# On Development of Neutrosophic Cubic Graphs with Applications in Decision Sciences

Authors: Fazeelat Sultana; Muhammad Gulistan; Peide Liu; Mumtaz Ali; Zahid Khan; Mohammed M. Al-Shamiri; Muhammad Azhar
Journal: Journal of Function Spaces
 (2022)

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## Abstract

In this study, the neutrosophic cubic graphs are further developed. We discussed and explored the open and the closed neighborhood for any vertex in neutrosophic cubic graphs, regular and totally regular neutrosophic cubic graphs, complete neutrosophic cubic graphs, balanced and strictly balanced neutrosophic cubic graphs, irregular and totally irregular neutrosophic cubic graphs, complement of a neutrosophic cubic graph, neighborly irregular and neighborly totally irregular neutrosophic cubic graphs, and highly irregular neutrosophic cubic graphs. It has been demonstrated that the proposed neutrosophic cubic graphs are associated with specific conditions. The comparison study of the proposed graphs with the existing cubic graphs has been carried out. Eventually, decision-making approaches for handling daily life problems such as effects of different factors on the neighboring countries of Pakistan and selection of a house based on the notions of proposed graphs are presented.


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## Body

## 1. Introduction

A human being has a higher position among all the creatures due to his ability to analyze and make decisions. The decisions are made by carefully scrutinizing the problem based on the experience and the current situation. In the past, this used to be a mental activity with its successful execution. With the advancement of science and technology, it is now possible to use some modern techniques to address this problem better. These methodologies rely on traditional knowledge virtually. The ability of humans has been mimicked out effectively by making use of artificial intelligence. Some artificial intelligence techniques have been used successfully to chalk out good decisions. In this approach, various instruments related to decision-making are used. There is a well-known approach called graph theory. Graph theory is the systematic and logical way to analyze and model many applications related to science and other social issues. Graph theory is an essential tool and has played a significant role in developing graph algorithms in computer-related applications. These algorithms are quite helpful in solving theoretical aspects of the problems. These techniques help in solving geometry, algebra, number theory, topology, and many other fields. But many issues are not practically solvable due to the crisp nature of the classical sets. So in 1965, Zadeh [1] introduced the notion of the fuzzy subset of a set. Many other extensions of fuzzy sets have been developed so far like interval-valued fuzzy sets [2] by Zadeh, intuitionistic fuzzy sets [3, 4] by Atanassov, and cubic sets [5, 6] by Jun et al. In [7], Akram et al. developed cubic KU-subalgebras. Smarandache extended the concept of Atanassov and gave the idea of neutrosophic sets [8, 9], and interval neutrosophic sets were introduced by Wang et al. [10]. Jun et al. gave the idea of a neutrosophic cubic set [11]. Rosenfeld [12] developed the fuzzy graphs in 1975. Bhattacharya [13] had started contributing in fuzzy graphs in 1987. Arya and Hazarika [14] developed functions with closed fuzzy graphs. Bhattacharya and Suraweera [15] had developed an algorithm to compute the supremum of max-min powers and a property of fuzzy graphs. Bhutani [16] studied automorphisms of fuzzy graphs. Cerruti [17] used graphs and fuzzy graphs in fuzzy information and decision processes. Chen [18] discuss matrix representations of fuzzy graphs. Crain [19] studied characterization of fuzzy interval graphs. After that, many others contributed to fuzzy graph’s theory like Mordeson and Nair’s contribution [20], Gani and Radha [21], Rashmanlou and Pal [22], Nandhini and Nandhini [23], Elmoasry et al. [24], and Akram et al. [25–27]. Other contributions are from Gani and Latha [28], Poulik et al. [29–32], Borzooei and Rashmanlou [33], Buckley [34], Rashmanlou and Pal [35], Mishra et al. [36], Pal et al. [37], Pramanik et al. [38, 39], Shannon and Atanassov [40], Parvathi et al. [41, 42], and Sahoo and Pal [43]. Akram [44, 45] initiated the concept of bipolar fuzzy graphs. Many others contributed on bipolar fuzzy graphs, like Rashmanlou et al. [46], Akram and Karunabigai [47], and Samanta and Pal [48]. Graphs in terms of neutrosophic’s set have been studied by Huang et al. [49], Naz et al. [50], Dey et al. [51], Broumi et al. [52], and Karaaslan and Dawaz [53]. Zuo et al. [54] discussed picture fuzzy graphs. Kandasamy et al. and Smarandache [55, 56] developed neutrosophic graphs for the first time. Broumi et al. [57–61] discussed different versions of neutrosophic graphs. More development on the neutrosophic graphs can be seen in [50, 62–64]. After reading the extensive literature at neutrosophic graphs, recently, Gulistan et al. [65] discussed the cubic graphs with the application, neutrosophic graphs, and presented the idea of neutrosophic cubic graphs and their structures in their work [66, 67].To further extend the work of Gulistan et al. [66, 67], in this paper we developed different types of neutrosophic cubic graphs including balanced, strictly balanced, complete, regular, totally regular, and irregular neutrosophic cubic graphs and complement of neutrosophic cubic graphs. Also, we established an open and close neighborhood of a vertex for neutrosophic cubic graphs and their application to the art of decision-making. The properties related to these newly suggested neutrosophic cubic graphs are also shown and how they are correlated. The arrangements of the paper are as follows: Section 2 is a review of basic concepts with their properties of neutrosophic cubic graphs. Section 3 describes different types of neutrosophic cubic graphs with examples. We also provide some results related to different types of neutrosophic cubic graphs. We present applications and a decision-making technique in Section 4. In Section 5, we provide a comparative analysis. Conclusions and suggested future work are presented in Section 6.


## 2. Preliminaries

This section consists of two parts: notations and predefined definitions.


### 2.1. Notations

Some notations with their descriptions are given in Table1.Table 1
Notations and their descriptions.
S.noNotationDescription1GAn arbitrary graph2AVertex set forG graph3BEdge set forG graph4FGFuzzy graph5θDegree6℘Density7αMembership function for vertex set inFG8βMembership function for edge set inFG9NNeighborhood10I^Represents an intervalal,ar with 0≤al≤ar≤111LCSNeutrosophic cubic set12LCGNeutrosophic cubic graph13ΦMembership function for vertex set inLCG14ΨMembership function for edge set inLCG.15NfGgFuzzy neighborhood of a vertexg in FG16ℕncggOpen neighborhood for any vertexg in LCG17ℕncggClosed neighborhood for any vertexg in LCG


### 2.2. Predefined Definitions

In this subsection, we added some important definitions which are directly used in our work.Definition 1 (see [66]).
A neutrosophic cubic graphLCG=Γ,Λ for a crisp graph G=A,B is a pair with
(1)Γ=Φg=αT^,αT,αI^,αI,αF^,αFgg∈A,Γ=g,αTl,αTrg,αTg,αIl,αIrg,αIg,αFl,αFrg,αFgg∈A,representing neutrosophic cubic vertex set A, and
(2)Λ=Ψg1g2=βT^,βT,βI^,βI,βF^,βFg1g2g1g2∈B=βTl,βTrg1g2,βTg1g2,βIl,βIrg1g2,βIg1g2,βFl,βFrg1g2,βFg1g2g1g2∈Bshows neutrosophic cubic edge set B such that
(3)βT^g1g2≤rminαT^g1,αT^g2,αTg1g2≤maxαTg1,αTg2,βI^g1g2≤rminαI^g1,αI^g2,αIg1g2≤maxαIg1,αIg2,βF^g1g2≤rmaxαF^g1,αF^g2,αFg1g2≤minαFg1,αFg2,for every vertex g1,g2∈A and edge g1g2∈B.


## 2.1. Notations

Some notations with their descriptions are given in Table1.Table 1
Notations and their descriptions.
S.noNotationDescription1GAn arbitrary graph2AVertex set forG graph3BEdge set forG graph4FGFuzzy graph5θDegree6℘Density7αMembership function for vertex set inFG8βMembership function for edge set inFG9NNeighborhood10I^Represents an intervalal,ar with 0≤al≤ar≤111LCSNeutrosophic cubic set12LCGNeutrosophic cubic graph13ΦMembership function for vertex set inLCG14ΨMembership function for edge set inLCG.15NfGgFuzzy neighborhood of a vertexg in FG16ℕncggOpen neighborhood for any vertexg in LCG17ℕncggClosed neighborhood for any vertexg in LCG


## 2.2. Predefined Definitions

In this subsection, we added some important definitions which are directly used in our work.Definition 1 (see [66]).
A neutrosophic cubic graphLCG=Γ,Λ for a crisp graph G=A,B is a pair with
(1)Γ=Φg=αT^,αT,αI^,αI,αF^,αFgg∈A,Γ=g,αTl,αTrg,αTg,αIl,αIrg,αIg,αFl,αFrg,αFgg∈A,representing neutrosophic cubic vertex set A, and
(2)Λ=Ψg1g2=βT^,βT,βI^,βI,βF^,βFg1g2g1g2∈B=βTl,βTrg1g2,βTg1g2,βIl,βIrg1g2,βIg1g2,βFl,βFrg1g2,βFg1g2g1g2∈Bshows neutrosophic cubic edge set B such that
(3)βT^g1g2≤rminαT^g1,αT^g2,αTg1g2≤maxαTg1,αTg2,βI^g1g2≤rminαI^g1,αI^g2,αIg1g2≤maxαIg1,αIg2,βF^g1g2≤rmaxαF^g1,αF^g2,αFg1g2≤minαFg1,αFg2,for every vertex g1,g2∈A and edge g1g2∈B.


## 3. Different Types of Neutrosophic Cubic Graphs

This section contains definitions for different neutrosophic cubic graphs with a good discussion on some of their related results.


### 3.1. The Open and the Closed Neighborhood for Any Vertex inLCG

In this subsection, we present the idea of open neighborhoodℕncgg, degree of open neighborhood θℕncgg, and closed neighborhood degree of neutrosophic cubic graph LCG.Definition 2.
An open neighborhoodℕncgg for any vertex g in LCG=Γ,Λ is given by
(4)ℕncgg=NTlg,NTrg,NTg,NIlg,NIrg,NIg,NFlg,NFrg,NFg,
where(5)NTlg=s∈A:βTlgs≤minαTlg,αTls,g≠s,NTrg=s∈A:βTrgs≤minαTrg,αTrs,g≠s,NTg=s∈A:βTgs≤minαTg,αTs,g≠s,NIlg=s∈A:βIlgs≤minαIlg,αIls,g≠s,NIrg=s∈A:βI+gs≤minαI+g,αI+s,g≠s,NIg=s∈A:βTgs≤minαTg,αTs,g≠s,NFlg=s∈A:βF−gs≤maxαF−g,αF−s,g≠s,NFrg=s∈A:βF+gs≤maxαF−g,αF−s,g≠s,NFg=s∈A:βFgs≤maxαFg,αFs,g≠s.
It consists of membership functions for all vertices adjacent tog, excluding s.Definition 3.
The degree of open neighborhoodθℕncgg for any vertex g in LCG is defined by
(6)θℕncgg=∑s∈ℕncgg∣s≠gΦs=θTlg,θTrg,θTg,θIlg,θIrg,θIg,θFlg,θFrg,θFg=∑s∈ℕncggs≠gαTls,∑s∈ℕncggs≠gαTry,∑s∈ℕncggs≠gαTy,∑s∈ℕncggs≠gαIls,∑s∈ℕncggs≠gαIrs,∑s∈ℕncggs≠gαIs,∑s∈ℕncggs≠gαFls,∑s∈ℕncggs≠gαFrs,∑s∈ℕncggs≠gαFs.Example 4.
LetG=A,B with vertices A=g1,g2,g3 and edges B=g1g2,g2g3,g1g3. Also, LCG=Γ,Λ such that
(7)Γ=g1,.1,.2,.5,.4,.5,.3,.6,.7,.2,g2,.2,.4,.1,.5,.6,.4,.1,.2,.3,g3,.3,.4,.2,.1,.3,.7,.4,.6,.3,Λ=g1g2,.1,.2,.5,.4,.5,.4,.6,.7,.2,g2g3,.2,.4,.2,.1,.3,.7,.4,.6,.3,g1g3,.1,.2,.5,.1,.3,.7,.6,.7,.2,then clearly, LCG=Γ,Λ is a neutrosophic cubic graph as shown in Figure 1.Figure 1
Represents a neutrosophic cubic graph.θℕncgg for each element g∈A is given by
(8)θℕncgg1=Φg2+Φg3=.2,.4,.1,.5,.6,.4,.1,.2,.3+.3,.4,.2,.1,.3,.7,.4,.6,.3=.5,.8,.3,.6,.9,1.1,.5,.8,.6,θℕncgg2=Φg1+Φg3=.1,.2,.5,.4,.5,.3,.6,.7,.2+.3,.4,.2,.1,.3,.7,.4,.6,.3=.4,.7,.6,.5,.8,1,1,1.3,.5,θℕncgg3=Φg1+Φg2=.1,.2,.5,.4,.5,.3,.6,.7,.2+.2,.4,.1,.5,.6,.4,.1,.2,.3=.3,.6,.6,.9,1.1,.7,.7,.9,.5.Definition 5.
A closed neighborhood degree for a vertexg in LCG=Γ,Λ is given by
(9)θℕncgg=∑s∈ℕncgg∪gΦsθTlg,θTrg,θTg,θTlg,θIrg,θIg,θFlg,θFrg,θFg=∑s∈ℕncgg∪gαTls,∑s∈ℕncgg∪gαTrs,∑s∈ℕncgg∪gαTs,∑s∈ℕncgg∪gαTls,∑s∈ℕncgg∪gαIrs,∑s∈ℕncgg∪gαIs,∑s∈ℕncgg∪gαFly,∑s∈ℕncgg∪gαFry,∑s∈ℕncgg∪gαFy.
It consists of the sum of membership functions of all vertices adjacent tox and membership function of x.Example 6.
Consider Example4, closed neighborhood degree θℕncgg for each element g∈A in LCG is given by
(10)θℕncgg1=Φg2+Φg3+Φg1=.2,.4,.1,.5,.6,.4,.1,.2,.3+.3,.4,.2,.1,.3,.7,.4,.6,.3+.1,.2,.5,.4,.5,.3,.6,.7,.2=.6,1,.8,1,1.4,1.4,1.1,1.5,.8;similarly,
(11)θℕncgg2=Φg1+Φg3+Φg2;also, we have
(12)θℕncgg3=Φg1+Φg2+Φg3.


