x1,μQ1y1,μQ2x2,μQ2y2eiminαQ1x1,αQ1y1,αQ2x2,αQ2y2,=minminμQ1m,μQ2x2,minμQ1m,μQ2y2eiminminαQ1m,αQ2x2,minαQ1m,αQ2y2,=minμQ1eμQ2x1,x2,μQ1⊙μQ2y1,y2eiminαQ1⊙αQ2x1,x2,αQ1⊙αQ2y1,y2,for allx1y1∈B1,and for allx2y2∈B2.(58)λZ1eλZ2x1,x2y1,y2eiγZ1⊙γZ2x1,x2y1,y2=minλZ1x1y1,λZ2x2y2eiminγZ1x1y1,γZ2x2y2,≤minminλQ1x1,λQ1y1,λQ2x2,λQ2y2eiminγQ1x1,γQ1y1,γQ2x2,γQ2y2,=minminλQ1m,λQ2x2,minλQ1m,λQ2y2eiminminγQ1m,γQ2x2,minγQ1m,γQ2y2,=minλQ1eλQ2x1,x2,λQ1⊙λQ2y1,y2eiminγQ1⊙γQ2x1,x2,γQ1⊙γQ2y1,y2,for allx1y1∈B1,and for allx2y2∈B2.(59)νZ1eνZ2x1,x2y1,y2eiβZ1⊙βZ2x1,x2y1,y2=maxνZ1x1y1,νZ2x2y2eimaxβZ1x1y1,βZ2x2y2,≤maxmaxνQ1x1,νQ1y1,νQ2x2,νQ2y2eimaxβQ1x1,βQ1y1,βQ2x2,βQ2y2,=maxmaxνQ1m,νQ2x2,maxνQ1m,νQ2y2eimaxmaxβQ1m,βQ2x2,maxβQ1m,βQ2y2,=maxνQ1⊙νQ2x1,x2,νQ1⊙νQ2y1,y2eimaxβQ1⊙βQ2x1,x2,βQ1⊙βQ2y1,y2,for allx1y1∈B1,and for allx2y2∈B2.
Hence,G1⊙G2.


## 4. Application of CSFG

SFSs are a useful extension of FSs, IFSs, and PFSs. CSFG is useful in decision-making problems. In the application, we create a few assumptions that are physically beneficial.The International Monetary Fund (IMF) wants to select countries in the world that have a target to deliver funds for minimum number of countries in the world. In such a way, most of the people in a country can take advantage of this project. The following are some of the criteria that were taken into consideration for this purpose:(1)
Elimination of price controls.(2)
Ceiling on government borrowing.(3)
Minimum level of the general government primary balance.(4)
Improving financial sector operations.(5)
Budget consistent with fiscal framework.(6)
Minimum level of international reserves.(7)
Building up social safety nets.(8)
Strengthening public financial management.The conditionality notion encompasses all of the activities connected with establishing IMF-supported programmes, which include macroeconomic and structural policies, as well as the precise methodologies that are used to monitor progress toward objectives set by a nation in conjunction with the IMF. A government that is experiencing balance of payment difficulties may benefit from using conditionality to help them resolve the problem without resorting to actions that are harmful to the national or international economy. The measures are also meant to preserve IMF resources by ensuring that the country’s balance of payments stays strong enough to enable it to repay the loan.Members of a team should choose four countries in which they are planning to release funds so that they may provide assistance to countries that meet the requirements of the IMF. They focus on the following two important situations:In the first scenario, we proceed as follows. LetP = \{Afghanistan, Pakistan, Bangladesh, Iran\} be the set of countries where the team wishes to deliver the fund to country as a node set. Let 70 percent of the team’s specialists feel that Afghanistan should be chosen, 5 percent of the specialists be neutral, and 20 percent of the specialists feel that there is no need to release the fund to Afghanistan after carefully analyzing the parameters. Therefore, we can determine the terms of all membership, neutral, and non-membership functions. It is necessary to compute the phase term, which defines the period, in order to do this. Let 30 percent of the specialists feel that in a particular time Afghanistan will fulfil the conditions, 10 percent of the specialists be neutral, and 20 percent of the experts have the opposite opinion. We will make model of this information as Afghanistan:0.7e0.3πi,0.05e0.1πi,0.2e0.2πi. So, this is their final opinion. Now, the team wants to go to Pakistan. Suppose that the model of this information is Pakistan:0.9e0.4πi,0.4e0.05πi,0.1e0.1πi. After this, they visit Bangladesh for their valuable mission. Suppose model information about Bangladesh is Bangladesh:0.4e0.2πi,0.2e0.10πi0.7e0.5πi, and finally they visit Iran, and the model of this information is Iran:0.6e0.4πi,0.3e0.4πi,0.5e0.3πi. We represent this model as(60)B=Afghanistan:0.7e0.3πi,0.05e0.1πi,0.2e0.2πiPakistan:0.9e0.4πi,0.4e0.05πi,0.1e0.1πiBangladesh:0.4e0.2πi,0.2e0.10πi0.7e0.5πiIran:0.6e0.4πi,0.3e0.4πi,0.5e0.3πiWe will use the score functionSQ =  μQ2−λQ2−νQ2 − 1/4π2αQ2−γQ2−βQ2.We will use the score function of four values to decide which option is the best one.(61)SAfghanistan=0.72−0.052−0.22+14π20.3π2−0.1π2−0.2π2=0.4575,SPakistan=0.92−0.42−0.12+14π20.4π2−0.05π2−0.1π2=0.6769,SBangladesh=0.42−0.22−0.72+14π20.2π2−0.1π2−0.5π2=−0.425,SIran=0.62−0.32−0.52+14π20.4π2−0.4π2−0.3π2=−0.0025.Pakistan is the best choice for IMF. Complex spherical fuzzy graph with no edge is shown in Figure20.Figure 20
Complex spherical fuzzy graph with no edge.Take P = \{Afghanistan, Pakistan, Bangladesh, Iran\} =R1,R2,R3,R4.Now the team of IMF looks into situation two as follows.If analyze the relationship betweenR1 and R2 for arc R1R2.Suppose the modelR1R2:0.6e0.5πi,0.3e0.2πi,0.1e0.1πi,0.4e0.4πi.All possible relationships of four countries are given below:(62)F=R1R2:0.7e0.3πi,0.05e0.05πi,0.2e0.2πi>undefinedR1R3:0.4e0.2πi,0.05e0.1πi,0.7e0.5πiR1R4:0.6e0.3πi,0.05e0.1πi,0.5e0.3πi>undefinedR2R3:0.4e0.2πi,0.2e0.05πi,0.7e0.5πiR2R4:0.6e0.4πi,0.3e0.05πi,0.5e0.3πi>undefinedR3R4:0.4e0.2πi,0.2e0.1πi,0.7e0.5πiScore function of edges for optimal solution is given by(63)SR1R2=0.72−0.052−0.22+14π20.3π2−0.05π2−0.2π2=0.4594,SR1R3=0.42−0.052−0.72+14π20.2π2−0.1π2−0.5π2=0.−3875,SR1R4=0.62−0.052−0.52+14π20.3π2−0.1π2−0.3π2=0.105,SR2R3=0.42−0.22−0.72+14π20.2π2−0.05π2−0.5π2=0.4575=−0.4231,SR2R4=0.62−0.32−0.52+14π20.4π2−0.05π2−0.3π2=0.4575=0.0369,SR3R4=0.42−0.22−0.72+14π20.2π2−0.1π2−0.5π2=−0.425.SR1R2 is the largest value and therefore more suitable for funding. Complex spherical fuzzy graph with edge is shown in Figure 21.Figure 21
Complex spherical fuzzy graph with edge.


