# Notes on Upper and Lower Truncation of Picture Fuzzy Graphs

Authors: Muhammad Shoaib; Waqas Mahmood; Ahmad N. Al-Kenani; Sahidul Islam
Journal: Discrete Dynamics in Nature and Society
 (2022)

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## Abstract

Due to the absence of a neutral function, there are drawbacks to the existing definition of an intuitionistic fuzzy graph (Int-FG). In that definition of Int-FG, membership function and nonmembership function are involved. In this study, lower truncation (Low-T) and upper truncation (Up-T) are applied to picture fuzzy graphs (Pic-FGs). Pic-FG has an additional neutral membership function. Furthermore, Low-T and Up-T of subdivision of the picture fuzzy line graph (Pic-FLnG) are also discussed. The degree of an edge in both Up-T and Low-T of Pic-FG is discussed. Some related theorems related to the degree of Up-T and Low-T of Pic-FG are discussed.


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## Body

## 1. Introduction

In 1965, Zadeh [1] considerably developed the fuzzy set (FS) idea as a generalization of the crisp set which deals with imprecise or vague data. In real-world situations, the theory of the crisp set may fail to deal with vague data. The true membership degree of FS lies in the closed interval [0, 1]. FS is widely used in the medical and biological sciences. To improve fuzzy theory, Atanassov [2] established the intuitionistic fuzzy set (Int-FS) as a generalization of FS. Int-FS is useful in various domains, having membership and nonmembership degrees whose sum does not exceed 1. It is observed that Int-FS cannot deal with neutrality. Cuong [3] proposed the Pic-FS, which gives more satisfactory results. A picture fuzzy set (Pic-FS) is an extension of Int-FS due to its neutrality degree as an extra term.Graphs have a wide range of applications in real-world problems and are used in computer science, sociology, circuit analysis, network traffic routing, biological networks, operations research, and social networks. Graphs are excellent techniques for information transfer with features such as entity relationships. Nodes can be represented by objects, and edges can be represented by relationships. The graph cannot deal with vague situations. To fulfill this limitation, Kauffman [4] initiated the concept of FG. Mahapatra et al. [5] worked on the colouring of the COVID-19-affected region based on fuzzy directed graphs. Gani and Malarvizhi [6] discussed the concept of truncation in FG. Mahapatra et al. [7] investigated Radio FG. Nowadays, a few authors are working in fuzzy analysis [8–12]. Mahapatra et al. [13] studied the edge colouring of FGs.Shao et al. [14] discussed the application of the water supplier system in Int-FG. Ghani et al. [15] presented the subdivision and middle Int-FG. The edge degree and edge regular properties of FG truncations are discussed by Radha and Kumaravel [16]. Mahapatra et al. [17] studied extended neutrosophic planar graphs. Mahapatra et al. [18] worked on predicting links in social networks by using a neutrosophic graph to look at the way people connect with each other. Furthermore, Mahapatra et al. [19] worked on a new way of link prediction in social networks. The concept of Pic-FG is foundation of Zuo et al. [20] which is extension of Int-FG. Shoaib et al. [21] studied some properties of Pic-FG. The following are the major points of this article:(i)
In this paper, we study the Low-T and Up-T of Pic-FG.(ii)
Furthermore, Low-T and Up-T of subdivision Pic-FLnG are discussed.(iii)
Pic-FG is the extension of Int-FG. In this study, we initiate the degree of an edge in truncations of Pic-FG.The following is the structure of this paper:We presented some basic definitions which will help to understand the paper in Section2. In Section 3, we study the Low-T and Up-T of Pic-FG. In Section 4, we present the Up-T and Low-T of Pic-FG. We study the degree of an edge in truncations of Pic-FLnG in Section 5. At the end, we write conclusion and some future plans in Section 6.The basic notations of this paper are shown in Table1.Table 1
Some basic notations.
NotationMeaningFuzzy graphFGIntuitionistic fuzzy graphInt-FGIntuitionistic fuzzy setInt-FSPicture fuzzy graphPic-FGPicture fuzzy line graphPic-FLnGPicture fuzzy setPic-FSLower truncationLow-TUpper truncationUp-TFuzzy setFSPicture fuzzy setPic-FSUnderlying crisp graphUnd-CGVertex setV-setEdge setE-setSubdivision picture fuzzy graphSud(G)Upper truncation picture fuzzy graphGmLower truncation picture fuzzy graphGmUpper truncation picture fuzzy line graphLnGmLower truncation picture fuzzy line graphLnGmSubdivision of upper truncation picture fuzzy graphSudGmSubdivision of lower truncation picture fuzzy graphSudGm


