# Some Results in Neutrosophic Cubic Graphs with an Application in

Authors: Zheng Kou; Maryam Akhoundi; M. Ghassemi; A. A. Talebi; G. Muhiuddin
Journal: Journal of Mathematics
 (2022)

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## Abstract

Neutrosophic cubic graph (NCG) belonging to FG family has good capabilities when facing
problems that cannot be expressed by FGs. When an element membership is not clear,
neutrality is a good option that can be well supported by a NCG. Hence, in this paper,
some types of edge irregular neutrosophic cubic graphs (EI-NCGs) such as neighborly edge
totally irregular (NETI), strongly edge irregular (SEI), and strongly edge totally
irregular (SETI) are introduced. A comparative study between NEI-NCGs and NETI-NCGs is
done. Finally, an application of neutrosophic cubic digraph to find the most effective
person in a school has been presented.


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## Body

## 1. Introduction

The fuzzy set theory was introduced by Zadeh [1]. It
focuses on the membership degree of an object in a particular set. Kaufmann [2] represented FGs based on Zadeh’s fuzzy relation
[3, 4].
Rosenfeld [5] described the structure of FGs obtaining
analogs of several graph theoretical concepts. Bhattacharya [6] gave some remarks on FGs. Several concepts on FGs were introduced by Mordeson
et al. [7]. The existence of a single degree for a
true membership could not resolve the ambiguity on uncertain issues, so the need for a
degree of membership was felt. Afterward, to overcome the existing ambiguities, Atanassov
[8] defined an extension of fuzzy set by introducing
non-membership function and defined intuitionistic fuzzy set (IFS). But after a while,
Atanassov and Gargov [9] developed IFS and presented
interval-valued intuitionistic fuzzy set (IVIFS). Hongmei and Lianhua [10] defined interval-valued fuzzy graph and studied its properties. Zhang
et al. [11] introduced bipolar fuzzy sets and
relations. Smarandache [12–14] gave the idea of neutrosophic sets. Kandasamy [15] defined neutrosophic graphs. Akram et al. [16–19]
studied new results in NGs. Jun et al. [20]
introduced cubic set. For more details about cubic sets and their applications in different
research areas, we refer the readers to [21–23]. Rashid et al. [24] investigated cubic graphs. Jun et al. [25, 26] gave the
idea of neutrosophic cubic set and defined different operations on it. Gulistan et al.
[27, 28]
presented complex bipolar fuzzy sets, NCGs, and some binary operations on it. Karunambigai
et al. [29] discussed edge regular-IFG. Gani and
Radha [30] studied the concept of regular fuzzy
graphs and defined degree of a vertex in FGs. Gani et al. [31] investigated the concept of IFGs, NI-FGs, and HI-FGs in 2008. Nandhini [32] described the concept of SI-FG and studied its
properties. Maheswari and Sekar defined the concepts of edge irregular-FGs and edge totally
irregular-FGs [33]. Also, they analyzed some
properties of NEI-FGs, NETI-FGs, SEI-FGs, and SETI-FGs [34, 35]. Rao et al. [36–38] studied dominating
set, equitable dominating set, valid degree, isolated vertex, and some properties of VGs
with novel application. Kou et al. [39] investigated
g-eccentric node and vague detour g-boundary nodes in VGs. Shi et al. [40, 41] introduced total
dominating set, perfect dominating set, and global dominating set in product vague graphs.
Rashmanlou et al. [42] presented some properties of
cubic graphs. Amanathulla et al. [43] studied on
distance two surjective labeling of paths and interval graphs. Bhattacharya and Pal [44] gave the fuzzy covering problem of fuzzy graphs and
its application. Borzooei et al. [45, 46] defined inverse fuzzy graphs and new results of
domination in vague graphs. Kalaiarasi et al. [47]
presented regular and irregular m-polar fuzzy graphs. Ramprasad et al. [48] investigated some properties of highly irregular,
edge regular, and totally edge regular m-polar fuzzy graphs. Poulik and Ghorai [49] defined certain indices of graphs under bipolar fuzzy
environment. Ullah et al. [50] introduced new results
on bipolar-valued hesitant fuzzy sets. Jan et al. [51] presented some root level modifications in interval valued fuzzy graphs. Broumi
et al. [52] introduced a novel system and method for
telephone network planning based on neutrosophic graph. Muhiuddin et al. [53, 54] presented
reinforcement number of a graph and new results in cubic graphs. Talebi et al. [55–57]
presented some properties of irregularity and edge irregularity on intuitionistic fuzzy
graphs and single valued neutrosophic graphs.NCGs have many applications in psychology and medical sciences and can play a significant
role in solving the vague and complex problems that exist around our lives. With the help of
this fuzzy graph, the most effective person in an organization can be determined according
to the amount of its performance in a specific period. Therefore, in this paper, some types
of EI-NCGs such as neighborly edge totally irregular (NETI)-NCGs, strongly edge irregular
(SEI)-NCGs, and strongly edge totally irregular (SETI)-NCGs are introduced. Also, we have
given some interesting results about EI-NCGs, and several examples are investigated.