### 3.2. Regular and Totally Regular Neutrosophic Cubic Graphs

In this subsection, we present the idea of regular and totally regular neutrosophic cubic graphs based on the open neighborhood degree and closed neighborhood degree.Definition 7.
If every vertex inLCG has the same open neighborhood degree n, i.e., if θℕncgg=n, for all g∈A, then LCG is called an n regular neutrosophic cubic graph.Definition 8.
If closed neighborhood degree is the same for all vertices inLCG, i.e., if θℕncgg=m, for all g∈A, then LCG is called an m-totally regular neutrosophic cubic graph.Example 9.
Consider Example4; here, LCG is a totally regular but not a regular neutrosophic cubic graph.Example 10.
ConsiderLCG=Γ,Λ for any graph G=A,B with A=v1,v2,v3,B=v1v2,v2v3,v1v3 and let
(13)Γ=v1,.3,.5,.6,.4,.5,.3,.6,.7,.2,v2,.3,.5,.6,.4,.5,.3,.6,.7,.2,v3,.3,.5,.6,.4,.5,.3,.6,.7,.3,Λ=v1v2,.2,.4,.6,.4,.5,.3,.6,.7,.2,v2v3,.1,.2,.6,.4,.5,.3,.6,.7,.3,v1v3,.3,.4,.6,.4,.5,.3,.6,.7,.3,then the neutrosophic cubic open neighborhood degree of each vertex is given by
(14)θℕncgv1=Φv2+Φv3=.6,1,1.2,.8,1,.6,1.2,1.4,.4,θℕncgv2=Φv1+Φv3=.6,1,1.2,.8,1,.6,1.2,1.4,.4,θℕncgv3=Φv1+Φv2=.6,1,1.2,.8,1,.6,1.2,1.4,.4;similarly, the neutrosophic cubic closed neighborhood degree of each vertex is given by
(15)θℕncgv1=Φv2+Φv3+Φv1=.9,1.5,1.8,1.2,1.5,.9,1.8,2.1,.7,θℕncgv2=Φv1+Φv3+Φv2=.9,1.5,1.8,1.2,1.5,.9,1.8,2.1,.7,θℕncgv3=Φv2+Φv1+Φv3=.9,1.5,1.8,1.2,1.5,.9,1.8,2.1,.7.Asθℕncgv1=θℕncgv2=θℕncgv3 also θℕncgv1=θℕncgv2=θℕncgv3. Hence, LCG is regular, also totally regular neutrosophic cubic graph, as shown in Figure 2.Example 11.
LetLCG=Γ,Λ be a neutrosophic cubic graph of G=A,B with A=v1,v2,v3,B=v1v2,v2v3,v1v3 such that Γ and Λ are given by Tables 2 and 3.Figure 2
Represents a regular and totally regular neutrosophic cubic graph.Table 2
A neutrosophic cubic membership for vertices.
Γ=AαTl,αTrαTαIl,αIrαIαFl,αFrαFv1.3,.5.4.1,.3.2.5,.6.5v2.2,.4.3:1,.4.3.6,.7.7v3.1,.3.2:2,.3.7.7,.8.6Table 3
A neutrosophic cubic membership for edges.
Λ=BβTl,βTrβTβTl,βIrβIβFl,βFrβFv1v2.2,.4.4.1,.3.2.3,.5.6v2v3.1,.3.3.1,.3.3.4,.6.5Then,(16)θℕncgv1=∑α∈ℕncgv1,α≠v1Φα=Φv2=.2,.4,.3,.1,.4,.3,.6,.7,.7,(17)θℕncgv2=∑α∈ℕncgv2,α≠v2Φα=Φv1+Φv3=.4,.8,.6,.3,.6,.3,1.2,1.4,1.1.Sinceθℕncgv1≠θℕncgv2 although θℕncgv1=θℕncgv3, so LCG is not regular as well as totally regular as shown in Figure 3.Theorem 12.
LetLCG=Γ,Λ be a neutrosophic cubic graph of G, with Γ showing LCS for vertex set A and Λ is LCS for edge set B. Then,
(18)Γ=Φg=gαTl,αTrg,αTg,αIl,αIr,αIg,αFl,αFrg,αFgg∈Ais a constant if and only if we have equivalence in the following:
(I)
LCG is regular(II)
LCG is totally regularProof.
Suppose(19)Γ=Φg=αTl,αTrg,αTg,αIl,αIrg,αIg,αFl,αFrg,αFg=kfor all g∈A, where k is some constant, then
(20)αTlg=t1,αTrg=t2,αTg=t,αIlg=i1,αIrg=i2,αIg=i,αFlg=f1,αFrg=f2,αFg=ffor all g∈A and for some constants t1,t2,t,i1,i2,i,f1,f2,f.
I⇒II Let LCG be a regular, then θℕncgg=n for all g∈A. So,
(21)θℕncgg=∑s∈ℕncgg,s≠gΦs=n=θTlg,θTg,θIlg,θIrg,θIgθFlg,θFrg,θFg.
Hence, for allg∈A, we have
(22)θTlg=n1t,θTrg=n2t,θTg=nt,θIlg=n1i,θIrg=n2i,θIg=ni,θFlg=n1f,θFrg=n2f,θFg=nf.
Thus,(23)θℕncgg=∑s∈ℕncgg∪gΦs=θTlg,θTrg,θTg,θIlg,θIrg,θIg,θFlg,θFrg,θFg=θTlg,θTrg,θTg,θIlg,θIrg,θIg,θFlg,θFrg,θFg+αTl,αTrg,αTg,αIl,αIrg,αIx,αFl,αFrg,αFg=n1t,n2t,nt,n1i,n2i,ni,n1f,n2f,nf+t1,t2,t,i1,i2,i,f1,f2,f,i.e.,
(24)θℕncgg=θℕncgg+Φg=n+k,a constant number, for all g∈A. Thus, LCG is totally regular.
II⇒I Suppose that LCG is totally regular. Then, for all g∈A,
(25)θℕncgg=m,θℕncgg=θℕncgg+Φg,i.e.,
(26)m=θTlg,θTrg,θTg,θIlg,θIrg,θIg,θFlg,θFrg,θFg.
Let(27)m=m1t,m2t,mt,m1i,m2i,mi,m1f,m2f,mf,where m1t,m2t,mt,m1i,m2i,mi,m1f,m2f,mf are all constants. Also, given that for every g∈A,
(28)Γ=Φg=αTl,αTrg,αTg,αIl,αIrg,αIg,αFl,αFrg,αFg=k,where k is a constant; also, let
(29)Φg=k=t1,t2,t,i1,i2,i,f1,f2,f.
Then,(30)m1t,m2t,mt,m1i,m2i,mi,m1f,m2f,mf=θℕncgg+t1,t2,t,i1,i2,i,f1,f2,ffor all g∈A. Hence, for all g∈A,
(31)θℕncgg=m1t,m2t,mt,m1i,m2i,mi,m1f,m2f,mf−t1,t2,t,i1,i2,i,f1,f2,f=m1t−t1,m2t−t2,mt−t,m1i−i1,m2i−i2,mi−i,m1f−f1,m2f−f2,mf−f,which a constant number. Thus, LCG is regular. So, I and II have equivalence. Conversely, if LCG is totally regular, then θℕncgg=m (a constant) for all g∈A; also, LCG is regular. So, θℕncgg=n say for every g∈A. Hence, for every g∈A,
(32)θℕncgg=θℕncgg+Φg⇒m=n+Φg⇒Φg=m−n,
a constant no for everyg∈A. Hence,
(33)Γ=Φg=αTl,αTrg,αTg,αIl,αIrg,αIg,αFl,αFrg,αFg
for allg∈A is a constant function.Theorem 13.
ConsiderLCG=Γ,Λ as a neutrosophic cubic graph with crisp graph G of an odd cycle. Then, LCG is regular if and only if Λ is a constant function.Proof.
Let(34)Λ=Ψxy=βTl,βTrxy,βTxy,βIl,βIrxy,βIxy,βFl,βFrxy,βFxy=C
be a constant function for allxy∈B. Then,
(35)C=c1t,c2t,ct,c1i,c2i,ci,c1f,c2f,cf.
Now,(36)θℕncgx=θTlx,θTrx,θTx,θIlx,θIrx,θIx,θFlx,θFrx,θFx=2c1t,2c2t,2ct,2c1i,2c2i,2ci,2c1f,2c2f,2cf=2C,∀x∈A.
SinceG is an odd cycle, LCG is regular. Conversely, suppose that LCG is an n- regular, where
(37)n=n1t,n2t,nt,n1i,n2i,ni,n1f,n2f,nf.
Lete1,e2,e3,⋯,e2n+1 be edges of LCG in that order. Let
(38)βTle1=k1,υTle2=n1t−k1,βTle3=n1t−n1t−k1=k1,υTle4=n1t−k1,and so on. Therefore,
(39)βTlei=k1,ifiisodd,n1t−k,ifiiseven.
This implies(40)βTle1=υTle2n+1=k1.
So, ife1 and e2n+1 are incident at a vertex v1, then
(41)θv1=n1t,θe1+θe2n+1=n1t.
Hence,(42)k1+k1=n1t,2k1=n1t,and so, k1=n1t/2, which shows that βTl is a constant function. Similarly, let
(43)βTre1=k2,βTre2=n2t−k2,βTre3=n2t−n2t−k2=k2,βTre4=n2t−k2,and so on. Therefore,
(44)βTrei=k2,ifiisodd,n2t−k2,ifiiseven.
Thus,βTre2=βTre2n=k2. So, if e2 and e2n are incident at a vertex v2, then
(45)θv2=n2t,θe2+θe2n=n2t.
Hence,(46)k2+k2=n2t,2k2=n2t,
and so,k2=n2t/2, which shows that υTr is a constant function.
Similar results hold for membership functionsβx,βI−x,βI+x,βIx,βF−x,βF+x,βFx. This shows that
(47)Λ=Ψxy=βTl,βTrxy,βTxy,βIl,βIrxy,βIxy,βFl,βFrxy,βFxy
is a constant function.Figure 3
Shows a neutrosophic cubic graph, which is not regular as well as totally regular.


### 3.3. Complete Neutrosophic Cubic Graphs

In this subsection, we present complete neutrosophic cubic graphLCG.Definition 14.
ConsiderLCG=Γ,Λ be a neutrosophic cubic graph for any arbitrary graph G=A,B. Then, LCG is complete if
(48)βT^v1v2=rminαT^v1,αTv2,βTv1v2=maxαTv1,αTv2,βI^v1v2=rminαI^v1,αIv2,βIv1v2=maxαIv1,αIv2,βF^v1v2=rmaxαF^v1,αFv2,βFv1v2=minαFv1,αFv2for all vertices v1,v2∈A and for all edges v1v2∈B.Example 15.
ConsiderLCG=Γ,Λ for a graph G=A,B with A=v1,v2,v3,B=v1v2,v2v3,v1v3 and let
(49)Γ=v1,.3,.4,.6,.2,.6,0.3,.6,.3,.3,v2,.2,.5,.5,.3,.7,0.4,.5,.4,.2,v3,.1,.2,.6,.4,.5,0.3,.6,.2,.3,(50)Λ=v1v2,.2,.4,.6,.2,.6,.3,.6,.4,.2,v2v3,.1,.2,.6,.3,.5,.3,.6,.4,.2,v1v3,.1,.2,.6,.2,.5,.3,.6,.3,.3,then
(51)βT^v1v2=.2,.4,rminαT^v1,αT^v2=.2,.4,βTv1v2=.6,maxαTv1,αTv2=.6,(52)βI^v1v2=.2,.6,rminαI^v1,αI^v2=.2,.6,βIv1v2=.3,maxαIv1,αIv2=.3.
Also,(53)βF^v1v2=rmaxαF^v1,αF^v2=.6,.4,βFv1v2=minαFv1,αFv2=.2;similar holds for other edges. Hence, LCG is a complete neutrosophic cubic graph, as shown in Figure 4.Definition 16.
LetLCG=Γ,Λ be a neutrosophic cubic graph for some graph G=V,E. The density of LCG is defined as
(54)℘LCG=℘TlLCG,℘TrLCG,℘TLCG,℘IlLCG,℘IrLCG,℘ILCG,℘FlLCG,℘FrLCG,℘FLCG,where
(55)℘TlLCG=2∑g1,g2∈VβTlg1g2∑g1,g2∈VminαTlg1,αTlg2,℘TrLCG=2∑g1,g2∈VβTrg1g2∑g1,g2∈VminαTrg1,αTrg2,℘TLCG=2∑g1,g2∈VβTg1g2∑g1,g2∈VminαTg1,αTg2,℘IlLCG=2∑g1,g2∈VβIlg1g2∑g1,g2∈VminαIlg1,αIlg2,℘IrLCG=2∑g1,g2∈VβIrg1g2∑g1,g2∈VminαIrg1,αIrg2,℘ILCG=2∑g1,g2∈VβIg1g2∑g1,g2∈VminαIg1,αIg2,℘FlLCG=2∑g1,g2∈VβFlg1g2∑g1,g2∈VminαFlg1,αFlg2,℘FrLCG=2∑g1,g2∈VβFrg1g2∑g1,g2∈VminαFrg1,αFrg2,℘FLCG=2∑g1,g2∈VβTlg1g2∑g1,g2∈VminαFlg1,αFlg2.Example 17.
ConsiderLCG=Γ,Λ for a graph G=A,B, with vertex set A=v1,v2,v3 and edge set B=v1v2,v2v3,v1v3. Also, let Γ and Λ be neutrosophic membership functions for vertices and edges, respectively, shown in Tables 4 and 5.Figure 4
Represents a complete neutrosophic cubic graph.Table 4
Neutrosophic membership functions for vertices.
Γ=AαTl,αTrαTαIl,αIrαIαFl,αFrαFv1.1,.5.6.5,.71.6,.7.9v2.2,.6.5.6,.8.8.7,.8.8v3.3,.7.8.7,.8.7.5,.6.7Table 5
Neutrosophic membership functions for edges.
Λ=BβTl,βTrβTβIl,βIrβIβFl,βFrβFv1v2.1,.2.5.4,.5.4.5,.6.2v2v3.2,.4.2.1,.3.5.4,.5.3v1v3.1,.2.5.1,.3.6.4,.6.2Since(56)2∑x,y∈AΨxy=2Ψv1v2+Ψv2v3+Ψv1v3=2∑x,y∈AβTl,βTrxy,βTxy,βIl,βIrxy,βIxy,βFl,βFrxy,βFxy=2.4,.8,1.2,.6,1.1,1.5,1.3,1.7,.7=.8,1.6,2.4,1.2,2.2,3,2.6,3.4,1.4.Also,(57)minΦv1,Φv2=.1,0.5,.5,.5,.7,.8,.6,.7,.8,minΦv2,Φv3=.2,.6,.5,.6,.8,.7,.5,.6,.7,minΦv1,Φv3=.1,.5,.6,.5,0.7,.7,.5,.6,.7.So,(58)∑x,y∈AminΦx,Φy=.4,1.6,1.6,1.6,2.2,2.2,1.6,1.9,2.2.Hence,℘LCG is given by
(59)℘LCG=2,1,1.6,.75,1,1.36,1.625,1.78,.63.