## 2. Preliminaries

Definition 1.
[20] Let X be a nonempty universal set, then, FS is defined as
{b:ϱ1b , ϱ1:V⟶0,1, b∈X}.Definition 2.
[20] Let X be a nonempty universal set, then, Int-FS is defined as follows:
<b:ϱ1x,τ1b>,b∈X is called Int-FS, where ϱ1:V⟶0,1 and τ1:V⟶0,1, and ϱ1 and τ1 satisfy the axiom
ϱ1b+τ1b≤1 (for all b ∈ X).Definition 3.
[20] Let X be a nonempty universal set, then, Pic-FS is defined as:
<b:ϱ1x,σ1b,τ1b>,b∈X is called Pic-FS, where ϱ1:V⟶0,1,σ1:V⟶0,1, and τ1:V⟶0,1 and are known as degree of truth, neutral, and falsity membership of b, respectively.
ϱ1, σ1, and τ1 satisfy the axiom(1)ϱ1b+σ1b+τ1b≤1.
The refusal membership degreeπ1b = 1 − (ϱ1b + σ1b + τ1b).Definition 4.
[20] A Pic-FG G=ϱ,σ,τ, which is of crisp graph G∗=ϱ∗,σ∗,τ∗ with (ϱ1, σ1, τ1) be a picture fuzzy subset on V and (ϱ2, σ2, τ2) be a picture fuzzy relation on V is defined as(i)
ϱ1, τ1, and σ1:V⟶0,1, where 0 ≤ϱ1 (b) + σ1 (b) + τ1 (b) ≤ 1∀b ε V.(ii)
The functionsϱ2, σ2, and τ2 :E ⊆ V×V ⟶ [0, 1] are defined by(2)ϱ2rk≤minϱ1a,ϱ1b,σ2rk≤minσ1a,ϕ1b,τ2rk≥maxτ1a,τ1b,0≤ϱ2rk+σ2rk+τ2rk≤1.Example 1.
Suppose a Pic-FG shown as in Figure1 having four vertices z, w, x, and y and four edges zy, wx, xz, and wy such that
U=\ltw/0.2,x/0.2,y/0.3,z/0.3,w/0.1,x/0.3,y/0.4,z/0.3,w/0.4,x/0.3,y/0.1,z/0.2> be the picture fuzzy V-set and
W=\ltwx/0.1,xz/0.1,zy/0.1,wy/0.1,wx/0.1,xz/0.2,zy/0.1,wy/0.1,wx/0.4,xz/0.5,zy/0.4,wy/0.4> be the picture fuzzy E-set.Figure 1
Pic-FG.