Finally, an application of neutrosophic cubic digraph to find the most effective person in a
school has been presented.


## 2. Preliminaries

Definition 1.
A graphG=V,E is a mathematical model consisting of a set of nodes
V and a set of edges E, where each is an unordered pair of distinct nodes.Definition 2 (see [5]).
A FGZ=V,ν,ξ is a non-empty set V together with a pair of functions ν:V⟶0,1 and ξ:V×V⟶0,1 so that ξxy≤minνx,νy, ∀x,y∈V.All the basic notations are shown in Table1.Table 1
Some basic notations.
NotationMeaningFGFuzzy graphNCGNeutrosophic cubic graphCNCGConnected neutrosophic cubic graphI-FGIrregular fuzzy graphEIEdge irregularNINeighborly irregularNEINeighborly edge irregularNETINeighborly edge totally irregularSIStrongly irregularSEIStrongly edge irregularSETIStrongly edge totally irregularHIHighly irregularHEIHighly edge irregularHETIHighly edge totally irregularTDTotal degreeTITotal irregularVDVarious degreeAEAdjacent edgeCFConstant function


## 3. New Concepts of Edge Irregular-NCGs

Definition 3.
LetG∗:V,E be a graph. By NCG of G∗, we mean a pair G:M,N where M=A,B=T˜A,TB,I˜A,IB,F˜A,FB is the NCS representation of V and N=C,D=T˜C,TD,I˜C,ID,F˜C,FD is the NCS representation of E so that(i)
T˜Cu,v≺¯rminT˜Au,T˜Av,TDu,v≤maxTBu,TBv.(ii)
I˜Cu,v≺¯rminI˜Au,I˜Av,IDu,v≤maxIBu,IBv.(iii)
F˜Cu,v≺¯rmaxF˜Au,F˜Av,FDu,v≤minFBu,FBv.Definition 4.
LetG:M,N be a NCG on G∗:V,E. Then, the degree of a node u is defined as dGu=dT˜Au,dTBu,dI˜Au,dIBu,dF˜Au,dFBu wheredT˜Au=Σv≠uT˜Cuv, dTBu=Σv≠uTDuv.dI˜Au=Σv≠uI˜Cuv, dIBu=Σv≠uIDuv.dF˜Au=Σv≠uF˜Cuv and dFBu=Σv≠uFDuv.Definition 5.
LetG:M,N be a NCG on G∗:V,E. The TD of a node u is defined by tdGu=tdT˜Au,tdTBu,tdI˜Au,tdIBu,tdF˜Au,tdFBu wheretdT˜Au=Σv≠uT˜Cuv+T˜Au, tdTBu=Σv≠uTDuv+TBu.tdI˜Au=Σv≠uI˜Cuv+I˜Au, tdIBu=Σv≠uIDuv+IBu.tdF˜Au=Σv≠uF˜Cuv+F˜Au and tdFBu=Σv≠uFDuv+FBu.Definition 6.
LetG:M,N be a NCG on G∗:V,E. Then:(i)
G is irregular, if there is a node that is neighbor to
nodes with VDs.(ii)
G is TI, if there is a node which is neighbor to nodes
with various TDs.Definition 7.
LetG:M,N be a CNCG. Then, G is called a(i)
NI-NCG if each pair of neighbor nodes has VDs.(ii)
NTI-NCG if each pair of neighbor nodes has various TDs.(iii)
SI-NCG if each pair of nodes has VDs.(iv)
STI-NCG if each pair of nodes has various TDs.(v)
HI-NCG if each node inG is neighbor to the nodes having VDs.(vi)
HTI-NCG if each node inG is neighbor to the nodes having various TDs.Definition 8.