### 3.4. Balanced and Strictly Balanced Neutrosophic Cubic Graphs

In this subsection, we use the density function℘ to discuss the idea of balanced and strictly balanced neutrosophic cubic graph LCG.Definition 18.
LCG is balanced if ℘H≤℘LCG for all subgraphs H of LCG.Definition 19.
LCG is strictly balanced if ℘H=℘LCG for all nonempty subgraphs H of LCG.Example 20.
ConsiderLCG as given in Example 17. Let H1=a,b,H2=a,c,H3=b,c. Then,
(60)℘H1=2ΨabminΦa,Φb=2.1,.2,.5,.4,.5,.4,.5,.6,.2.1,.5,.5,.5,.7,.8,.6,.7,.8=.2,.4,1,.8,1,.8,1,1.2,.4.1,.5,.5,.5,.7,.8,.6,0.7,.8=2,.8,2,1.6,1.4,1,1.66,1.7,.5,℘H2=2ΨacminΦa,Φc=2.1,.2,.5,.1,.3,.6,.4,.6,.2.1,.5,.6,.5,.7,.7,.5,.6,.7=.2,.4,1,.2,.6,1.2,.8,1.2,.4.1,.5,.6,.5,.7,.7,.5,.6,.7=2,1.33,.4,.33,.75,1.42,1.6,1.66,.85;also,
(61)℘H3=2ΨbcminΦb,Φc=2.2,.4,.2,.1,.3,.5,.4,.5,.3.2,.6,.5,.6,.8,.7,.5,.6,.7=.4,.8,.2,.2,.6,1,.8,1,.6.2,.6,.5,.6,.8,.7,.5,.6,.7=2,1.33,.4,.33,.75,1.42,1.6,1.66,.85.
Hence,(62)℘H1=2,.8,2,1.6,1.4,1,1.66,1.7,.5,℘H2=2,.8,1.66,.4,.85,1.7,1.6,2,.57,℘H3=2,1.33,.4,.33,.75,1.42,1.6,1.66,.85,as ℘H≨℘LCG, for all nonempty subgraphs H of LCG. Hence, LCG is not balanced.Remark 21.
All regular neutrosophic cubic graphs are not necessarily balanced.


### 3.5. Irregular and Totally Irregular Neutrosophic Cubic Graphs

In this subsection, we use the neighborhood degrees to discuss the idea of irregular and totally irregular neutrosophic cubic graphLCG.Definition 22.
If there is at least one vertex inLCG adjacent to vertices having different open neighborhood degrees, then LCG is called irregular, i.e., if θℕncgv≠n for all v∈A.Example 23.
ConsiderLCG=Γ,Λ for some graph G=A,B, with A=v1,v2,v3,v4 and
(63)B=v1v2,v2v3,v3v4,v2v4,and let
(64)Γ=v1,.2,.5,.4,.1,.3,.6,.6,.7,.8,v2,.3,.4,.3,.2,.5,.5,.6,.8,.7,v3,.1,.3,.5,.3,.4,.2,.5,.7,.9,v4,.2,.4,.2,.4,.5,.3,.7,.8,.6,(65)Λ=v1v2,.1,.4,.4,.1,.5,.6,.6,.7,.8,v2v3,.1,.3,.3,.2,.4,.5,.6,.8,.9,v3v4,.1,.3,.5,.2,.4,.4,.5,.7,.9,v2v4,.1,.4,.3,.2,.4,.5,.5,.7,1,then
(66)θℕncgv1=Φv2=.3,.4,.3,.2,.5,.5,.6,.8,.7,θℕncgv2=Φv3+Φv4=.1,.3,.5,.3,.4,.2,.5,.7,.9+.2,.4,.2,.4,.5,.3,.7,.8,.6=.3,.7,.7,.7,.9,.5,1.2,1.5,1.5,θℕncgv3=Φv2+Φv4=.3,.4,.3,.2,.5,.5,.6,.8,.7+.2,.4,.2,.4,.5,.3,.7,.8,.6=.5,.8,.5,.6,1,.8,1.3,1.6,1.3;hence,
(67)θℕncgv1≠θℕncgv2≠θℕncgv3≠θℕncgv4.Hence,LCG is irregular as shown in Figure 5.Definition 24.
A connectedLCG is totally irregular, if at least one vertex is adjacent to the vertices having different closed neighborhood degrees.Example 25.
ConsiderLCG=Γ,Λ for G=A,B, with
(68)A=v1,v2,v3,v4,v5,B=v1v2,v2v3,v2v4,v3v4,v1v3,v1v4,v4v5,
and let(69)Γ=v1,.3,.5,.4,.2,.3,.1,.3,.6,.4,v2,.2,.4,.3,.3,.4,.2,.6,.7,.5,v3,.2,.6,.5,.2,.5,.4,.5,.7,.4,v4,.3,.5,.4,.3,.6,.5,.4,.6,.3,v5,.2,.2,.1,.2,.3,.4,.3,.5,.2,(70)Λ=v1v2,.2,.4,.3,.2,.3,.1,.3,.5,.3,v2v3,.2,.4,.3,.2,.4,.2,.5,.6,.2,v2v4,.1,.4,.2,.3,.4,.2,.3,.5,.2,v3v4,.2,.5,.3,.2,.5,.3,.4,.5,.3,v3v1,.2,.4,.1,.2,.3,.1,.3,.6,.4,v1v4,.2,.3,.3,.2,.3,.1,.3,.5,.3,v4v5,.1,.2,.1,.2,.3,.3,.3,.4,.2.
Then,(71)θℕncgv1=Φv2+Φv3+Φv4+Φv1=.2,.4,.3,.3,.4,.2,.6,.7,.5+.2,.6,.5,.2,.5,.4,.5,.7,.4+.3,.5,.4,.3,.6,.5,.4,.6,.3+.3,.5,.4,.2,.3,.1,.3,.6,.4=1,2,1.6,1,1.8,1.2,1.8,2.6,1.6,θℕncgv2=Φv1+Φv4+Φv3+Φv2=1,2,1.6,1,1.8,1.2,1.8,2.6,1.6,θℕncgv3=Φv2+Φv1+Φv4+Φv3=1,2,1.6,1,1.8,1.2,1.8,2.6,1.6,θℕncgv4=Φv1+Φv2+Φv3+Φv5+Φv4=.3,.5,.4,.2,.3,.1,.3,.6,.4+.2,.4,.3,.3,.4,.2,.6,.7,.5+.2,.6,.5,.2,.5,.4,.5,.7,.4+.2,.2,.1,.2,.3,.4,.3,.5,.2+.3,.5,.4,.3,.6,.5,.4,.6,.3=1.2,2.2,1.7,1.2,2.1,1.6,2.1,3.1,1.8,θℕncgv5=Φℕncgv4=.3,.5,.4,.3,.6,.5,.4,.6,.3.Figure 5
Represents irregular neutrosophic cubic graph.Clearly,LCG is totally irregular as in Figure 6.Figure 6
Represents totally irregular neutrosophic cubic graph.


### 3.6. Complement of a Neutrosophic Cubic Graph

Complement of a neutrosophic cubic graph is a very important concept we discuss here.Definition 26.
The complement ofLCG=Γ,Λ is a neutrosophic cubic graph LCG¯=Γ¯,Λ¯, where
(72)Γ¯=Φ¯g=g,α¯Tl,α¯Trg,α¯Tg,α¯Il,α¯Ir,α¯Ig,α¯Fl,α¯Frg,α¯Fgg∈A,Λ¯=Ψ¯g1g2=g1g2,β¯Tl,β¯Trg1g2,β¯Tg1g2,β¯Il,β¯Irg1g2,β¯Ig1g2,β¯Fl,β¯Frg1g2,β¯Fg1g2g1g2∈B,since
(73)Ψ¯xy=minΦx,Φy−Ψxy,or for truth membership functions, we have
(74)β¯Tlxy=minαTlx,αTly−βTlxy,β¯Txy=minαTx,αTy−βTxy,β¯Trxy=minαTrx,αTry−βTrxy;
similarly, for indeterminate membership functions, we have(75)β¯Ilxy=minαIlx,αIly−βIlxy,β¯Irxy=minαIrx,αIry−βIrxy,β¯Ixy=minαIx,αIy−βIxy;also, for falsity membership functions, we have similar results.Proposition 27.
For self-complementaryLCG=Γ,Λ, we have
(76)℘LCG=1,1,1,1,1,1,1,1,1.Proof.
GivenLCG is self-complementary, so Ψ¯xy=Ψxy; also, by definition of a self-complementary neutrosophic cubic graph, we have
(77)Ψ¯xy=minΦx,Φy−Ψxy.
Dividing both sides of equation (77) by minΦx,Φy,
(78)ΨxyminΦx,Φy=1,1,1,1,1,1,1,1,1−ΨxyminΦx,Φy,we get
(79)∑x,y∈AΨxyminΦx,Φy=1,1,1,1,1,1,1,1,1−∑x,y∈AΨxyminΦx,Φy.
Hence,(80)2∑x,y∈AΨxyminΦx,Φy=1,1,1,1,1,1,1,1,1,so
(81)℘LCS=1,1,1,1,1,1,1,1,1.Proposition 28.
LetLCG=Γ,Λ be strictly balanced, let LCG¯ be its complement, then ℘LCG+℘LCG¯=2,2,2,2,2,2,2,2,2.Proof.
LetLCG be strictly balanced; let LCG¯ be its complement. Let H be a nonempty subgraph of LCG. Since LCG is strictly balanced, ℘LCG=℘H for every subset H⊆LCG and for any x,y∈A. In LCG¯, we have Ψ¯xy=minΦx,Φy−Ψxy, and so, for truth membership functions, we have
(82)β¯Tlxy=minαTlx,αTly−βTlxy,forallx,y∈A.
Dividing equation (82) by minαTlx,αTly, we get
(83)β¯TlxyminαTlx,αTly=1−βTlxyminαTlx,αTly.
Hence,(84)∑x,y∈Aβ¯TlxyminαTlx,αTly=1−∑x,y∈EβTlxyminαTlx,αTly.
Multiplying both sides by 2,we get(85)2∑x,y∈Aβ¯TlxyminαTlx,αTly=2−2∑x,y∈EβTlxyminαTlx,αTly.
Hence,℘TlLCG¯=2−℘TlLCG. Similarly for right end point of interval in truth valued membership functions, we have ℘TrLCG¯=2−℘TrLCG. Similar results hold for the rest of membership functions. Hence,
(86)℘ℓCG¯=2,2,2,2,2,2,2,2,2−℘ℓCG,℘ℓCG¯+℘ℓCG=2,2,2,2,2,2,2,2,2.
This completes the proof.


### 3.7. Neighborly Irregular and Neighborly Totally Irregular Neutrosophic Cubic Graphs

In this subsection, we use the neighborhood degrees to discuss the idea of neighborly irregular and neighborly totally irregular neutrosophic cubic graphs.Definition 29.
A connectedLCG is neighborly irregular if every two adjacent vertices in LCG have different closed neighborhood degrees.Example 30.
ConsiderLCG=Γ,Λ for any G=A,B with A=a,b,c,d and B=ab,bc,cd,da and let neutrosophic cubic membership functions for vertices and edges be represented in Tables 6 and 7, respectively.Table 6
Neutrosophic membership functions for vertices.
Γ=AαTl,αTrαTαIl,αIrαIαFl,αFrαFa.2,.6.3.1,.3.2.5,.7.5b.3,.7.2.2,.4.3.4,.5.6c.4,.8.3.3,.5.4.6,.7.4d.5,.7.4.1,.2.3.7,.8.7Table 7
Neutrosophic membership functions for edges.
Λ=BβTl,βTrβTβIl,βIrβIβFl,βFrβFab.1,.4.2.1,.2.1.3,.5.5bc.2,.5.1.2,.3.2.4,.5.3cd.3,.6.3.1,.2.3.5,.6.4da.2,.5.2.1,.2.2.4,.5.5Then, clearlyLCG is neighborly irregular, as shown in Figure 7.Definition 31.
LCG is said to be neighborly totally irregular if every two adjacent vertices of LCG have different closed neighborhood degrees.Example 32.
ConsiderLCG=Γ,Λ for any G=A,B with A=v1,v2,v3,v4 and B=v1v2,v2v3,v3v4,v1v4. Also, let
(87)Γ=v1,.2,.6,.4,.2,.4,.5,.4,.6,.5,v2,.3,.5,.3,.4,.5,.5,.5,.7,.4,v3,.4,.7,.5,.3,.5,.2,.3,.6,.1,v4,.5,.6,.2,.2,.5,.3,.6,.8,.2,(88)Λ=v1v2,.2,.5,.4,.2,.4,.4,.5,.7,.4,v2v3,.3,.5,.5,.3,.4,.4,.5,.6,.1,v3v4,.2,.6,.5,.2,.5,.3,.5,.7,.1,v1v4,.1,.4,.3,.2,.4,.5,.5,.7,.2.Figure 7
Represents neighborly irregular neutrosophic cubic graph.Then, clearlyLCG is neighborly totally irregular, as shown in Figure 8.Figure 8
Represents neighborly totally irregular neutrosophic cubic graph.