## 3. Truncation and Subdivision Pic-FG

Definition 5.
Letϱ1,σ1,τ1 be a picture fuzzy subset of a set V. Low-Ts and Up-Ts of ϱ1,σ1,τ1 at the level m, 0≤m≤1, are the picture fuzzy subsets ϱ1,σ1,τ1m and ϱ1,σ1,τ1m defined by:(3)ϱ1,σ1,τ1mb=ϱ1,σ1,τ1bifb∈ϱ1m,b∈τ1m,andb∈σ1m,0ifb∉ϱ1m,b∉σ1morb∉τ1m,ϱ1,σ1,τ1mb=mifb∈ϱ1m,b∈σ1mandb∈τ1m,ϱ1,σ1,τ1bifb∉ϱ1m,b∉ϱ1m,orb∉τ1m,where(4)ϱ1m=b∈Vϱ1b≥m,σ1m=b∈Vσ1b≥m,τ1m=b∈Vτ1b≤m.
ConsiderVm=ϱ1,σ1,τ1mandEm=ϱ2,σ2,τ2m.
LetGm=ϱm,σm,τm be Pic-FG with underlying crisp graph (Und-CG) G∗:Vm,Em. It is known as Low-T of the Pic-FG G at level m, where Vm and Em can be a proper subset of V and E, respectively. Take Vm=V and Em=E, then, Gm:ϱm,σm,τm is an Pic-FG with Und-CG Gm∗:Vm,Em. It is called as Up-T of the Pic-FG G at level m.Theorem 1.
For any levelm, 0<m≤1, the Low-T of SudG is equal as the subdivision of Gm,i.e.,SudGm=SudGm.Proof.
SupposeG:ϱ,σ,τ is a Pic-FG with its underlying V-set V and crisp graph G∗:ϱ∗,σ∗,τ∗ as V,E.
We claim that V-setSudGm = V-set SudGm. Let b is a node in Pic-FG. Then, by using the definition of Low-T of SudG, we have(5)ϱ1,σ1,τ1Sudmb=ϱ1,σ1,τ1Sudb,ifb∈ϱ1Sudm,b∈σ1Sudmandb∈τ1Sudm,ϱ1,σ1,τ1Sudmb=ϱ1,σ1,τ1Sudb,ifb∉ϱ1Sudm,b∉σ1Sudmandb∉τ1Sudm.
By using the definition of Sud(G),(6)ϱ1Sudb=ϱ1b,ifb∈V,ϱ2b,ifb∈E,(7)σ1Sudb=σ1b,ifb∈V,σ2b,ifb∈E,(8)τ1Sudb=τ1b,ifb∈V,τ2b,ifb∈E.Case 1.
Consider the nodeb in SudGm, such that b∈ϱ1Sudm, b∈σ1Sudm, and b∈τ1Sudm.(9)ϱ1Sudb≥m,σ1Sudb≥mandτ1Sudb≤m.Subcase 1.
Ifb∈ V, by equations (6)–(9), ϱ1b≥m, σ1b≥m, and τ1b≤m. Hence, b∈ϱ1m, b∈σ1m, and b∈τ1m. By the definition of Gm, ϱ1,σ1,τ1mb=ϱ1,σ1,τ1b, if b∈ϱ1m, b∈σ1m, and b∈τ1m, i.e., b is vertex in Gm such that(10)ϱ1,σ1,τ1mb=ϱ1,σ1,τ1b.Subcase 2.
Ifb∈E, let b=f; equations (6) and (7) ⇒ϱ1,σ1,τ1Sudf=ϱ2,σ2,τ2f, by equation (8), ϱ2f≥m,σ2f≥m,τ2f≤m. Hence, f∈ϱ2,σ2,τ2m. In Gm,ϱ2,σ2,τ2mf=ϱ2,σ2,τ2f as f∈ϱ2,σ2,τ2m, that is, b=f∈E is an edge in Gm which taken together as the nodes of SudGm, by Subcases 1 and 2, b is node of SudGm,ϱ1,σ1,τ1Sudm=ϱ1,σ1,τ1Sudb=ϱ1,σ1,τ1mSudm if b∈ϱ1Sudm, σ1Sudm and b∈τ1Sudm.Case 2.
Suppose the nodeb in SudGm in such a way u∉ϱ1Sudm, u∉σ1Sudm, and b∉τ1Sudm; equation (5) ⇒ϱ1,σ1,τ1Sudmb=0, i.e., ϱ1,σ1,τ1Sudmb=0 as ϱ1,σ1,τ1Sudb<m.Subcase 3.
Letu∈V; equations (6) and (7) ⇒ϱ1,σ1,τ1Sudb=ϱ1,σ1,τ1b. Hence, ϱ1,σ1,τ1b<m. u∉ϱ1m, u∉σ1m (or) b∉τ1m, by definition of Gmb is a node of Gm, ϱ1,τ1b=0; i.e., u is node in Gm such that ϱ1,τ1mb=0.Subcase 4.