LetG:M,N be a NCG. The degree of an edge uv is defined as dGuv=dT˜Cuv,dTDuv,dI˜Cuv,dIDuv,dF˜Cuv,dFDuv wheredT˜Cuv=dT˜Au+dT˜Av−2T˜Cuv, dTDuv=dTBu+dTBv−2TDuv.dI˜Cuv=dI˜Au+dI˜Av−2I˜Cuv, dIDuv=dIBu+dIBv−2IDuv.dF˜Cuv=dF˜Au+dF˜Av−2F˜Cuv and dFDuv=dFBu+dFBv−2FDuv.Definition 9.
LetG:M,N be a NCG. The TD of an edge uv is defined as tdGuv=tdT˜Cuv,tdTDuv,tdI˜Cuv,tdIDuv,tdF˜Cuv,tdFDuv wheretdT˜Cuv=dT˜Au+dT˜Av−T˜Cuv=dT˜Cuv+T˜Cuv.tdTDuv=dTBu+dTBv−TDuv=dTDuv+TDuv.tdI˜Cuv=dI˜Au+dI˜Av−I˜Cuv=dI˜Cuv+I˜Cuv.tdIDuv=dIBu+dIBv−IDuv=dIDuv+IDuv.tdF˜Cuv=dF˜Au+dF˜Av−F˜Cuv=dF˜Cuv+F˜Cuv.tdFDuv=dFBu+dFBv−FDuv=dFDuv+FDuv.Definition 10.
LetG:M,N be a CNCG on G∗:V,E. Then, G is called a(1)
NEI-NCG if each pair of AEs has VDs.(2)
NETI-NCG if each pair of AEs has various TDs.Example 1.
Consider a graph which is both NEI-NCG and NETI-NCG.
ConsiderG∗:V,E where V=u,v,w,x and E=uv,vw,wx,xu are defined as(1)M=u,0.3,0.5,0.6,0.4,0.7,0.3,0.6,0.8,0.5,v,0.4,0.6,0.4,0.2,0.5,0.2,0.3,0.7,0.9,w,0.3,0.5,0.6,0.4,0.7,0.3,0.6,0.8,0.5,x,0.4,0.6,0.4,0.2,0.5,0.2,0.3,0.7,0.9,(2)N=uv,0.2,0.4,0.5,0.1,0.2,0.3,0.3,0.5,0.4,vw,0.1,0.3,0.4,0.3,0.4,0.2,0.4,0.6,0.3,wx,0.2,0.4,0.5,0.1,0.2,0.3,0.3,0.5,0.4,xu,0.1,0.3,0.4,0.3,0.4,0.2,0.4,0.6,0.3.
From Figure1,(3)dGu=dGv=dGw=dGx=0.3,0.7,0.9,0.4,0.6,0.5,0.7,1.1,0.7,(4)dGuv=dGwx=0.2,0.6,0.8,0.6,0.8,0.4,0.8,1.2,0.6,dGvw=dGxu=0.4,0.8,1.0,0.2,0.4,0.6,0.6,1.0,0.8.
Clearly,G is a NEI-NCG.(5)tdGuv=tdGwx=0.4,1.0,1.3,0.7,1.0,0.7,1.1,1.7,1.0,tdGvw=tdGxu=0.5,1.1,0.5,0.5,0.8,0.8,1.0,1.6,1.1.
So,G is a NETI-NCG.
Therefore,G is both NEI-NCG and NETI-NCG.Figure 1
NEI-NCG and NETI-NCG.Example 2.
NEI-NCG need not to be NETI-NCG.
LetG be a NCG and G∗ be a star that includes four nodes where
V=u,v,w,x and E=ux,vx,wx are defined as(6)M=u,0.3,0.4,0.4,0.2,0.3,0.3,0.4,0.5,0.6,v,0.5,0.6,0.6,0.4,0.5,0.5,0.6,0.7,0.8,w,0.4,0.5,0.5,0.5,0.4,0.9,0.7,0.8,0.7,x,0.2,0.3,0.3,0.4,0.5,0.7,0.3,0.4,0.9,(7)N=ux,0.2,0.3,0.3,0.1,0.2,0.2,0.4,0.5,0.6,vx,0.1,0.2,0.2,0.4,0.5,0.5,0.6,0.7,0.8,wx,0.0,0.1,0.1,0.3,0.4,0.6,0.5,0.6,0.7.