### 3.8. Highly Irregular Neutrosophic Cubic Graphs

In this subsection, we use the neighborhood degrees to discuss the idea of highly irregular neutrosophic cubic graphs.Definition 33.
Consider a connectedLCG; then, LCG is highly irregular if every vertex in LCG is adjacent to vertices with different neighborhood degrees.Example 34.
ConsiderLCG=Γ,Λ for a graph G=A,B with A=v1,v2,v3,v4,v5,v6 and B=v1v2,v2v3,v2v6,v3v4,v3v5,v4v5,v1v5 and let
(89)Γ=v1,.2,.5,.3,.1,.3,.4,.4,.5,.3,v2,.1,.4,.2,.2,.4,.2,.3,.6,.4,v3,.3,.5,.6,.2,.5,.3,.5,.7,.4,v4,.4,.6,.3,.3,.6,.5,.4,.6,.3,v5,.2,.4,.5,.2,.3,.4,.2,.5,.2,v6,.1,.4,.3,.2,.4,.2,.3,.5,.2.
By routine computations, we have(90)θℕncgv1=Φv2+Φv5=.1,.4,.2,.2,.4,.2,.3,.6,.4+.2,.4,.5,.2,.3,.4,.2,.5,.2=.3,.8,.7,.4,.7,.6,.5,1.1,.6,θℕncgv2=Φv1+Φv6+Φv3=.2,.5,.3,.1,.3,.4,.4,.5,.3+.1,.4,.3,.2,.4,.2,.3,.5,.2+.3,.5,.6,.2,.5,.3,.5,.7,.4=.6,1.4,1.2,.5,1.2%,.9,1.2,1.7,.9,θℕncgv4=Φv3+Φv5=.3,.5,.6,.2,.5,.3,.5,.7,.4+.2,.4,.5,.2,.3,.4,.2,.5,.2=.5,.9,1.1,.4,.8,.7,.7,1.2,.6,θℕncgv5=Φv3+Φv4+Φv1=.3,.5,.6,.2,.5,.3,.5,.7,.4+.4,.6,.3,.3,.6,.5,.4,.6,.3+.2,.5,.3,.1,.3,.4,.4,.5,.3=.9,1.6,1.2,.6,1.4%,1.2,1.3,1.8,1,θℕncgv6=Φv2.1,.4,.2,.2,.4,.2,.3,.6,.4.Clearly,LCG as shown in Figure 9 is highly irregular.Theorem 35.
LCG is highly irregular and neighborly irregular if and only if open neighborhood degrees for all vertices of LCG are different.Proof.
SupposeLCG has n vertices v1,v2,⋯,vn. Also, let LCG be highly irregular and neighborly irregular.
Claim 1. The open neighborhood degrees for all vertices inLCG are different. Let
(91)θℕncgvi=λiTl,λiTr,λiT,λiIl,λiIr,λiI,λiFl,λiFr,λiF
for alli=1,2,3,⋯,n. Let the adjacent vertices of v1 be v2,v3,⋯,vn with open neighborhood degrees:
(92)θℕncgvi=λiTl,λiTr,λiT,λiIl,λiIr,λiI,λiFl,λiFr,λiFfor all i=2,3,⋯,n, respectively. Then, as LCG is highly irregular, we have
(93)λ2Tl≠λ3Tl≠⋯≠λnTl,λ2Tr≠λ3Tr≠⋯≠λnTr,λ2T≠λ3T≠⋯≠λnTfor all i=2,3,⋯,n. Similar holds for indeterminacy and falsity membership functions. Also, LCG is neighborly irregular, so we have
(94)λ1Tl≠λ2Tl≠,⋯,≠λnTl,λ1Tr≠λ2Tr≠⋯≠λnTr,λ1T≠λ2T≠⋯≠λnT.Similar holds for indeterminacy and falsity membership functions. Hence, open neighborhood degrees of all vertices of LCG are different.
Claim 2.LCG is highly irregular and neighborly irregular. Let
(95)θℕncgvi=λiTl,λiTr,λiT,λiIl,λiIr,λiI,λiFl,λiFr,λiFfor all i=1,2,3,⋯,n are degrees for all vertices of LCG. Given open neighborhood degrees of all vertices of LCG are different, so
(96)θℕncgv1≠θℕncgv2≠⋯≠θℕncgvn,λ1Tl≠λ2Tl≠⋯≠λnTl,λ1Tr≠λ2Tr≠⋯≠λnTr,λ1T≠λ2T≠⋯≠λnTfor all i=1,2,3,⋯,n. Similar holds for indeterminacy and falsity membership functions. So, any two vertices in LCG have different open neighborhood degrees, and for every vertex, adjacent vertices have different open neighborhood degrees, which proves the result.Remark 36.
A completeLCG may not be neighborly irregular.Example 37.
LetLCG=Γ,Λ for any G=A,B, with A=v1,v2,v3 and B=v1v2,v2v3,v1v3 such that
(97)Γ=v1,.4,.8,.4,.3,.5%,.4,.6,.7,.3,v2,.2,.4,.2,.4,.7%,.5,.5,.8,.2,v3,.2,.4,.2,.4,.7%,.5,.5,.8,.2,(98)Λ=v1v2,.2,.4,.2,.3,.5%,.4,.5,.7,.2,v2v3,.2,.4,.2,.4,.7%,.5,.5,.8,.2,v1v3,.2,.4,.2,.3,.5%,.4,.5,.7,.2.
By simple computation, we get(99)θℕncgv1=.4,.8,.4,.8,1.4,1,1,1.6,.4,θℕncgv2=.6,1.2,.6,.7,1.2,.9,1.1,1.5,.5,θℕncgv3=.6,1.2,.6,.7,1.2,.9,1.1,1.5,.5.Figure 9
Represents highly irregular neutrosophic cubic graph.Here,θℕncgv2=θℕncgv3, so the neighborhood degree is not different. Hence, LCG is not neighborly irregular; also, we have similar holds for all vertices and edges. So, LCG is complete. Hence, a complete LCG may not be neighborly irregular as shown in Figure 10.Theorem 38.
IfLCG is neighborly irregular and
(100)Γ=Φx=αTl,αTrx,αTx,αIl,αIrx,αIx,αFl,αFrx,αFxfor all x∈A is a constant function, then it is neighborly totally irregular.Proof.
Assume thatLCG is a neighborly irregular. Then, open neighborhood degrees of every two adjacent vertices are different. Let vi,vj∈A be adjacent vertices with different open neighborhood degrees. Then, θℕncgvi≠θℕncgvj for all i≠j; let θℕncgvi=d1&θℕncgvj=d2 then d1≠d2. Also, as
(101)Γ=Φx=αTl,αTrx,αTx,αIl,αIrx,αIx,αFl,αFrx,αFxis constant for all x∈A. Hence, Φvi=Φvj=k; suppose that LCG is not neighborly totally irregular, then
(102)θℕncgvi=θℕncgvj,for some i≠j but θℕncgvi=θℕncgvi+Φvi and θℕncgvj=θℕncgvj+Φvj using these values in equation (102), we get
(103)θℕncgvi+Φvi=θℕncgvj+Φvj⇒d1+k=d2+k⇒d1=d2,as cancellation law holds in 0,1, which contradicts, as
(104)d1≠d2.
Hence,(105)θℕncgvi≠θℕncgvj,so LCG is neighborly totally irregular. This proves the result.Theorem 39.
IfLCG is neighborly totally irregular and
(106)Γ=Φx=αTl,αTrx,αTx,αIl,αIrx,αIx,αFl,αFrx,αFx,x∈A,is a constant function, then it is neighborly irregular.Proof.
Assume thatLCG is a neighborly totally irregular. Then, closed neighborhood degrees of every two adjacent vertices are distinct. Let vi,vj∈A be adjacent vertices with distinct closed neighborhood degrees. Then, for all i≠j,
(107)θℕncgvi≠θℕncgvj,let
(108)θℕncgvi=f1&θℕncgvj=f2,then f1≠f2. Also, as
(109)Γ=Φx=αTl,αTrx,αTx,αIl,αIrx,αIx,αFl,αFrx,αFx;suppose that LCG is not neighborly irregular, then
(110)θℕncgvi=θℕncgvj=w;say, for some i≠j but
(111)θℕncgvi=θℕncgvi+Φvi,(112)θℕncgvj=θℕncgvj+Φvj,using these values in equation (112), we get
(113)θℕncgvi+Φvi=θℕncgvj+Φvj=w+r,so
(114)θℕncgvi=θℕncgvj,for some i≠j which is a contradiction to the fact that LCG is a neighborly totally irregular neutrosophic cubic graph. Hence,
(115)θℕncgvi≠θℕncgvj,so LCG is neighborly irregular. This proves the result.Proposition 40.
IfLCG is neighborly irregular as well as neighborly totally irregular, then
(116)Γ=Φx=αTl,αTrx,αTx,αIl,αIrx,αIx,αFl,αFrx,αFxneed not be a constant function.Remark 41.
IfLCG is neighborly irregular, then a neutrosophic cubic subgraph H of LCG may not be neighborly irregular.Remark 42.
IfLCG is neighborly totally irregular, then a neutrosophic cubic subgraph H of LCG may not be neighborly totally irregular.Figure 10
Represents a complete neutrosophic cubic graph, but not neighborly irregular.


## 3.1. The Open and the Closed Neighborhood for Any Vertex inLCG

In this subsection, we present the idea of open neighborhoodℕncgg, degree of open neighborhood θℕncgg, and closed neighborhood degree of neutrosophic cubic graph LCG.Definition 2.
An open neighborhoodℕncgg for any vertex g in LCG=Γ,Λ is given by
(4)ℕncgg=NTlg,NTrg,NTg,NIlg,NIrg,NIg,NFlg,NFrg,NFg,
where(5)NTlg=s∈A:βTlgs≤minαTlg,αTls,g≠s,NTrg=s∈A:βTrgs≤minαTrg,αTrs,g≠s,NTg=s∈A:βTgs≤minαTg,αTs,g≠s,NIlg=s∈A:βIlgs≤minαIlg,αIls,g≠s,NIrg=s∈A:βI+gs≤minαI+g,αI+s,g≠s,NIg=s∈A:βTgs≤minαTg,αTs,g≠s,NFlg=s∈A:βF−gs≤maxαF−g,αF−s,g≠s,NFrg=s∈A:βF+gs≤maxαF−g,αF−s,g≠s,NFg=s∈A:βFgs≤maxαFg,αFs,g≠s.
It consists of membership functions for all vertices adjacent tog, excluding s.Definition 3.
The degree of open neighborhoodθℕncgg for any vertex g in LCG is defined by
(6)θℕncgg=∑s∈ℕncgg∣s≠gΦs=θTlg,θTrg,θTg,θIlg,θIrg,θIg,θFlg,θFrg,θFg=∑s∈ℕncggs≠gαTls,∑s∈ℕncggs≠gαTry,∑s∈ℕncggs≠gαTy,∑s∈ℕncggs≠gαIls,∑s∈ℕncggs≠gαIrs,∑s∈ℕncggs≠gαIs,∑s∈ℕncggs≠gαFls,∑s∈ℕncggs≠gαFrs,∑s∈ℕncggs≠gαFs.Example 4.
LetG=A,B with vertices A=g1,g2,g3 and edges B=g1g2,g2g3,g1g3. Also, LCG=Γ,Λ such that
(7)Γ=g1,.1,.2,.5,.4,.5,.3,.6,.7,.2,g2,.2,.4,.1,.5,.6,.4,.1,.2,.3,g3,.3,.4,.2,.1,.3,.7,.4,.6,.3,Λ=g1g2,.1,.2,.5,.4,.5,.4,.6,.7,.2,g2g3,.2,.4,.2,.1,.3,.7,.4,.6,.3,g1g3,.1,.2,.5,.1,.3,.7,.6,.7,.2,then clearly, LCG=Γ,Λ is a neutrosophic cubic graph as shown in Figure 1.Figure 1
Represents a neutrosophic cubic graph.θℕncgg for each element g∈A is given by
(8)θℕncgg1=Φg2+Φg3=.2,.4,.1,.5,.6,.4,.1,.2,.3+.3,.4,.2,.1,.3,.7,.4,.6,.3=.5,.8,.3,.6,.9,1.1,.5,.8,.6,θℕncgg2=Φg1+Φg3=.1,.2,.5,.4,.5,.3,.6,.7,.2+.3,.4,.2,.1,.3,.7,.4,.6,.3=.4,.7,.6,.5,.8,1,1,1.3,.5,θℕncgg3=Φg1+Φg2=.1,.2,.5,.4,.5,.3,.6,.7,.2+.2,.4,.1,.5,.6,.4,.1,.2,.3=.3,.6,.6,.9,1.1,.7,.7,.9,.5.Definition 5.
A closed neighborhood degree for a vertexg in LCG=Γ,Λ is given by
(9)θℕncgg=∑s∈ℕncgg∪gΦsθTlg,θTrg,θTg,θTlg,θIrg,θIg,θFlg,θFrg,θFg=∑s∈ℕncgg∪gαTls,∑s∈ℕncgg∪gαTrs,∑s∈ℕncgg∪gαTs,∑s∈ℕncgg∪gαTls,∑s∈ℕncgg∪gαIrs,∑s∈ℕncgg∪gαIs,∑s∈ℕncgg∪gαFly,∑s∈ℕncgg∪gαFry,∑s∈ℕncgg∪gαFy.
It consists of the sum of membership functions of all vertices adjacent tox and membership function of x.Example 6.
Consider Example4, closed neighborhood degree θℕncgg for each element g∈A in LCG is given by
(10)θℕncgg1=Φg2+Φg3+Φg1=.2,.4,.1,.5,.6,.4,.1,.2,.3+.3,.4,.2,.1,.3,.7,.4,.6,.3+.1,.2,.5,.4,.5,.3,.6,.7,.2=.6,1,.8,1,1.4,1.4,1.1,1.5,.8;similarly,
(11)θℕncgg2=Φg1+Φg3+Φg2;also, we have
(12)θℕncgg3=Φg1+Φg2+Φg3.