Considerb∈E, b=f; equations (6) and (7) ⇒ϱ1,σ1,τ1b=ϱ2,σ2,τ2b. Hence, ϱ1,σ1,τ1b<m, i.e., f∈ϱ2,σ2,τ2m. So, by definition of Gm,ϱ2,σ2,τ2f=0; i.e., f is edge in Gm such that ϱ2,σ2,τ2mf=0. Therefore, from Subcases 3 and 4 and by definition of SudGmu is vertex of SudGm such that ϱ1,σ1,τ1mSudb=0, so, by Cases 1 and 2, ϱ1,σ1,τ1Sudmm=ϱ1,σ1,τ1mSudb∈b∈V∪E.Claim 1.
Now, our claim is E-setSudGm is equal E-set SudGm.
Takeb,f to be edge in SudGm. By using the definition of SudGm,(11)ϱ2,σ2,τ2Sudmb,f=ϱ1,σ1,τ1Sudb,fifb,f∈ϱ2,σ2,τ2Sudm,(12)ϱ2,σ2,τ2Sudmb,f=0ifb,f∉ϱ2Sudm,b,f∉σ2Sudmorb,fτ2Sudm.
By using the definition ofSudG,(13)ϱ2Sudf,b=ϱ1Sudb∧ϱ1Sudf,ifu∈V,f∈Eandbwhich lies onf,(14)ϱ2Sudf,b=0,otherwise,(15)σ2Sudf,b=σ1Sudb∧σ1Sudf,ifb∈V,f∈Eandblies onf,(16)σ2Sudf,b=0,otherwise,(17)τ2Sudf,b=τ1Sudb∨τ1Sudf,ifb∈V,f∈Eandblies onf,(18)τ2Sudf,b=0,otherwise.Case 3.
Letb∈V,f∈E be such that b,f∈ϱ2,τ2Sudm.(19)ϱ2,σ2,τ2Sudmb,f=ϱ2Sudb,f≥m,σ2Sudb,f≥m,τ2Sudb,f≤m.
Asm>0, equation (12) ⇒ϱ1Sudb∧ϱ1Sudf≥m, equation (12) ⇒σ1Sudb∧σ1Sudf≥m, and equation (14) ⇒(20)τ1Sudb∨τ1Sudf≤m⇒ϱ1b∧ϱ1f≥m,σ1b∧σ1f≥m,τ1b∨τ1f≤m,⇒ϱ1b,ϱ2b≥m,σ1b,σ2b≥m,blie onf,τ1b,τ2b≤mblie on f,⇒b∈ϱ1,σ1,τ1m,f∈τ2,σ2,τ2mandblies on f.
By definition ofGm,(21)1,σ1,τ1mb=1,σ1,τ1b,2,σ2,τ2mf=2,σ2,τ2fasb∈1,σ1,τ1m,f∈2,σ2,τ2m.
By equations (12), (14), and (16), we have,(22)ϱ2Sudb,f=ϱ1mb∧ϱ2mbasblie onf=ϱ1mSudb∧ϱ2mSudbasblie onf=ϱ2mSudb,f,σ2Sudb,f=σ1mb∧σ2mbasblie onf=σ1mSudb∧σ2mSudbasblie onf=σ2mSudb,f,τ2Sudb,f=τ1mb∧τ2mbasblie onf=τ1mSudb∧τ2mSudbasblie onf=τ2mSudb,f.
So,b,f is an edge in SudGm and(23)ϱ2,σ2,τ2Sudmb,f=ϱ1,σ1,τ1mSudb,fifb,f∈ϱ2,σ2,τ2Sudm.Case 4.
Supposeb,f is an edge in SudGm be such that(24)ϱ2,σ2,τ2Sudmb,f=0,⇒by7b,f∉ϱSudm,b,f∉σ2Sudm,b,f∉τ2Sudm⇒ϱ2Sudb,f<m,σ2Sudb,f<morτ2Sudb,f>m.Subcase 5.
Ifb lies on f, equations (12) and (14)(25)⇒ϱ1Sudb∧ϱ1Sudf<m,σ1Sudb∧σ1Sudf<m,τ1Sudb∨τ1Sudf>m,⇒ϱ1b∧ϱ2f<m,σ1b∧σ2f<m,τ1b∨τ2f>m,⇒ϱ2f<morτ2f>m.
So,f∉ϱ2m,f∉ϱ2morf∉τ2m,i.e.,ϱ2,σ2,τ2mf=0. Considering(26)ϱ2mSudb,f=ϱ1mSudb∧ϱ2mSudb=ϱ1mb∧ϱ2mb=0sinceϱ2mf,andblies onf,σ2mSudb,f=σ1mSudb∧σ2mSudb=σ1mb∧σ2mb=0sinceσ2mf,andblies onf,τ2mSudb,f=τ1mSudb∧τ2mSudb=τ1mb∧τ2mb=0sinceτ2mf,andblies onf.Subcase 6.
Ifb does not lie on f, ϱ2,σ2,τ2mSudb,f=0. It yields, by using Subcases 5 and 6, if b,f is an arc in SudGm such that ϱ2,σ2,τ2Sudmb,f=0, then, ϱ2,σ2,τ2mSudb,f=0.(27)ϱ2,σ2,τ2Sudmb,f=ϱ2,σ2,τ2mSudb,f.
Ifb,f∉ϱ2Sudm,b,f∉σ2Sudmorb,f∉τ2Sudm. Hence, by Cases 3 and 4,ϱ2,σ2,τ2Sudmb,f=ϱ2,σ2,τ2mSudb,f. So, SudGm=SudGm.Theorem 2.
For any level0<m≤1, SudGm=SudGm.