From Figure2,(8)dGu=0.2,0.3,0.3,0.1,0.2,0.2,0.4,0.5,0.6,dGv=0.1,0.2,0.2,0.4,0.5,0.5,0.6,0.7,0.8,dGw=0.0,0.1,0.1,0.3,0.4,0.6,0.5,0.6,0.7,dGx=0.3,0.6,0.6,0.8,1.1,1.3,1.5,1.8,2.1,(9)dGux=0.1,0.3,0.3,0.7,0.9,1.1,1.1,1.3,1.5,dGvx=0.2,0.4,0.4,0.4,0.6,0.8,0.9,1.1,1.3,dGwx=0.3,0.5,0.5,0.5,0.7,0.7,0.1,1.2,1.4.
tdGux=tdGvx=tdGwx=0.3,0.6,0.6,0.8,1.1,1.3,1.5,1.8,2.1.
Here,dGux≠dGvx≠dGwx. Hence, G is a NEI-NCG. But G is not a NETI-NCG, since all edges have same TDs.Figure 2
G is NEI-NCG but it is not NETI-NCG.Example 3.
NETI-NCG does not need to be NEI-NCG. The following shows this subject.
LetG:M,N be a NCG so that G∗:V,E is a path that consists of four nodes where
V=u,v,w,x and E=uv,vw,wx are defined as(10)M=u,0.4,0.6,0.4,0.2,0.3,0.5,0.5,0.7,0.2,v,0.4,0.7,0.2,0.2,0.6,0.3,0.7,0.9,0.4,w,0.5,0.6,0.7,0.3,0.5,0.9,0.4,0.7,0.3,x,0.3,0.5,0.5,0.2,0.4,0.7,0.1,0.5,0.6,(11)N=uv,0.2,0.3,0.3,0.1,0.2,0.4,0.3,0.4,0.1,vw,0.4,0.6,0.6,0.2,0.4,0.8,0.6,0.8,0.2,wx,0.2,0.3,0.3,0.1,0.2,0.4,0.3,0.4,0.1.
From Figure3,(12)dGu=dGx=0.2,0.3,0.3,0.1,0.2,0.4,0.3,0.4,0.1,dGv=dGw=0.6,0.9,0.9,0.3,0.6,1.2,0.9,1.2,0.3,(13)dGuv=dGvw=dGwx=0.4,0.6,0.6,0.2,0.4,0.8,0.6,0.8,0.2,(14)tdGuv=tdGwx=0.6,0.9,0.9,0.3,0.6,1.20.9,1.2,0.3,tdGvw=0.8,1.2,1.2,0.4,0.8,1.61.2,1.6,0.4.
Here,dGuv=dGvw=dGwx. Hence, G is not a NEI-NCG. But G is a NETI-NCG, since tdGuv≠tdGvw and tdGvw≠tdGwx.Figure 3
G is NETI-NCG but it is not NEI-NCG.Theorem 1.
LetG be a CNCG on G∗ and N be a CF. Then, G is a NEI-NCG, iff G is a NETI-NCG.Proof.
Assume thatN=C,D=T˜C,TD,I˜C,ID,F˜C,FD is a CF, and let Nuv=R˜,S, ∀uv in E, where R˜,S=R˜T,ST,R˜I,SI,R˜F,SF is constant.
Letuv and vw be pair of AEs in E. Then,(15)dGuv≠dGvw⟺dGuv+R˜,S≠dGvw+R˜,S⟺dT˜Cuv,dTDuv,dI˜Cuv,dIDuv,dF˜Cuv,dFDuv+R˜T,ST,R˜I,SI,R˜F,SF≠dT˜Cvw,dTDvw,dI˜Cvw,dIDvw,dF˜Cvw,dFDvw+R˜T,ST,R˜I,SI,R˜F,SF(16)⟺dT˜Cuv+R˜T,dTDuv+ST,dI˜Cuv+R˜I,dIDuv+SI,dF˜Cuv+R˜F,dFDuv+SF≠dT˜Cvw+R˜T,dTDvw+ST,dI˜Cvw+R˜I,dIDvw+SI,dF˜Cvw+R˜F,dFDvw+SF(17)⟺dT˜Cuv+T˜C,dTDuv+TD,dI˜Cuv+I˜C,dIDuv+ID,dF˜Cuv+F˜C,dFDuv+FD≠dT˜Cvw+T˜C,dTDvw+TD,dI˜Cvw+I˜C,dIDvw+ID,dF˜Cvw+F˜C,dFDvw+FD(18)⟺tdT˜Cuv,tdTDuv,tdI˜Cuv,tdIDuv,tdF˜Cuv,tdFDuv≠tdT˜Cvw,tdTDvw,tdI˜Cvw,tdIDvw,tdF˜Cvw,tdFDvw⟺tdGuv≠tdGvw.