## 3.2. Regular and Totally Regular Neutrosophic Cubic Graphs

In this subsection, we present the idea of regular and totally regular neutrosophic cubic graphs based on the open neighborhood degree and closed neighborhood degree.Definition 7.
If every vertex inLCG has the same open neighborhood degree n, i.e., if θℕncgg=n, for all g∈A, then LCG is called an n regular neutrosophic cubic graph.Definition 8.
If closed neighborhood degree is the same for all vertices inLCG, i.e., if θℕncgg=m, for all g∈A, then LCG is called an m-totally regular neutrosophic cubic graph.Example 9.
Consider Example4; here, LCG is a totally regular but not a regular neutrosophic cubic graph.Example 10.
ConsiderLCG=Γ,Λ for any graph G=A,B with A=v1,v2,v3,B=v1v2,v2v3,v1v3 and let
(13)Γ=v1,.3,.5,.6,.4,.5,.3,.6,.7,.2,v2,.3,.5,.6,.4,.5,.3,.6,.7,.2,v3,.3,.5,.6,.4,.5,.3,.6,.7,.3,Λ=v1v2,.2,.4,.6,.4,.5,.3,.6,.7,.2,v2v3,.1,.2,.6,.4,.5,.3,.6,.7,.3,v1v3,.3,.4,.6,.4,.5,.3,.6,.7,.3,then the neutrosophic cubic open neighborhood degree of each vertex is given by
(14)θℕncgv1=Φv2+Φv3=.6,1,1.2,.8,1,.6,1.2,1.4,.4,θℕncgv2=Φv1+Φv3=.6,1,1.2,.8,1,.6,1.2,1.4,.4,θℕncgv3=Φv1+Φv2=.6,1,1.2,.8,1,.6,1.2,1.4,.4;similarly, the neutrosophic cubic closed neighborhood degree of each vertex is given by
(15)θℕncgv1=Φv2+Φv3+Φv1=.9,1.5,1.8,1.2,1.5,.9,1.8,2.1,.7,θℕncgv2=Φv1+Φv3+Φv2=.9,1.5,1.8,1.2,1.5,.9,1.8,2.1,.7,θℕncgv3=Φv2+Φv1+Φv3=.9,1.5,1.8,1.2,1.5,.9,1.8,2.1,.7.Asθℕncgv1=θℕncgv2=θℕncgv3 also θℕncgv1=θℕncgv2=θℕncgv3. Hence, LCG is regular, also totally regular neutrosophic cubic graph, as shown in Figure 2.Example 11.
LetLCG=Γ,Λ be a neutrosophic cubic graph of G=A,B with A=v1,v2,v3,B=v1v2,v2v3,v1v3 such that Γ and Λ are given by Tables 2 and 3.Figure 2
Represents a regular and totally regular neutrosophic cubic graph.Table 2
A neutrosophic cubic membership for vertices.
Γ=AαTl,αTrαTαIl,αIrαIαFl,αFrαFv1.3,.5.4.1,.3.2.5,.6.5v2.2,.4.3:1,.4.3.6,.7.7v3.1,.3.2:2,.3.7.7,.8.6Table 3
A neutrosophic cubic membership for edges.
Λ=BβTl,βTrβTβTl,βIrβIβFl,βFrβFv1v2.2,.4.4.1,.3.2.3,.5.6v2v3.1,.3.3.1,.3.3.4,.6.5Then,(16)θℕncgv1=∑α∈ℕncgv1,α≠v1Φα=Φv2=.2,.4,.3,.1,.4,.3,.6,.7,.7,(17)θℕncgv2=∑α∈ℕncgv2,α≠v2Φα=Φv1+Φv3=.4,.8,.6,.3,.6,.3,1.2,1.4,1.1.Sinceθℕncgv1≠θℕncgv2 although θℕncgv1=θℕncgv3, so LCG is not regular as well as totally regular as shown in Figure 3.Theorem 12.
LetLCG=Γ,Λ be a neutrosophic cubic graph of G, with Γ showing LCS for vertex set A and Λ is LCS for edge set B. Then,
(18)Γ=Φg=gαTl,αTrg,αTg,αIl,αIr,αIg,αFl,αFrg,αFgg∈Ais a constant if and only if we have equivalence in the following:
(I)
LCG is regular(II)
LCG is totally regularProof.
Suppose(19)Γ=Φg=αTl,αTrg,αTg,αIl,αIrg,αIg,αFl,αFrg,αFg=kfor all g∈A, where k is some constant, then
(20)αTlg=t1,αTrg=t2,αTg=t,αIlg=i1,αIrg=i2,αIg=i,αFlg=f1,αFrg=f2,αFg=ffor all g∈A and for some constants t1,t2,t,i1,i2,i,f1,f2,f.
I⇒II Let LCG be a regular, then θℕncgg=n for all g∈A. So,
(21)θℕncgg=∑s∈ℕncgg,s≠gΦs=n=θTlg,θTg,θIlg,θIrg,θIgθFlg,θFrg,θFg.
Hence, for allg∈A, we have
(22)θTlg=n1t,θTrg=n2t,θTg=nt,θIlg=n1i,θIrg=n2i,θIg=ni,θFlg=n1f,θFrg=n2f,θFg=nf.
Thus,(23)θℕncgg=∑s∈ℕncgg∪gΦs=θTlg,θTrg,θTg,θIlg,θIrg,θIg,θFlg,θFrg,θFg=θTlg,θTrg,θTg,θIlg,θIrg,θIg,θFlg,θFrg,θFg+αTl,αTrg,αTg,αIl,αIrg,αIx,αFl,αFrg,αFg=n1t,n2t,nt,n1i,n2i,ni,n1f,n2f,nf+t1,t2,t,i1,i2,i,f1,f2,f,i.e.,
(24)θℕncgg=θℕncgg+Φg=n+k,a constant number, for all g∈A. Thus, LCG is totally regular.
II⇒I Suppose that LCG is totally regular. Then, for all g∈A,
(25)θℕncgg=m,θℕncgg=θℕncgg+Φg,i.e.,
(26)m=θTlg,θTrg,θTg,θIlg,θIrg,θIg,θFlg,θFrg,θFg.
Let(27)m=m1t,m2t,mt,m1i,m2i,mi,m1f,m2f,mf,where m1t,m2t,mt,m1i,m2i,mi,m1f,m2f,mf are all constants. Also, given that for every g∈A,
(28)Γ=Φg=αTl,αTrg,αTg,αIl,αIrg,αIg,αFl,αFrg,αFg=k,where k is a constant; also, let
(29)Φg=k=t1,t2,t,i1,i2,i,f1,f2,f.
Then,(30)m1t,m2t,mt,m1i,m2i,mi,m1f,m2f,mf=θℕncgg+t1,t2,t,i1,i2,i,f1,f2,ffor all g∈A. Hence, for all g∈A,
(31)θℕncgg=m1t,m2t,mt,m1i,m2i,mi,m1f,m2f,mf−t1,t2,t,i1,i2,i,f1,f2,f=m1t−t1,m2t−t2,mt−t,m1i−i1,m2i−i2,mi−i,m1f−f1,m2f−f2,mf−f,which a constant number. Thus, LCG is regular. So, I and II have equivalence. Conversely, if LCG is totally regular, then θℕncgg=m (a constant) for all g∈A; also, LCG is regular. So, θℕncgg=n say for every g∈A. Hence, for every g∈A,
(32)θℕncgg=θℕncgg+Φg⇒m=n+Φg⇒Φg=m−n,
a constant no for everyg∈A. Hence,
(33)Γ=Φg=αTl,αTrg,αTg,αIl,αIrg,αIg,αFl,αFrg,αFg
for allg∈A is a constant function.Theorem 13.
ConsiderLCG=Γ,Λ as a neutrosophic cubic graph with crisp graph G of an odd cycle. Then, LCG is regular if and only if Λ is a constant function.Proof.
Let(34)Λ=Ψxy=βTl,βTrxy,βTxy,βIl,βIrxy,βIxy,βFl,βFrxy,βFxy=C
be a constant function for allxy∈B. Then,
(35)C=c1t,c2t,ct,c1i,c2i,ci,c1f,c2f,cf.
Now,(36)θℕncgx=θTlx,θTrx,θTx,θIlx,θIrx,θIx,θFlx,θFrx,θFx=2c1t,2c2t,2ct,2c1i,2c2i,2ci,2c1f,2c2f,2cf=2C,∀x∈A.
SinceG is an odd cycle, LCG is regular. Conversely, suppose that LCG is an n- regular, where
(37)n=n1t,n2t,nt,n1i,n2i,ni,n1f,n2f,nf.
Lete1,e2,e3,⋯,e2n+1 be edges of LCG in that order. Let
(38)βTle1=k1,υTle2=n1t−k1,βTle3=n1t−n1t−k1=k1,υTle4=n1t−k1,and so on. Therefore,
(39)βTlei=k1,ifiisodd,n1t−k,ifiiseven.
This implies(40)βTle1=υTle2n+1=k1.
So, ife1 and e2n+1 are incident at a vertex v1, then
(41)θv1=n1t,θe1+θe2n+1=n1t.
Hence,(42)k1+k1=n1t,2k1=n1t,and so, k1=n1t/2, which shows that βTl is a constant function. Similarly, let
(43)βTre1=k2,βTre2=n2t−k2,βTre3=n2t−n2t−k2=k2,βTre4=n2t−k2,and so on. Therefore,
(44)βTrei=k2,ifiisodd,n2t−k2,ifiiseven.
Thus,βTre2=βTre2n=k2. So, if e2 and e2n are incident at a vertex v2, then
(45)θv2=n2t,θe2+θe2n=n2t.
Hence,(46)k2+k2=n2t,2k2=n2t,
and so,k2=n2t/2, which shows that υTr is a constant function.
Similar results hold for membership functionsβx,βI−x,βI+x,βIx,βF−x,βF+x,βFx. This shows that
(47)Λ=Ψxy=βTl,βTrxy,βTxy,βIl,βIrxy,βIxy,βFl,βFrxy,βFxy
is a constant function.Figure 3
Shows a neutrosophic cubic graph, which is not regular as well as totally regular.


## 3.3. Complete Neutrosophic Cubic Graphs

In this subsection, we present complete neutrosophic cubic graphLCG.Definition 14.
ConsiderLCG=Γ,Λ be a neutrosophic cubic graph for any arbitrary graph G=A,B. Then, LCG is complete if
(48)βT^v1v2=rminαT^v1,αTv2,βTv1v2=maxαTv1,αTv2,βI^v1v2=rminαI^v1,αIv2,βIv1v2=maxαIv1,αIv2,βF^v1v2=rmaxαF^v1,αFv2,βFv1v2=minαFv1,αFv2for all vertices v1,v2∈A and for all edges v1v2∈B.Example 15.
ConsiderLCG=Γ,Λ for a graph G=A,B with A=v1,v2,v3,B=v1v2,v2v3,v1v3 and let
(49)Γ=v1,.3,.4,.6,.2,.6,0.3,.6,.3,.3,v2,.2,.5,.5,.3,.7,0.4,.5,.4,.2,v3,.1,.2,.6,.4,.5,0.3,.6,.2,.3,(50)Λ=v1v2,.2,.4,.6,.2,.6,.3,.6,.4,.2,v2v3,.1,.2,.6,.3,.5,.3,.6,.4,.2,v1v3,.1,.2,.6,.2,.5,.3,.6,.3,.3,then
(51)βT^v1v2=.2,.4,rminαT^v1,αT^v2=.2,.4,βTv1v2=.6,maxαTv1,αTv2=.6,(52)βI^v1v2=.2,.6,rminαI^v1,αI^v2=.2,.6,βIv1v2=.3,maxαIv1,αIv2=.3.
Also,(53)βF^v1v2=rmaxαF^v1,αF^v2=.6,.4,βFv1v2=minαFv1,αFv2=.2;similar holds for other edges. Hence, LCG is a complete neutrosophic cubic graph, as shown in Figure 4.Definition 16.
LetLCG=Γ,Λ be a neutrosophic cubic graph for some graph G=V,E. The density of LCG is defined as
(54)℘LCG=℘TlLCG,℘TrLCG,℘TLCG,℘IlLCG,℘IrLCG,℘ILCG,℘FlLCG,℘FrLCG,℘FLCG,where
(55)℘TlLCG=2∑g1,g2∈VβTlg1g2∑g1,g2∈VminαTlg1,αTlg2,℘TrLCG=2∑g1,g2∈VβTrg1g2∑g1,g2∈VminαTrg1,αTrg2,℘TLCG=2∑g1,g2∈VβTg1g2∑g1,g2∈VminαTg1,αTg2,℘IlLCG=2∑g1,g2∈VβIlg1g2∑g1,g2∈VminαIlg1,αIlg2,℘IrLCG=2∑g1,g2∈VβIrg1g2∑g1,g2∈VminαIrg1,αIrg2,℘ILCG=2∑g1,g2∈VβIg1g2∑g1,g2∈VminαIg1,αIg2,℘FlLCG=2∑g1,g2∈VβFlg1g2∑g1,g2∈VminαFlg1,αFlg2,℘FrLCG=2∑g1,g2∈VβFrg1g2∑g1,g2∈VminαFrg1,αFrg2,℘FLCG=2∑g1,g2∈VβTlg1g2∑g1,g2∈VminαFlg1,αFlg2.Example 17.
ConsiderLCG=Γ,Λ for a graph G=A,B, with vertex set A=v1,v2,v3 and edge set B=v1v2,v2v3,v1v3. Also, let Γ and Λ be neutrosophic membership functions for vertices and edges, respectively, shown in Tables 4 and 5.Figure 4
Represents a complete neutrosophic cubic graph.Table 4
Neutrosophic membership functions for vertices.
Γ=AαTl,αTrαTαIl,αIrαIαFl,αFrαFv1.1,.5.6.5,.71.6,.7.9v2.2,.6.5.6,.8.8.7,.8.8v3.3,.7.8.7,.8.7.5,.6.7Table 5
Neutrosophic membership functions for edges.
Λ=BβTl,βTrβTβIl,βIrβIβFl,βFrβFv1v2.1,.2.5.4,.5.4.5,.6.2v2v3.2,.4.2.1,.3.5.4,.5.3v1v3.1,.2.5.1,.3.6.4,.6.2Since(56)2∑x,y∈AΨxy=2Ψv1v2+Ψv2v3+Ψv1v3=2∑x,y∈AβTl,βTrxy,βTxy,βIl,βIrxy,βIxy,βFl,βFrxy,βFxy=2.4,.8,1.2,.6,1.1,1.5,1.3,1.7,.7=.8,1.6,2.4,1.2,2.2,3,2.6,3.4,1.4.Also,(57)minΦv1,Φv2=.1,0.5,.5,.5,.7,.8,.6,.7,.8,minΦv2,Φv3=.2,.6,.5,.6,.8,.7,.5,.6,.7,minΦv1,Φv3=.1,.5,.6,.5,0.7,.7,.5,.6,.7.So,(58)∑x,y∈AminΦx,Φy=.4,1.6,1.6,1.6,2.2,2.2,1.6,1.9,2.2.Hence,℘LCG is given by
(59)℘LCG=2,1,1.6,.75,1,1.36,1.625,1.78,.63.