## 4. Truncation and Pic-FLnG

Theorem 3.
For any level set0<m≤1, LnGm=LnGm.Proof.
LetG:ϱ,σ,τ is a Pic-FG with its underlying set V and crisp graph G∗:ϱ∗,σ∗,τ∗ as V,E.Case 5.
TakeLnGm:α1,τ1m,α2,τ2m,α3,τ3m by applying the definition of equations (6) and (7),(28)α1,τ1mSa=α1,τ1mSaifSa∈α1m,Sa∈τ1m,0,ifSa∉α1morSa∉τ1m.
Hence,Sa∈α1m⇒α1Sa≥tandSa∈τ1m⇒τ1Sa≤t.(29)⇒ϱ2a≥m,σ2a≥mandτ2a≤m⇒a∈ϱ2m,σ2m,anda∈τ2m,⇒ϱ2,σ2,τ2ma=ϱ2,σ2,τ2a,Sa∉α1m⇒α1Sa≤morSa∉τ1m⇒τ1Sa≥m,⇒ϱ2a≤m,σ2a≤morτ2a≥m⇒a∉ϱ2m,a∉σ2manda∉τ2m,⇒ϱ2,σ2,τ2ma=0.
So, a node inLnGm gives an edge in Gm which in turn in form of a node in LnGm.(30)⇒V−setLnGm⊆V−setLnGm.
Similarly, it can be proved that(31)⇒vertex setLnGm⊇vertex setLnGm,α2,τ2mSa=α2,τ2mSaifSa∈α2m,Sa∈τ2m,0,ifSa∉α2morSa∉τ2m.
Hence,Sa∈α2m⇒α2Sa≥tandSa∈τ2m⇒τ2Sa≤m.(32)⇒ϱ2a≥m,σ2a≥mandτ2a≤m⇒a∈ϱ2m,σ2m,anda∈τ2m,⇒ϱ2,σ2,τ2ma=ϱ2,σ2,τ2a,Sa∉α2m⇒α2Sa≤morSa∉τ2m⇒τ2Sa≥m,⇒ϱ2a≤m,σ2a≤morτ2a≥m⇒a∉ϱ2m,a∉σ2manda∉τ2m,⇒ϱ2,σ2,τ2ma=0.
So, a node inLnGm gives an edge in Gm which in turn in form of a node in LnGm.(33)⇒V−setLnGm⊆V−setLnGm.
Similarly, it can be proven that(34)⇒V−setLnGm⊇V−setLnGm.Case 6.
LetSa,Ss be node in LnGm, where a,s∈E and they consists common node in G.(35)α2,τ2mSa,Ss=α2,τ2Sa,SsifSa,Ss∈α2,τ2m,0,otherwise.Subcase 7.
IfSa,Ss∈α2,τ2m⇒α2Sa,Ss≥m.(36)τ2Sa,Ss≤m⇒α1Sa∧α1Ss≥m,τ1Sa∨τ1Ss≤m,⇒ϱ2a∧ϱ2s≥m,σ2a∧σ2s≥m,τ2a∨τ2s≤m,i.e.,a,s∈ϱ2m,a,s∈σ2m, anda,s∈τ2m.If are the end vertices ofaands,respectively,asa,s∈ϱ2,σ2,τ2m,⇒ϱ1,σ1,τ1ba,ϱ1,σ1,τ1va,ϱ1,σ1,τ1bs,ϱ1,σ1,τ1vs≥t,⇒ba,va,bs,vs∈ϱ1,σ1,τ1m,ba,va,bs,vs∈ϱ1,σ1,τ1m.
We know that by using equations (6) as union of Sa and Ss is nonempty, a=ba,wa,va and s=bs,wsvs have a common node, where a,s∈ϱ1,σ1,τ1m. So, the respective Sa and Ss will be node in LnGm and union of Sa and Ss is nonempty. Therefore, Sa,Ss will be arc in LnGm such that the membership value of Sa,Ss is LnGmα2=ϱ2a∧ϱ2s=α2mSa,Ss, indeterminate value of Sa,Ss is LnGmα2=σ2a∧σ2s=α2mSa,Ss, and the nonmembership value of Sa,Ss is LnGmσ2=τ2a∧τ2s=σ2mSa,Ss.Subcase 8.
Let(37)α2,τ2mSa,Ss=0,i.eSa,Ss∉α2morSa,Ss∉τ2m,⇒α1Sa,Ss<morα2Sa,Ss>m,⇒α1Sa∧α1Ss<morα1Sa∨α1Ss>m,⇒ϱ2a∧ϱ2s<m,σ2a∧σ2s<m,orτ2a∨τ2s>m,⇒The edgesa,s∈Ehave a node in common,but at least either aors∉ϱ2,σ2,τ2m.
So, the respectiveSa and Ss will not become node in LnGm, and Sa,Ss will not become an arc in LnGm. So, by using Subcases 7 and 8,(38)⇒E−setLnGm⊆E−setLnGm.
Similarly, it can be proven that(39)⇒E−setLnGm⊇E−setLnGm.
By using equations (18), (21), and (27), LnGm=LnGmTheorem 4.
For any level0<m≤1, LnGm=LnGm.


## 5. Degree of an Edge in Truncations of Pic-FG

### 5.1. Degree of an Edge in Lower Truncation of Pic-FG

(40)dGmba=dϱGmba,dσGmba,dτGmba,dϱGmba=∑bw∈Em,w≠aϱ2mbw+∑≀∈Em,w≠bϱ2m≀,∀ba∈Em,=∑bw∈Em,w≠aϱ2bw−∑bw∈Em,w≠vandϱ2bw<mϱ2bw+∑≀∈Em,w≠bϱ2≀−∑≀∈Em,w≠bandϱ2≀<mϱ2≀,∀ba∈Em,=dϱGba−∑bw∈Em,w≠vandϱ2bw<mϱ2bw−∑≀∈Em,w≠bandϱ2≀<mϱ2≀,∀ba∈Em,dσGmba=∑bw∈Em,w≠aσ2mbw+∑≀∈Em,w≠bσ2m≀,∀ba∈Em,=∑bw∈Em,w≠aσ2bw−∑bw∈Em,w≠vandσ2bw<mσ2bw+∑≀∈Em,w≠bσ2≀−∑≀∈Em,w≠bandσ2≀<mσ2≀,∀ba∈Em,=dσGba−∑bw∈Em,w≠vandσ2bw<mσ2bw−∑≀∈Em,w≠bandσ2≀<mσ2≀,∀ba∈Em,dτGmba=∑bw∈Em,w≠aτ2mbw+∑≀∈Em,w≠bτ2m≀,∀ba∈Em,=∑bw∈Em,w≠aτ2bw−∑bw∈Em,w≠vandτ2bw>mτ2bw+∑≀∈Em,w≠bτ2≀−∑≀∈Em,w≠bandτ2≀>mτ2≀,∀ba∈Em,=dτGba−∑bw∈Em,w≠vandτ2bw>mτ2bw−∑≀∈Em,w≠bandτ2≀>mτ2≀,∀ba∈Em.