Therefore, adjacent edges have various degrees if and only if they have various total
degrees. So,G is a NEI-NCG iff G is a NETI-NCG.Remark 1.
LetG be a CNCG on G∗. If G is both NEI-NCG and NETI-NCG, then
N does not need to be a CF.Example 4.
LetG:M,N be a NCG and G∗:V,E be a path that consists of four nodes where
V=u,v,w,x and E=uv,vw,wx are defined as(19)M=u,0.3,0.5,0.3,0.2,0.6,0.6,0.4,0.7,0.2,v,0.2,0.4,0.5,0.3,0.5,0.4,0.6,0.8,0.3,w,0.2,0.3,0.2,0.4,0.5,0.7,0.5,0.6,0.4,x,0.3,0.6,0.6,0.2,0.4,0.1,0.7,0.9,0.6,(20)N=uv,0.2,0.3,0.4,0.1,0.4,0.5,0.6,0.7,0.2,vw,0.1,0.2,0.3,0.3,0.4,0.6,0.5,0.6,0.1,wx,0.2,0.3,0.4,0.1,0.4,0.5,0.6,0.7,0.2.
From Figure4,(21)dGu=dGx=0.2,0.3,0.4,0.1,0.4,0.5,0.6,0.7,0.2,dGv=dGw=0.3,0.5,0.7,0.4,0.8,1.1,1.1,1.3,0.3,(22)dGuv=dGwx=0.1,0.2,0.3,0.3,0.4,0.6,0.5,0.6,0.1,dGvw=0.4,0.6,0.8,0.2,0.8,1.0,1.2,1.4,0.4,(23)tdGuv=tdGwx=0.3,0.5,0.7,0.4,0.8,1.1,1.1,1.3,0.3,tdGvw=0.5,0.8,1.1,0.5,1.2,1.7,1.7,2.0,0.4.
Here,dGuv≠dGvw and dGvw≠dGwx. Hence, G is a NEI-NCG. Also, tdGuv≠tdGvw and tdGvw≠tdGwx. Hence, G is a NETI-NCG. But N is not CF.Figure 4
N is not a CF.Theorem 2.
LetG be a CNCG on G∗ and N be a CF. If G is a SI-NCG, then G is a NEI-NCG.Proof.
LetG:M,N be a CNCG. Assume that N=C,D=T˜C,TD,I˜C,ID,F˜C,FD is a CF, and let Nuv=R˜,S, ∀uv in E, where R˜,S=R˜T,ST,R˜I,SI,R˜F,SF is constant.
Letuv and vw be any two AEs in G and G be a SI-NCG. Then, each pair of nodes in
G has VDs, and hence(24)dGu≠dGv≠dGw⟹dT˜Au,dTBu,dI˜Au,dIBu,dF˜Au,dFBu≠dT˜Av,dTBv,dI˜Av,dIBv,dF˜Av,dFBv≠dT˜Aw,dTBw,dI˜Aw,dIBw,dF˜Aw,dFBw(25)⟹dT˜Au,dTBu,dI˜Au,dIBu,dF˜Au,dFBu+dT˜Av,dTBv,dI˜Av,dIBv,dF˜Av,dFBv−2R˜T,ST,R˜I,SI,R˜F,SF≠dT˜Av,dTBv,dI˜Av,dIBv,dF˜Av,dFBv+dT˜Aw,dTBw,dI˜Aw,dIBw,dF˜Aw,dFBw−2R˜T,ST,R˜I,SI,R˜F,SF(26)⟹dT˜Au+dT˜Av−2R˜T,dTBu+dTBv−2ST,dI˜Au+dI˜Av−2R˜I,dIBu+dIBv−2SI,dF˜Au+dF˜Av−2R˜F,dFBu+dFBv−2SF≠dT˜Av+dT˜Aw−2R˜T,dTBv+dTBw−2ST,dI˜Av+dI˜Aw−2R˜I,dIBv+dIBw−2SI,dF˜Av+dF˜Aw−2R˜F,dFBv+dFBw−2SF(27)⟹dT˜Au+dT˜Av−2T˜Cuv,dTBu+dTBv−2TDuv,dI˜Au+dI˜Av−2I˜Cuv,dIBu+dIBv−2IDuv,dF˜Au+dF˜Av−2F˜Cuv,dFBu+dFBv−2FDuv≠dT˜Av+dT˜Aw−2T˜Cvw,dTBv+dTBw−2TDvw,dI˜Av+dI˜Aw−2I˜Cvw,dIBv+dIBw−2IDvw,dF˜Av+dF˜Aw−2F˜Cvw,dFBv+dFBw−2FDvw(28)⟹dT˜Cuv,dTDuv,dI˜Cuv,dIDuv,dF˜Cuv,dFDuv≠dT˜Cvw,dTDvw,dI˜Cvw,dIDvw,dF˜Cvw,dFDvw⟹dGuv≠dGvw.