## 3.4. Balanced and Strictly Balanced Neutrosophic Cubic Graphs

In this subsection, we use the density function℘ to discuss the idea of balanced and strictly balanced neutrosophic cubic graph LCG.Definition 18.
LCG is balanced if ℘H≤℘LCG for all subgraphs H of LCG.Definition 19.
LCG is strictly balanced if ℘H=℘LCG for all nonempty subgraphs H of LCG.Example 20.
ConsiderLCG as given in Example 17. Let H1=a,b,H2=a,c,H3=b,c. Then,
(60)℘H1=2ΨabminΦa,Φb=2.1,.2,.5,.4,.5,.4,.5,.6,.2.1,.5,.5,.5,.7,.8,.6,.7,.8=.2,.4,1,.8,1,.8,1,1.2,.4.1,.5,.5,.5,.7,.8,.6,0.7,.8=2,.8,2,1.6,1.4,1,1.66,1.7,.5,℘H2=2ΨacminΦa,Φc=2.1,.2,.5,.1,.3,.6,.4,.6,.2.1,.5,.6,.5,.7,.7,.5,.6,.7=.2,.4,1,.2,.6,1.2,.8,1.2,.4.1,.5,.6,.5,.7,.7,.5,.6,.7=2,1.33,.4,.33,.75,1.42,1.6,1.66,.85;also,
(61)℘H3=2ΨbcminΦb,Φc=2.2,.4,.2,.1,.3,.5,.4,.5,.3.2,.6,.5,.6,.8,.7,.5,.6,.7=.4,.8,.2,.2,.6,1,.8,1,.6.2,.6,.5,.6,.8,.7,.5,.6,.7=2,1.33,.4,.33,.75,1.42,1.6,1.66,.85.
Hence,(62)℘H1=2,.8,2,1.6,1.4,1,1.66,1.7,.5,℘H2=2,.8,1.66,.4,.85,1.7,1.6,2,.57,℘H3=2,1.33,.4,.33,.75,1.42,1.6,1.66,.85,as ℘H≨℘LCG, for all nonempty subgraphs H of LCG. Hence, LCG is not balanced.Remark 21.
All regular neutrosophic cubic graphs are not necessarily balanced.


## 3.5. Irregular and Totally Irregular Neutrosophic Cubic Graphs

In this subsection, we use the neighborhood degrees to discuss the idea of irregular and totally irregular neutrosophic cubic graphLCG.Definition 22.
If there is at least one vertex inLCG adjacent to vertices having different open neighborhood degrees, then LCG is called irregular, i.e., if θℕncgv≠n for all v∈A.Example 23.
ConsiderLCG=Γ,Λ for some graph G=A,B, with A=v1,v2,v3,v4 and
(63)B=v1v2,v2v3,v3v4,v2v4,and let
(64)Γ=v1,.2,.5,.4,.1,.3,.6,.6,.7,.8,v2,.3,.4,.3,.2,.5,.5,.6,.8,.7,v3,.1,.3,.5,.3,.4,.2,.5,.7,.9,v4,.2,.4,.2,.4,.5,.3,.7,.8,.6,(65)Λ=v1v2,.1,.4,.4,.1,.5,.6,.6,.7,.8,v2v3,.1,.3,.3,.2,.4,.5,.6,.8,.9,v3v4,.1,.3,.5,.2,.4,.4,.5,.7,.9,v2v4,.1,.4,.3,.2,.4,.5,.5,.7,1,then
(66)θℕncgv1=Φv2=.3,.4,.3,.2,.5,.5,.6,.8,.7,θℕncgv2=Φv3+Φv4=.1,.3,.5,.3,.4,.2,.5,.7,.9+.2,.4,.2,.4,.5,.3,.7,.8,.6=.3,.7,.7,.7,.9,.5,1.2,1.5,1.5,θℕncgv3=Φv2+Φv4=.3,.4,.3,.2,.5,.5,.6,.8,.7+.2,.4,.2,.4,.5,.3,.7,.8,.6=.5,.8,.5,.6,1,.8,1.3,1.6,1.3;hence,
(67)θℕncgv1≠θℕncgv2≠θℕncgv3≠θℕncgv4.Hence,LCG is irregular as shown in Figure 5.Definition 24.
A connectedLCG is totally irregular, if at least one vertex is adjacent to the vertices having different closed neighborhood degrees.Example 25.
ConsiderLCG=Γ,Λ for G=A,B, with
(68)A=v1,v2,v3,v4,v5,B=v1v2,v2v3,v2v4,v3v4,v1v3,v1v4,v4v5,
and let(69)Γ=v1,.3,.5,.4,.2,.3,.1,.3,.6,.4,v2,.2,.4,.3,.3,.4,.2,.6,.7,.5,v3,.2,.6,.5,.2,.5,.4,.5,.7,.4,v4,.3,.5,.4,.3,.6,.5,.4,.6,.3,v5,.2,.2,.1,.2,.3,.4,.3,.5,.2,(70)Λ=v1v2,.2,.4,.3,.2,.3,.1,.3,.5,.3,v2v3,.2,.4,.3,.2,.4,.2,.5,.6,.2,v2v4,.1,.4,.2,.3,.4,.2,.3,.5,.2,v3v4,.2,.5,.3,.2,.5,.3,.4,.5,.3,v3v1,.2,.4,.1,.2,.3,.1,.3,.6,.4,v1v4,.2,.3,.3,.2,.3,.1,.3,.5,.3,v4v5,.1,.2,.1,.2,.3,.3,.3,.4,.2.
Then,(71)θℕncgv1=Φv2+Φv3+Φv4+Φv1=.2,.4,.3,.3,.4,.2,.6,.7,.5+.2,.6,.5,.2,.5,.4,.5,.7,.4+.3,.5,.4,.3,.6,.5,.4,.6,.3+.3,.5,.4,.2,.3,.1,.3,.6,.4=1,2,1.6,1,1.8,1.2,1.8,2.6,1.6,θℕncgv2=Φv1+Φv4+Φv3+Φv2=1,2,1.6,1,1.8,1.2,1.8,2.6,1.6,θℕncgv3=Φv2+Φv1+Φv4+Φv3=1,2,1.6,1,1.8,1.2,1.8,2.6,1.6,θℕncgv4=Φv1+Φv2+Φv3+Φv5+Φv4=.3,.5,.4,.2,.3,.1,.3,.6,.4+.2,.4,.3,.3,.4,.2,.6,.7,.5+.2,.6,.5,.2,.5,.4,.5,.7,.4+.2,.2,.1,.2,.3,.4,.3,.5,.2+.3,.5,.4,.3,.6,.5,.4,.6,.3=1.2,2.2,1.7,1.2,2.1,1.6,2.1,3.1,1.8,θℕncgv5=Φℕncgv4=.3,.5,.4,.3,.6,.5,.4,.6,.3.Figure 5
Represents irregular neutrosophic cubic graph.Clearly,LCG is totally irregular as in Figure 6.Figure 6
Represents totally irregular neutrosophic cubic graph.


## 3.6. Complement of a Neutrosophic Cubic Graph

Complement of a neutrosophic cubic graph is a very important concept we discuss here.Definition 26.
The complement ofLCG=Γ,Λ is a neutrosophic cubic graph LCG¯=Γ¯,Λ¯, where
(72)Γ¯=Φ¯g=g,α¯Tl,α¯Trg,α¯Tg,α¯Il,α¯Ir,α¯Ig,α¯Fl,α¯Frg,α¯Fgg∈A,Λ¯=Ψ¯g1g2=g1g2,β¯Tl,β¯Trg1g2,β¯Tg1g2,β¯Il,β¯Irg1g2,β¯Ig1g2,β¯Fl,β¯Frg1g2,β¯Fg1g2g1g2∈B,since
(73)Ψ¯xy=minΦx,Φy−Ψxy,or for truth membership functions, we have
(74)β¯Tlxy=minαTlx,αTly−βTlxy,β¯Txy=minαTx,αTy−βTxy,β¯Trxy=minαTrx,αTry−βTrxy;
similarly, for indeterminate membership functions, we have(75)β¯Ilxy=minαIlx,αIly−βIlxy,β¯Irxy=minαIrx,αIry−βIrxy,β¯Ixy=minαIx,αIy−βIxy;also, for falsity membership functions, we have similar results.Proposition 27.
For self-complementaryLCG=Γ,Λ, we have
(76)℘LCG=1,1,1,1,1,1,1,1,1.Proof.
GivenLCG is self-complementary, so Ψ¯xy=Ψxy; also, by definition of a self-complementary neutrosophic cubic graph, we have
(77)Ψ¯xy=minΦx,Φy−Ψxy.
Dividing both sides of equation (77) by minΦx,Φy,
(78)ΨxyminΦx,Φy=1,1,1,1,1,1,1,1,1−ΨxyminΦx,Φy,we get
(79)∑x,y∈AΨxyminΦx,Φy=1,1,1,1,1,1,1,1,1−∑x,y∈AΨxyminΦx,Φy.
Hence,(80)2∑x,y∈AΨxyminΦx,Φy=1,1,1,1,1,1,1,1,1,so
(81)℘LCS=1,1,1,1,1,1,1,1,1.Proposition 28.
LetLCG=Γ,Λ be strictly balanced, let LCG¯ be its complement, then ℘LCG+℘LCG¯=2,2,2,2,2,2,2,2,2.Proof.
LetLCG be strictly balanced; let LCG¯ be its complement. Let H be a nonempty subgraph of LCG. Since LCG is strictly balanced, ℘LCG=℘H for every subset H⊆LCG and for any x,y∈A. In LCG¯, we have Ψ¯xy=minΦx,Φy−Ψxy, and so, for truth membership functions, we have
(82)β¯Tlxy=minαTlx,αTly−βTlxy,forallx,y∈A.
Dividing equation (82) by minαTlx,αTly, we get
(83)β¯TlxyminαTlx,αTly=1−βTlxyminαTlx,αTly.
Hence,(84)∑x,y∈Aβ¯TlxyminαTlx,αTly=1−∑x,y∈EβTlxyminαTlx,αTly.
Multiplying both sides by 2,we get(85)2∑x,y∈Aβ¯TlxyminαTlx,αTly=2−2∑x,y∈EβTlxyminαTlx,αTly.
Hence,℘TlLCG¯=2−℘TlLCG. Similarly for right end point of interval in truth valued membership functions, we have ℘TrLCG¯=2−℘TrLCG. Similar results hold for the rest of membership functions. Hence,
(86)℘ℓCG¯=2,2,2,2,2,2,2,2,2−℘ℓCG,℘ℓCG¯+℘ℓCG=2,2,2,2,2,2,2,2,2.
This completes the proof.


## 3.7. Neighborly Irregular and Neighborly Totally Irregular Neutrosophic Cubic Graphs

In this subsection, we use the neighborhood degrees to discuss the idea of neighborly irregular and neighborly totally irregular neutrosophic cubic graphs.Definition 29.
A connectedLCG is neighborly irregular if every two adjacent vertices in LCG have different closed neighborhood degrees.Example 30.
ConsiderLCG=Γ,Λ for any G=A,B with A=a,b,c,d and B=ab,bc,cd,da and let neutrosophic cubic membership functions for vertices and edges be represented in Tables 6 and 7, respectively.Table 6
Neutrosophic membership functions for vertices.
Γ=AαTl,αTrαTαIl,αIrαIαFl,αFrαFa.2,.6.3.1,.3.2.5,.7.5b.3,.7.2.2,.4.3.4,.5.6c.4,.8.3.3,.5.4.6,.7.4d.5,.7.4.1,.2.3.7,.8.7Table 7
Neutrosophic membership functions for edges.
Λ=BβTl,βTrβTβIl,βIrβIβFl,βFrβFab.1,.4.2.1,.2.1.3,.5.5bc.2,.5.1.2,.3.2.4,.5.3cd.3,.6.3.1,.2.3.5,.6.4da.2,.5.2.1,.2.2.4,.5.5Then, clearlyLCG is neighborly irregular, as shown in Figure 7.Definition 31.
LCG is said to be neighborly totally irregular if every two adjacent vertices of LCG have different closed neighborhood degrees.Example 32.
ConsiderLCG=Γ,Λ for any G=A,B with A=v1,v2,v3,v4 and B=v1v2,v2v3,v3v4,v1v4. Also, let
(87)Γ=v1,.2,.6,.4,.2,.4,.5,.4,.6,.5,v2,.3,.5,.3,.4,.5,.5,.5,.7,.4,v3,.4,.7,.5,.3,.5,.2,.3,.6,.1,v4,.5,.6,.2,.2,.5,.3,.6,.8,.2,(88)Λ=v1v2,.2,.5,.4,.2,.4,.4,.5,.7,.4,v2v3,.3,.5,.5,.3,.4,.4,.5,.6,.1,v3v4,.2,.6,.5,.2,.5,.3,.5,.7,.1,v1v4,.1,.4,.3,.2,.4,.5,.5,.7,.2.Figure 7
Represents neighborly irregular neutrosophic cubic graph.Then, clearlyLCG is neighborly totally irregular, as shown in Figure 8.Figure 8
Represents neighborly totally irregular neutrosophic cubic graph.