### 5.2. Degree of an Edge in Upper Truncation of Pic-FG

(41)dGmba=dϱGmba,dσGmba,dτGmba,dϱGmba=∑bw∈Em,w≠aϱ2mbw+∑≀∈Em,w≠bϱ2m≀,∀ba∈Em,=∑bw∈Em,w≠aϱ2bw−∑bw∈Em,w≠vandϱ2bw≥mϱ2bw−m+∑≀∈Em,w≠bϱ2≀−∑≀∈Em,w≠bandϱ2≀≥mϱ2≀−m,∀ba∈Em,=dϱGba−∑bw∈Em,w≠vandϱ2bw>mϱ2bw−m−∑≀∈Em,w≠bandϱ2≀>mϱ2≀−m,∀ba∈Em,dσGmba=∑bw∈Em,w≠aσ2mbw+∑≀∈Em,w≠bσ2m≀,∀ba∈Em=∑bw∈Em,w≠aσ2bw−∑bw∈Em,w≠vandσ2bw≥mσ2bw−m+∑≀∈Em,w≠bσ2≀−∑≀∈Em,w≠bandσ2≀≥mσ2≀−m,∀ba∈Em,=dσGba−∑bw∈Em,w≠vandσ2bw>mσ2bw−m−∑≀∈Em,w≠bandσ2≀>mσ2≀−m,∀ba∈Em,dτGmba=∑bw∈Em,w≠aτ2mbw+∑≀∈Em,w≠bτ2m≀,∀ba∈Em=∑bw∈Em,w≠aτ2bw−∑bw∈Em,w≠vandτ2bw≤mτ2bw−m+∑≀∈Em,w≠bτ2≀−∑≀∈Em,w≠bandτ2≀≤mτ2≀−m,∀ba∈Em,=dτGba−∑bw∈Em,w≠vandτ2bw<mτ2bw−m−∑≀∈Em,w≠bandτ2≀<mτ2≀−m,∀ba∈Em.Theorem 5.
LetG be Pic-FG such that ϱ2ba≥m, σ2ba≥mτ2ba≤m, ∀ ba ∈E, where 0<m≤1. Then, for any ba∈Et, dGtba = dGba.Proof.
Since(42)dϱGmba=dϱGba−∑bw∈Em,w≠vandϱ2bw<mϱ2bw−∑≀∈Em,w≠bandϱ2≀<mϱ2≀,∀ba∈Em,⇒dϱGmba=dϱGba,dσGmba=dσGba−∑bw∈Em,w≠vandσ2bw<mσ2bw−∑≀∈Em,w≠bandσ2≀<mσ2≀,∀ba∈Em,⇒dσGmba=dσGba,dτGmba=dτGba−∑bw∈Em,w≠vandτ2bw>mτ2bw−∑≀∈Em,w≠bandτ2≀>mτ2≀,∀ba∈Em,⇒dτGmba=dτGba.Theorem 6.
LetG be a Pic-FG such that ϱ2ba=i1, σ2ba=i2, and τ2ba=i3, ∀ba∈E, where i1, i2 and i3 are constants. Then, for any ba∈Em.(43)dGmba=dGba,ifi1<m,i2<m,andi3>m,dGba−i−mdG∗ba,ifi1≥m,i2≥m,andi3≤m.Proof.
Leti1<m.(44)dϱGmba=dϱGba−∑bw∈Em,w≠vandϱ2bw>mϱ2bw−m−∑≀∈Em,w≠bandϱ2≀>mϱ2≀−m,=dϱGba−∑bw∈Em,w≠vandi1>mi1−m−∑≀∈Em,w≠bandi1>mi1−m,⇒dϱGmba=dϱGba.
Leti2<m.(45)dσGmba=dσGba−∑bw∈Em,w≠vandσ2bw>mσ2bw−m−∑≀∈Em,w≠bandσ2≀>mσ2≀−m,=dσGba−∑bw∈Em,w≠vandi1>mi1−m−∑≀∈Em,w≠bandi1>mi1−m,⇒dσGmba=dσGba.
Leti3>m.(46)dτGmba=dτGba−∑bw∈Em,w≠vandτ2bw<mτ2bw−m−∑≀∈Em,w≠bandτ2≀<mτ2≀−m,=dτGba−∑bw∈Em,w≠vandi1<mi1−m−∑≀∈Em,w≠bandi1<mi1−m,⇒dτGmba=dτGba.
Hence,dGmba = dGba.
Leti1≥m.(47)dϱGmba=dϱGba−∑bw∈Em,w≠vandϱ2bw>mϱ2bw−m−∑≀∈Em,w≠bandϱ2≀>mϱ2≀−m,=dϱGba−∑bw∈Em,w≠vandi1>mi1−m−∑≀∈Em,w≠bandi1>mi1−m,=dϱGba−i1−mdG∗b−1−i1−mdG∗a−1,=dϱGba−i1−mdG∗b+dG∗a+2,=dϱGba−i1−mdG∗ba.
Leti2≥m.(48)dσGmba=dσGba−∑bw∈Em,w≠vandσ2bw>mσ2bw−m−∑≀∈Em,w≠bandσ2≀>mσ2≀−m,=dσGba−∑bw∈Em,w≠vandi1>mi1−m−∑≀∈Em,w≠bandi1>mi1−m,=dσGba−i1−mdG∗b−1−i1−mdG∗a−1,=dσGba−i1−mdG∗b+dG∗a+2,=dσGba−i1−mdG∗ba.
Leti3≤m.(49)dτGmba=dτGba−∑bw∈Em,w≠vandτ2bw<mτ2bw−m−∑≀∈Em,w≠bandτ2≀<mτ2≀−m=dτGba−∑bw∈Em,w≠vandi1<mi1−m−∑≀∈Em,w≠bandi1<mi1−m=dτGba−i1−mdG∗b−1−i1−mdG∗a−1=dτGba−i1−mdG∗b+dG∗a+2=dτGba−i1−mdG∗ba.
Hence,dGmba=dGba−i−mdG∗ba.