### 4.1. Comparative Analysis

In a spherical fuzzy graphical model, there exist just three standard membership grades of each vertex and edge. CSFG can be used to get better approximation. In this work, different types of degrees of vertices have been used. The degree of vertices in SFG gives the total contribution of the amplitude in the system. However, the degree of vertices in CSFG gives the total information and contribution of the amplitude and phase terms.


### 4.2. Advantages and Limitations

The main advantages of the proposed method are as follows:The communication relationship between a few countries has been examined in this article.This method can be used to explain IMF-country communication in the world when dealing with complex spherical fuzzy information.Some of the limitations of this work are as follows:The focus of this research was just on CSFGs and their related network systems.In a connected complex spherical fuzzy graphical system, this method is only applicable if three kinds of directional thinking exist.It is not always possible to collect real data.


## 4.1. Comparative Analysis

In a spherical fuzzy graphical model, there exist just three standard membership grades of each vertex and edge. CSFG can be used to get better approximation. In this work, different types of degrees of vertices have been used. The degree of vertices in SFG gives the total contribution of the amplitude in the system. However, the degree of vertices in CSFG gives the total information and contribution of the amplitude and phase terms.


## 4.2. Advantages and Limitations

The main advantages of the proposed method are as follows:The communication relationship between a few countries has been examined in this article.This method can be used to explain IMF-country communication in the world when dealing with complex spherical fuzzy information.Some of the limitations of this work are as follows:The focus of this research was just on CSFGs and their related network systems.In a connected complex spherical fuzzy graphical system, this method is only applicable if three kinds of directional thinking exist.It is not always possible to collect real data.


## 5. Conclusion and Future Works

CSFG is more flexible than the SFG and CPFG. CSFG is the extension of CPFG due to fulfilling the criteria of complex neutrality grade. We have defined the order, size, union, join, and ring sum of CSFG. The degree and the total degree of CSFG are determined. We also discussed four properties on CSFG known as a strong product, composition, Cartesian product, and semi-strong product of CSFG. In the end, we have established the application of CSFG. In the future, we will apply some new operations to CSFG. We will check edge regularity of CSFGs. Furthermore, we will introduce complex Dombi spherical fuzzy graph.



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