## 3.8. Highly Irregular Neutrosophic Cubic Graphs

In this subsection, we use the neighborhood degrees to discuss the idea of highly irregular neutrosophic cubic graphs.Definition 33.
Consider a connectedLCG; then, LCG is highly irregular if every vertex in LCG is adjacent to vertices with different neighborhood degrees.Example 34.
ConsiderLCG=Γ,Λ for a graph G=A,B with A=v1,v2,v3,v4,v5,v6 and B=v1v2,v2v3,v2v6,v3v4,v3v5,v4v5,v1v5 and let
(89)Γ=v1,.2,.5,.3,.1,.3,.4,.4,.5,.3,v2,.1,.4,.2,.2,.4,.2,.3,.6,.4,v3,.3,.5,.6,.2,.5,.3,.5,.7,.4,v4,.4,.6,.3,.3,.6,.5,.4,.6,.3,v5,.2,.4,.5,.2,.3,.4,.2,.5,.2,v6,.1,.4,.3,.2,.4,.2,.3,.5,.2.
By routine computations, we have(90)θℕncgv1=Φv2+Φv5=.1,.4,.2,.2,.4,.2,.3,.6,.4+.2,.4,.5,.2,.3,.4,.2,.5,.2=.3,.8,.7,.4,.7,.6,.5,1.1,.6,θℕncgv2=Φv1+Φv6+Φv3=.2,.5,.3,.1,.3,.4,.4,.5,.3+.1,.4,.3,.2,.4,.2,.3,.5,.2+.3,.5,.6,.2,.5,.3,.5,.7,.4=.6,1.4,1.2,.5,1.2%,.9,1.2,1.7,.9,θℕncgv4=Φv3+Φv5=.3,.5,.6,.2,.5,.3,.5,.7,.4+.2,.4,.5,.2,.3,.4,.2,.5,.2=.5,.9,1.1,.4,.8,.7,.7,1.2,.6,θℕncgv5=Φv3+Φv4+Φv1=.3,.5,.6,.2,.5,.3,.5,.7,.4+.4,.6,.3,.3,.6,.5,.4,.6,.3+.2,.5,.3,.1,.3,.4,.4,.5,.3=.9,1.6,1.2,.6,1.4%,1.2,1.3,1.8,1,θℕncgv6=Φv2.1,.4,.2,.2,.4,.2,.3,.6,.4.Clearly,LCG as shown in Figure 9 is highly irregular.Theorem 35.
LCG is highly irregular and neighborly irregular if and only if open neighborhood degrees for all vertices of LCG are different.Proof.
SupposeLCG has n vertices v1,v2,⋯,vn. Also, let LCG be highly irregular and neighborly irregular.
Claim 1. The open neighborhood degrees for all vertices inLCG are different. Let
(91)θℕncgvi=λiTl,λiTr,λiT,λiIl,λiIr,λiI,λiFl,λiFr,λiF
for alli=1,2,3,⋯,n. Let the adjacent vertices of v1 be v2,v3,⋯,vn with open neighborhood degrees:
(92)θℕncgvi=λiTl,λiTr,λiT,λiIl,λiIr,λiI,λiFl,λiFr,λiFfor all i=2,3,⋯,n, respectively. Then, as LCG is highly irregular, we have
(93)λ2Tl≠λ3Tl≠⋯≠λnTl,λ2Tr≠λ3Tr≠⋯≠λnTr,λ2T≠λ3T≠⋯≠λnTfor all i=2,3,⋯,n. Similar holds for indeterminacy and falsity membership functions. Also, LCG is neighborly irregular, so we have
(94)λ1Tl≠λ2Tl≠,⋯,≠λnTl,λ1Tr≠λ2Tr≠⋯≠λnTr,λ1T≠λ2T≠⋯≠λnT.Similar holds for indeterminacy and falsity membership functions. Hence, open neighborhood degrees of all vertices of LCG are different.
Claim 2.LCG is highly irregular and neighborly irregular. Let
(95)θℕncgvi=λiTl,λiTr,λiT,λiIl,λiIr,λiI,λiFl,λiFr,λiFfor all i=1,2,3,⋯,n are degrees for all vertices of LCG. Given open neighborhood degrees of all vertices of LCG are different, so
(96)θℕncgv1≠θℕncgv2≠⋯≠θℕncgvn,λ1Tl≠λ2Tl≠⋯≠λnTl,λ1Tr≠λ2Tr≠⋯≠λnTr,λ1T≠λ2T≠⋯≠λnTfor all i=1,2,3,⋯,n. Similar holds for indeterminacy and falsity membership functions. So, any two vertices in LCG have different open neighborhood degrees, and for every vertex, adjacent vertices have different open neighborhood degrees, which proves the result.Remark 36.
A completeLCG may not be neighborly irregular.Example 37.
LetLCG=Γ,Λ for any G=A,B, with A=v1,v2,v3 and B=v1v2,v2v3,v1v3 such that
(97)Γ=v1,.4,.8,.4,.3,.5%,.4,.6,.7,.3,v2,.2,.4,.2,.4,.7%,.5,.5,.8,.2,v3,.2,.4,.2,.4,.7%,.5,.5,.8,.2,(98)Λ=v1v2,.2,.4,.2,.3,.5%,.4,.5,.7,.2,v2v3,.2,.4,.2,.4,.7%,.5,.5,.8,.2,v1v3,.2,.4,.2,.3,.5%,.4,.5,.7,.2.
By simple computation, we get(99)θℕncgv1=.4,.8,.4,.8,1.4,1,1,1.6,.4,θℕncgv2=.6,1.2,.6,.7,1.2,.9,1.1,1.5,.5,θℕncgv3=.6,1.2,.6,.7,1.2,.9,1.1,1.5,.5.Figure 9
Represents highly irregular neutrosophic cubic graph.Here,θℕncgv2=θℕncgv3, so the neighborhood degree is not different. Hence, LCG is not neighborly irregular; also, we have similar holds for all vertices and edges. So, LCG is complete. Hence, a complete LCG may not be neighborly irregular as shown in Figure 10.Theorem 38.
IfLCG is neighborly irregular and
(100)Γ=Φx=αTl,αTrx,αTx,αIl,αIrx,αIx,αFl,αFrx,αFxfor all x∈A is a constant function, then it is neighborly totally irregular.Proof.
Assume thatLCG is a neighborly irregular. Then, open neighborhood degrees of every two adjacent vertices are different. Let vi,vj∈A be adjacent vertices with different open neighborhood degrees. Then, θℕncgvi≠θℕncgvj for all i≠j; let θℕncgvi=d1&θℕncgvj=d2 then d1≠d2. Also, as
(101)Γ=Φx=αTl,αTrx,αTx,αIl,αIrx,αIx,αFl,αFrx,αFxis constant for all x∈A. Hence, Φvi=Φvj=k; suppose that LCG is not neighborly totally irregular, then
(102)θℕncgvi=θℕncgvj,for some i≠j but θℕncgvi=θℕncgvi+Φvi and θℕncgvj=θℕncgvj+Φvj using these values in equation (102), we get
(103)θℕncgvi+Φvi=θℕncgvj+Φvj⇒d1+k=d2+k⇒d1=d2,as cancellation law holds in 0,1, which contradicts, as
(104)d1≠d2.
Hence,(105)θℕncgvi≠θℕncgvj,so LCG is neighborly totally irregular. This proves the result.Theorem 39.
IfLCG is neighborly totally irregular and
(106)Γ=Φx=αTl,αTrx,αTx,αIl,αIrx,αIx,αFl,αFrx,αFx,x∈A,is a constant function, then it is neighborly irregular.Proof.
Assume thatLCG is a neighborly totally irregular. Then, closed neighborhood degrees of every two adjacent vertices are distinct. Let vi,vj∈A be adjacent vertices with distinct closed neighborhood degrees. Then, for all i≠j,
(107)θℕncgvi≠θℕncgvj,let
(108)θℕncgvi=f1&θℕncgvj=f2,then f1≠f2. Also, as
(109)Γ=Φx=αTl,αTrx,αTx,αIl,αIrx,αIx,αFl,αFrx,αFx;suppose that LCG is not neighborly irregular, then
(110)θℕncgvi=θℕncgvj=w;say, for some i≠j but
(111)θℕncgvi=θℕncgvi+Φvi,(112)θℕncgvj=θℕncgvj+Φvj,using these values in equation (112), we get
(113)θℕncgvi+Φvi=θℕncgvj+Φvj=w+r,so
(114)θℕncgvi=θℕncgvj,for some i≠j which is a contradiction to the fact that LCG is a neighborly totally irregular neutrosophic cubic graph. Hence,
(115)θℕncgvi≠θℕncgvj,so LCG is neighborly irregular. This proves the result.Proposition 40.
IfLCG is neighborly irregular as well as neighborly totally irregular, then
(116)Γ=Φx=αTl,αTrx,αTx,αIl,αIrx,αIx,αFl,αFrx,αFxneed not be a constant function.Remark 41.
IfLCG is neighborly irregular, then a neutrosophic cubic subgraph H of LCG may not be neighborly irregular.Remark 42.
IfLCG is neighborly totally irregular, then a neutrosophic cubic subgraph H of LCG may not be neighborly totally irregular.Figure 10
Represents a complete neutrosophic cubic graph, but not neighborly irregular.


## 4. Applications

As neutrosophic cubic graph theory is a developing field of modern mathematics, it has many applications in different fields. In this section, we discuss applications of neutrosophic cubic graphs in finding the effects of different factors in the neighboring countries of Pakistan. Further, we used our proposed model in decision-making while selecting a house in a certain locality.We will use the following proposed algorithm in the following real-life problems.Step 1.
Calculate the memberships, indetermined-memberships and falsity membership for corresponding vertex in vertex setA in interval form as well as in the ordinary fuzzy set.Step 2.
Calculate the neutrosophic cubic open neighborhood degree of a vertex.Step 3.
Calculate the neutrosophic cubic closed neighborhood degree of the same vertex.Step 4.
Comparison between degrees provided in Steps2 and 3.The frame diagram to clarify the organization of the proposed method is given in Figure11.Figure 11
Frame diagram of the proposed method.


### 4.1. Effects of Different Factors on the Neighboring Countries of Pakistan

Suppose we are interested to check the effects (e.g., time/durations/situations) on different factors in the neighboring countries of Pakistan. These factors may be the population, literacy, health conditions, etc., of these countries. So, we take Pakistan and its neighboring countries as a set of vertices and link between these countries through roads as our edge set. Hence, graphG=A,B has set of vertices A=Pak,Ir,In,Ch,Af, where Pak stands for Pakistan, Ir for Iran, In for India, Ch for China, and Af for Afghanistan. Let the set of edges be E=A network of roads between these countries, so we can define membership function for each vertex v∈A to denote strength or degree of these vertices as Φv=Pop,PCI,LR,WTRU,HE,PSI; here, we have three different categories/situations/time/duration say past, future, and present. Also, here, interval membership represents past and future for truth and indeterminate membership, respectively, and present time represents falsity memberships.
(i)
Pop represents interval membership for the population of a country in the duration (1st July 2018, 1st July 2019)/max population of the corresponding country in the same duration. Here, interval represents past and future to represent truth membership and indetermined-membership for members of the vertex set A(ii)
PCI is for per capita income of a country which represents falsity membership for the corresponding vertex in vertex setA(iii)
LR represents interval membership for literacy rate of a country in the duration [2011, 2014]. Here again, we have interval to represent past and future for truth membership and indetermined-memberships for members of vertex setA(iv)
WTRU is the position of the corresponding country in the world’s top-ranking universities/max number of universities in these countries(v)
HE represents interval membership for %age of health expenditure of a country in the duration [2010, 2015] where interval shows past and future to represent truth membership and indetermined-membership for members of vertex setA(vi)
PSI is the number of popular sports interest/max number of sports played in the corresponding country as health depends on sportsBy data collection for different time intervals, we have neutrosophic cubic membership for vertex setA represented in Table 8.Table 8
A neutrosophic cubic representation for vertex setA.
APop [past, future]PCI (present)LR [past, future]WTRU (present)HE [past, future]PSI (present)Af.026,.026.06.32,.38.01.92,.82.13Ch[1, 1]1.92,.961.49,.55.67In.94,.95.212.69,.72.21.43,.471Ir.057,.058.57.86,.95.123.81,.69.2Pak.148,.151.162.55,.57.06.31,.26.33The neutrosophic cubic open neighborhood degree of a vertex (say Pak) is(117)NPak=ΦAf+ΦCh+ΦIn+ΦIr=2.0303,2.0373,1.841,2.785,3.015,1.343,2.639,2.531,2;also, the neutrosophic cubic closed neighborhood degree for the same vertex is defined as
(118)NPak=ΦPak+ΦAf+ΦCh+ΦIn+ΦIr=2.1789,2.1883,2.003,3.332,3.585,1.403,2.941,2.792,2.33.The neutrosophic cubic open neighborhood degree of a vertex (say Pak) is less than the neutrosophic cubic closed neighborhood degree of a vertex (say, Pak). Thus, we may conclude that the vertex (say, Pak) has more closed neighborhoods than the open neighborhoods that can change their loyalties according to time as shown in Figure12. Similarly, we may check other countries.Figure 12
Represents neighborhood of Pakistan.