## 5.1. Degree of an Edge in Lower Truncation of Pic-FG

(40)dGmba=dϱGmba,dσGmba,dτGmba,dϱGmba=∑bw∈Em,w≠aϱ2mbw+∑≀∈Em,w≠bϱ2m≀,∀ba∈Em,=∑bw∈Em,w≠aϱ2bw−∑bw∈Em,w≠vandϱ2bw<mϱ2bw+∑≀∈Em,w≠bϱ2≀−∑≀∈Em,w≠bandϱ2≀<mϱ2≀,∀ba∈Em,=dϱGba−∑bw∈Em,w≠vandϱ2bw<mϱ2bw−∑≀∈Em,w≠bandϱ2≀<mϱ2≀,∀ba∈Em,dσGmba=∑bw∈Em,w≠aσ2mbw+∑≀∈Em,w≠bσ2m≀,∀ba∈Em,=∑bw∈Em,w≠aσ2bw−∑bw∈Em,w≠vandσ2bw<mσ2bw+∑≀∈Em,w≠bσ2≀−∑≀∈Em,w≠bandσ2≀<mσ2≀,∀ba∈Em,=dσGba−∑bw∈Em,w≠vandσ2bw<mσ2bw−∑≀∈Em,w≠bandσ2≀<mσ2≀,∀ba∈Em,dτGmba=∑bw∈Em,w≠aτ2mbw+∑≀∈Em,w≠bτ2m≀,∀ba∈Em,=∑bw∈Em,w≠aτ2bw−∑bw∈Em,w≠vandτ2bw>mτ2bw+∑≀∈Em,w≠bτ2≀−∑≀∈Em,w≠bandτ2≀>mτ2≀,∀ba∈Em,=dτGba−∑bw∈Em,w≠vandτ2bw>mτ2bw−∑≀∈Em,w≠bandτ2≀>mτ2≀,∀ba∈Em.


## 5.2. Degree of an Edge in Upper Truncation of Pic-FG

(41)dGmba=dϱGmba,dσGmba,dτGmba,dϱGmba=∑bw∈Em,w≠aϱ2mbw+∑≀∈Em,w≠bϱ2m≀,∀ba∈Em,=∑bw∈Em,w≠aϱ2bw−∑bw∈Em,w≠vandϱ2bw≥mϱ2bw−m+∑≀∈Em,w≠bϱ2≀−∑≀∈Em,w≠bandϱ2≀≥mϱ2≀−m,∀ba∈Em,=dϱGba−∑bw∈Em,w≠vandϱ2bw>mϱ2bw−m−∑≀∈Em,w≠bandϱ2≀>mϱ2≀−m,∀ba∈Em,dσGmba=∑bw∈Em,w≠aσ2mbw+∑≀∈Em,w≠bσ2m≀,∀ba∈Em=∑bw∈Em,w≠aσ2bw−∑bw∈Em,w≠vandσ2bw≥mσ2bw−m+∑≀∈Em,w≠bσ2≀−∑≀∈Em,w≠bandσ2≀≥mσ2≀−m,∀ba∈Em,=dσGba−∑bw∈Em,w≠vandσ2bw>mσ2bw−m−∑≀∈Em,w≠bandσ2≀>mσ2≀−m,∀ba∈Em,dτGmba=∑bw∈Em,w≠aτ2mbw+∑≀∈Em,w≠bτ2m≀,∀ba∈Em=∑bw∈Em,w≠aτ2bw−∑bw∈Em,w≠vandτ2bw≤mτ2bw−m+∑≀∈Em,w≠bτ2≀−∑≀∈Em,w≠bandτ2≀≤mτ2≀−m,∀ba∈Em,=dτGba−∑bw∈Em,w≠vandτ2bw<mτ2bw−m−∑≀∈Em,w≠bandτ2≀<mτ2≀−m,∀ba∈Em.Theorem 5.
LetG be Pic-FG such that ϱ2ba≥m, σ2ba≥mτ2ba≤m, ∀ ba ∈E, where 0<m≤1. Then, for any ba∈Et, dGtba = dGba.Proof.
Since(42)dϱGmba=dϱGba−∑bw∈Em,w≠vandϱ2bw<mϱ2bw−∑≀∈Em,w≠bandϱ2≀<mϱ2≀,∀ba∈Em,⇒dϱGmba=dϱGba,dσGmba=dσGba−∑bw∈Em,w≠vandσ2bw<mσ2bw−∑≀∈Em,w≠bandσ2≀<mσ2≀,∀ba∈Em,⇒dσGmba=dσGba,dτGmba=dτGba−∑bw∈Em,w≠vandτ2bw>mτ2bw−∑≀∈Em,w≠bandτ2≀>mτ2≀,∀ba∈Em,⇒dτGmba=dτGba.Theorem 6.
LetG be a Pic-FG such that ϱ2ba=i1, σ2ba=i2, and τ2ba=i3, ∀ba∈E, where i1, i2 and i3 are constants. Then, for any ba∈Em.(43)dGmba=dGba,ifi1<m,i2<m,andi3>m,dGba−i−mdG∗ba,ifi1≥m,i2≥m,andi3≤m.Proof.
Leti1<m.(44)dϱGmba=dϱGba−∑bw∈Em,w≠vandϱ2bw>mϱ2bw−m−∑≀∈Em,w≠bandϱ2≀>mϱ2≀−m,=dϱGba−∑bw∈Em,w≠vandi1>mi1−m−∑≀∈Em,w≠bandi1>mi1−m,⇒dϱGmba=dϱGba.
Leti2<m.(45)dσGmba=dσGba−∑bw∈Em,w≠vandσ2bw>mσ2bw−m−∑≀∈Em,w≠bandσ2≀>mσ2≀−m,=dσGba−∑bw∈Em,w≠vandi1>mi1−m−∑≀∈Em,w≠bandi1>mi1−m,⇒dσGmba=dσGba.
Leti3>m.(46)dτGmba=dτGba−∑bw∈Em,w≠vandτ2bw<mτ2bw−m−∑≀∈Em,w≠bandτ2≀<mτ2≀−m,=dτGba−∑bw∈Em,w≠vandi1<mi1−m−∑≀∈Em,w≠bandi1<mi1−m,⇒dτGmba=dτGba.
Hence,dGmba = dGba.
Leti1≥m.(47)dϱGmba=dϱGba−∑bw∈Em,w≠vandϱ2bw>mϱ2bw−m−∑≀∈Em,w≠bandϱ2≀>mϱ2≀−m,=dϱGba−∑bw∈Em,w≠vandi1>mi1−m−∑≀∈Em,w≠bandi1>mi1−m,=dϱGba−i1−mdG∗b−1−i1−mdG∗a−1,=dϱGba−i1−mdG∗b+dG∗a+2,=dϱGba−i1−mdG∗ba.
Leti2≥m.(48)dσGmba=dσGba−∑bw∈Em,w≠vandσ2bw>mσ2bw−m−∑≀∈Em,w≠bandσ2≀>mσ2≀−m,=dσGba−∑bw∈Em,w≠vandi1>mi1−m−∑≀∈Em,w≠bandi1>mi1−m,=dσGba−i1−mdG∗b−1−i1−mdG∗a−1,=dσGba−i1−mdG∗b+dG∗a+2,=dσGba−i1−mdG∗ba.
Leti3≤m.(49)dτGmba=dτGba−∑bw∈Em,w≠vandτ2bw<mτ2bw−m−∑≀∈Em,w≠bandτ2≀<mτ2≀−m=dτGba−∑bw∈Em,w≠vandi1<mi1−m−∑≀∈Em,w≠bandi1<mi1−m=dτGba−i1−mdG∗b−1−i1−mdG∗a−1=dτGba−i1−mdG∗b+dG∗a+2=dτGba−i1−mdG∗ba.
Hence,dGmba=dGba−i−mdG∗ba.


## 6. Conclusion

Graph theory has many applications in other fields of mathematics. Int-FGs are the backbone of many real systems, such as scheduling, networks, and image segmentation. However, due to the missing neutrality function, Int-FG is limited to representing some systems. To improve real systems, Pic-FG works better. Pic-FG is a generalization of Int-FG due to the extra neutrality membership function. To improve the fuzzy theory, we have introduced the Up-T and Low-T of Pic-FG and Pic-FLnG. In this paper, the degree of an edge in truncations of Pic-FG is also discussed. We have noticed some limitations to this work. The focus of this research was only on Pic-FGs and their related network systems. It is not always possible to collect the real data. In the future, we will apply some operations to bipolar Pic-FG.



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*Source: 7646828-2022-04-26.xml*