### 4.2. Decision-Making while Selecting a House

Suppose we are interested to purchase a house in a housing society. Then, we have to consider certain features before making our final decision like availability of mosque, workplace, school, college, university, clinic/hospital, market, park, and gym, width/condition of roads, and the distance of the house and all these facilities. We also keep in view past, future, and present situations of all these attributes, or we keep in view trends and demands and check effects of duration on these areas. So, we take a survey of different areas in a locality and take a set of different houses with different features as our set of vertices and link or distance between these as our edge set. Leth1,h2,h3,h4 be different choices of houses, and we define neutrosophic cubic membership function of a house h∈V as
(119)M=Φh=school,university,mosque,workplace,hospital,gym,shops,market,park;here, interval membership represents past and future for truth and indeterminate membership, respectively, and present time represents falsity memberships, and
(120)N=Ψh1h2=distancebetweenthesehouses.Let(121)Φh1=.6,.7,1,.7,.2,.3,.8,1,1,Φh2=.2,.3,.2,.5,.6,.3,.4,.7,.9,Φh3=.5,.6,.5,.3,.9,.4,0,.2,.7,Φh4=.1,.5,.4,.6,.8,.1,.6,.5,.4.Then, the neutrosophic cubic open neighborhood degree of each vertex (house) is given for househ1; we have
(122)θℕncgh1=Φh2+Φh3+Φh4=.2,.3,.2,.5,.6,.3,.4,.7,.9+.5,.6,.5,.3,.9,.4,,.2,.7+.1,.5,0.4,.6,.8,.1,.6,.5,.4=.8,1.4,1.1,1.5,2.3,.8,1.0,1.4,2.0;similarly, for house h2, we have
(123)θℕncgh2=Φh1+Φh4=.6,.7,1,.7,.2,.3,.8,1,1+.1,.5,.4,.6,.8,.1,.6,.5,.4=.7,1.2,1.4,1.3,1.0,.4,1.4,1.5,1.4,and for house h3, we have
(124)θℕncgh3=Φh1+Φh4=.6,.7,1,.7,.2,.3,.8,1,1+.1,.5,.4,.6,.8,.1,.6,.5,.4=.7,1.2,1.4,1.3,1.0,.4,1.4,1.5,1.4;also, for house h4, we have
(125)θℕncgh4=Φh3+Φh1+Φh2=.5,.6,.5,.3,.9,.4,0,.2,.7+.6,.7,1,.7,.2,.3,.8,1,1+.2,.3,.2,.5,.6,.3,.4,.7,.9=1.3,1.6,1.7,1.5,1.7,1.0,1.4,1.9,2.6.Also, the neutrosophic cubic closed neighborhood degree of each vertex (house) is given for househ1; we have
(126)θℕncgh1=Φh1+Φh2+Φh3+Φh4=.6,.7,1,.7,.2,.3,.8,1,1+.2,.3,.2,.5,.6,.3,.4,.7,.9+.5,.6,.5,.3,.9,.4,0,.2,.7+.1,.5,.4,.6,.8,.1,.6,0.5,.4=1.4,2.1,2.1,2.1,2.5,1.1,1.8,2.4,3.0,and for househ2, we have
(127)θℕncgh2=Φh2+Φh1+Φh4=.2,.3,.2,.5,.6,.3,.4,.7,.9+.6,.7,1,.7,.2,.3,.8,1,1+.1,.5,.4,.6,.8,.1,.6,.5,.4=.9,1.5,1.6,1.8,1.6,.7,1.8,2.2,2.3,and for househ3, we have
(128)θℕncgh3=Φh1+Φh3+Φh4=.6,.7,1,.7,.2,.3,.8,1,1+.5,.6,.5,.3,.9,.4,0,.2,.7+.1,.5,.4,.6,.8,.1,.6,.5,.4=1.2,1.8,1.9,1.6,1.9,.8,1.4,1.7,2.1.Also, for househ4, we have
(129)θℕncgh4=Φh1+Φh2+Φh3+Φh4=.6,.7,1,.7,.2,.3,.8,1,1+.2,.3,.2,.5,.6,.3,.4,.7,.9+.5,.6,.5,.3,.9,.4,0,.2,.7+.1,.5,0.4,.6,.8,.1,.6,.5,.4=1.4,2.1,2.1,2.1,2.5,1.1,1.8,2.4,3.0.We compare neutrosophic cubic open neighborhood and observe that neutrosophic cubic open neighborhood ofh2 and h3 is the same, but comparison of neutrosophic cubic open neighborhood of h1andh4 shows that h1andh4 are more effective than h2andh3. Also, we observe that neutrosophic cubic open neighborhood of h4 is effective than h1. Moreover, one can observe that the neutrosophic cubic closed neighborhood degree of houses h1andh4 is the same, but in view of neutrosophic cubic open neighborhood, h4 is the best choice in all respects as compared to other houses h2andh3. So, choice of house h4 is the best choice for our selection of an ideal house. Position of four houses with different facilities is shown in Figure 13.Figure 13
Represents position of four houses with different facilities.


## 4.1. Effects of Different Factors on the Neighboring Countries of Pakistan

Suppose we are interested to check the effects (e.g., time/durations/situations) on different factors in the neighboring countries of Pakistan. These factors may be the population, literacy, health conditions, etc., of these countries. So, we take Pakistan and its neighboring countries as a set of vertices and link between these countries through roads as our edge set. Hence, graphG=A,B has set of vertices A=Pak,Ir,In,Ch,Af, where Pak stands for Pakistan, Ir for Iran, In for India, Ch for China, and Af for Afghanistan. Let the set of edges be E=A network of roads between these countries, so we can define membership function for each vertex v∈A to denote strength or degree of these vertices as Φv=Pop,PCI,LR,WTRU,HE,PSI; here, we have three different categories/situations/time/duration say past, future, and present. Also, here, interval membership represents past and future for truth and indeterminate membership, respectively, and present time represents falsity memberships.
(i)
Pop represents interval membership for the population of a country in the duration (1st July 2018, 1st July 2019)/max population of the corresponding country in the same duration. Here, interval represents past and future to represent truth membership and indetermined-membership for members of the vertex set A(ii)
PCI is for per capita income of a country which represents falsity membership for the corresponding vertex in vertex setA(iii)
LR represents interval membership for literacy rate of a country in the duration [2011, 2014]. Here again, we have interval to represent past and future for truth membership and indetermined-memberships for members of vertex setA(iv)
WTRU is the position of the corresponding country in the world’s top-ranking universities/max number of universities in these countries(v)
HE represents interval membership for %age of health expenditure of a country in the duration [2010, 2015] where interval shows past and future to represent truth membership and indetermined-membership for members of vertex setA(vi)
PSI is the number of popular sports interest/max number of sports played in the corresponding country as health depends on sportsBy data collection for different time intervals, we have neutrosophic cubic membership for vertex setA represented in Table 8.Table 8
A neutrosophic cubic representation for vertex setA.
APop [past, future]PCI (present)LR [past, future]WTRU (present)HE [past, future]PSI (present)Af.026,.026.06.32,.38.01.92,.82.13Ch[1, 1]1.92,.961.49,.55.67In.94,.95.212.69,.72.21.43,.471Ir.057,.058.57.86,.95.123.81,.69.2Pak.148,.151.162.55,.57.06.31,.26.33The neutrosophic cubic open neighborhood degree of a vertex (say Pak) is(117)NPak=ΦAf+ΦCh+ΦIn+ΦIr=2.0303,2.0373,1.841,2.785,3.015,1.343,2.639,2.531,2;also, the neutrosophic cubic closed neighborhood degree for the same vertex is defined as
(118)NPak=ΦPak+ΦAf+ΦCh+ΦIn+ΦIr=2.1789,2.1883,2.003,3.332,3.585,1.403,2.941,2.792,2.33.The neutrosophic cubic open neighborhood degree of a vertex (say Pak) is less than the neutrosophic cubic closed neighborhood degree of a vertex (say, Pak). Thus, we may conclude that the vertex (say, Pak) has more closed neighborhoods than the open neighborhoods that can change their loyalties according to time as shown in Figure12. Similarly, we may check other countries.Figure 12
Represents neighborhood of Pakistan.


## 4.2. Decision-Making while Selecting a House

Suppose we are interested to purchase a house in a housing society. Then, we have to consider certain features before making our final decision like availability of mosque, workplace, school, college, university, clinic/hospital, market, park, and gym, width/condition of roads, and the distance of the house and all these facilities. We also keep in view past, future, and present situations of all these attributes, or we keep in view trends and demands and check effects of duration on these areas. So, we take a survey of different areas in a locality and take a set of different houses with different features as our set of vertices and link or distance between these as our edge set. Leth1,h2,h3,h4 be different choices of houses, and we define neutrosophic cubic membership function of a house h∈V as
(119)M=Φh=school,university,mosque,workplace,hospital,gym,shops,market,park;here, interval membership represents past and future for truth and indeterminate membership, respectively, and present time represents falsity memberships, and
(120)N=Ψh1h2=distancebetweenthesehouses.Let(121)Φh1=.6,.7,1,.7,.2,.3,.8,1,1,Φh2=.2,.3,.2,.5,.6,.3,.4,.7,.9,Φh3=.5,.6,.5,.3,.9,.4,0,.2,.7,Φh4=.1,.5,.4,.6,.8,.1,.6,.5,.4.Then, the neutrosophic cubic open neighborhood degree of each vertex (house) is given for househ1; we have
(122)θℕncgh1=Φh2+Φh3+Φh4=.2,.3,.2,.5,.6,.3,.4,.7,.9+.5,.6,.5,.3,.9,.4,,.2,.7+.1,.5,0.4,.6,.8,.1,.6,.5,.4=.8,1.4,1.1,1.5,2.3,.8,1.0,1.4,2.0;similarly, for house h2, we have
(123)θℕncgh2=Φh1+Φh4=.6,.7,1,.7,.2,.3,.8,1,1+.1,.5,.4,.6,.8,.1,.6,.5,.4=.7,1.2,1.4,1.3,1.0,.4,1.4,1.5,1.4,and for house h3, we have
(124)θℕncgh3=Φh1+Φh4=.6,.7,1,.7,.2,.3,.8,1,1+.1,.5,.4,.6,.8,.1,.6,.5,.4=.7,1.2,1.4,1.3,1.0,.4,1.4,1.5,1.4;also, for house h4, we have
(125)θℕncgh4=Φh3+Φh1+Φh2=.5,.6,.5,.3,.9,.4,0,.2,.7+.6,.7,1,.7,.2,.3,.8,1,1+.2,.3,.2,.5,.6,.3,.4,.7,.9=1.3,1.6,1.7,1.5,1.7,1.0,1.4,1.9,2.6.Also, the neutrosophic cubic closed neighborhood degree of each vertex (house) is given for househ1; we have
(126)θℕncgh1=Φh1+Φh2+Φh3+Φh4=.6,.7,1,.7,.2,.3,.8,1,1+.2,.3,.2,.5,.6,.3,.4,.7,.9+.5,.6,.5,.3,.9,.4,0,.2,.7+.1,.5,.4,.6,.8,.1,.6,0.5,.4=1.4,2.1,2.1,2.1,2.5,1.1,1.8,2.4,3.0,and for househ2, we have
(127)θℕncgh2=Φh2+Φh1+Φh4=.2,.3,.2,.5,.6,.3,.4,.7,.9+.6,.7,1,.7,.2,.3,.8,1,1+.1,.5,.4,.6,.8,.1,.6,.5,.4=.9,1.5,1.6,1.8,1.6,.7,1.8,2.2,2.3,and for househ3, we have
(128)θℕncgh3=Φh1+Φh3+Φh4=.6,.7,1,.7,.2,.3,.8,1,1+.5,.6,.5,.3,.9,.4,0,.2,.7+.1,.5,.4,.6,.8,.1,.6,.5,.4=1.2,1.8,1.9,1.6,1.9,.8,1.4,1.7,2.1.Also, for househ4, we have
(129)θℕncgh4=Φh1+Φh2+Φh3+Φh4=.6,.7,1,.7,.2,.3,.8,1,1+.2,.3,.2,.5,.6,.3,.4,.7,.9+.5,.6,.5,.3,.9,.4,0,.2,.7+.1,.5,0.4,.6,.8,.1,.6,.5,.4=1.4,2.1,2.1,2.1,2.5,1.1,1.8,2.4,3.0.We compare neutrosophic cubic open neighborhood and observe that neutrosophic cubic open neighborhood ofh2 and h3 is the same, but comparison of neutrosophic cubic open neighborhood of h1andh4 shows that h1andh4 are more effective than h2andh3. Also, we observe that neutrosophic cubic open neighborhood of h4 is effective than h1. Moreover, one can observe that the neutrosophic cubic closed neighborhood degree of houses h1andh4 is the same, but in view of neutrosophic cubic open neighborhood, h4 is the best choice in all respects as compared to other houses h2andh3. So, choice of house h4 is the best choice for our selection of an ideal house. Position of four houses with different facilities is shown in Figure 13.Figure 13
Represents position of four houses with different facilities.


## 5. Comparison Analysis

In this paper, our focus is to introduce some different types of neutrosophic cubic graphs. These include balanced, strictly balanced, complete, regular, totally regular, and irregular neutrosophic cubic graphs. In this regard, we explained the open and closed neighborhood of a vertex of the neutrosophic cubic graph and its role in the art of decision-making. Many of these graphs have already been discussed from a different perspective by the other researchers, for example, Poulik et al. [29–32], Akram [44, 45], and Gulistan et al. [65]. We have tried to discuss them concerning the neutrosophic cubic graphs. The neutrosophic cubic graphs are the generalization of different versions of the fuzzy graph which is extended to the neutrosophic cubic graph. The idea is summarized in the form of a flow chart (Figure 14).Figure 14
Flow chart of comparison analysis.This flow chart shows under certain conditions neutrosophic cubic graphs reduced to crisp graphs. So, under certain conditions, all the different types described are reduced for neutrosophic graphs, cubic graphs, intuitionistic graphs, fuzzy graphs, and crisp graphs.


## 6. Conclusion and Future Work

In this article, we provided different types of neutrosophic cubic graphs with examples and give many results which correlate with these neutrosophic cubic graphs. We used the idea of the neutrosophic cubic open neighborhood degree and neutrosophic cubic closed neighborhood degree of the same vertex in two real-life problems. We concluded the following: (1) As the neutrosophic cubic open neighborhood degree of a vertex (say Pak) is less than the neutrosophic cubic closed neighborhood degree of a vertex (say, Pak), the vertex (say, Pak) has more closed neighborhoods than the open neighborhoods. (2) Also, we observe that househ4 is the best choice for our selection of an ideal house using the idea of neutrosophic cubic open neighborhood degree and neutrosophic cubic closed neighborhood degree of the same vertex. The limitation of the presented method is the data collection which is not an easy task. In the future, we aim to make more different types of graphs such as line, planer, and directed neutrosophic cubic graphs. We are also aiming to have more real-life applications of neutrosophic cubic graphs.



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