Therefore, each pair of AEs has VDs. Hence,G is a NEI-NCG.Theorem 3.
LetG be a CNCG on G∗ and N be a CF. If G is a SI-NCG, then G is a NETI-NCG.Remark 2.
Converse of Theorems3 is not generally
true.Example 5.
LetG:M,N be a NCG so that G∗:V,E is a path on four nodes where V=u,v,w,x and E=uv,vw,wx are defined as(29)M=u,0.2,0.4,0.5,0.1,0.50.3,0.4,0.7,0.4,v,0.4,0.5,0.6,0.3,0.6,0.7,0.5,0.8,0.3,w,0.3,0.4,0.4,0.2,0.7,0.6,0.1,0.6,0.2,x,0.2,0.5,0.3,0.3,0.4,0.1,0.6,0.9,0.5,(30)N=uv,0.2,0.3,0.4,0.1,0.4,0.5,0.5,0.7,0.2,vw,0.2,0.3,0.4,0.1,0.4,0.5,0.5,0.7,0.2,wx,0.2,0.3,0.4,0.1,0.4,0.5,0.5,0.7,0.2.
From Figure5,(31)dGu=dGx=0.2,0.3,0.4,0.1,0.4,0.5,0.5,0.7,0.2,dGv=dGw=0.4,0.6,0.8,0.2,0.8,1.0,1.0,1.4,0.4.
Here,G is not a SI-NCG.dGuv=dGwx=0.2,0.3,0.4,0.1,0.4,0.5,0.5,0.7,0.2.dGvw=0.4,0.6,0.8,0.2,0.8,1.0,1.0,1.4,0.4.tdGuv=tdGwx=0.4,0.6,0.8,0.2,0.8,1.0,1.0,1.4,0.4.tdGvw=0.6,0.9,1.2,0.3,1.2,1.5,1.5,2.1,0.6.
It is noted thatdGuv≠dGvw and dGvw≠dGwx. Also, tdGuv≠tdGvw and tdGvw≠tdGwx. Hence, G is both NEI-NCG and NETI-NCG. But G is not a SI-NCG.Figure 5
G is both NEI-NCG and NETI-NCG, but it is not SI-NCG.Theorem 4.
LetG be a CNCG and N be a CF. Then, G is a HI-NCG if and only if G is a NEI-NCG.Proof.
LetG:M,N be a CNCG. Assume that N=C,D=T˜C,TD,I˜C,ID,F˜C,FD is a CF, and let Nuv=R˜,S, ∀uv in E, in which R˜,S=R˜T,ST,R˜I,SI,R˜F,SF is constant.
Letuv and vw be any two AEs in G. Then,(32)dGu≠dGw⟺dT˜Au,dTBu,dI˜Au,dIBu,dF˜Au,dFBu≠dT˜Aw,dTBw,dI˜Aw,dIBw,dF˜Aw,dFBw⟺dT˜Au,dTBu,dI˜Au,dIBu,dF˜Au,dFBu+dT˜Av,dTBv,dI˜Av,dIBv,dF˜Av,dFBv−2R˜T,ST,R˜I,SI,R˜F,SF(33)≠dT˜Av,dTBv,dI˜Av,dIBv,dF˜Av,dFBv+dT˜Aw,dTBw,dI˜Aw,dIBw,dF˜Aw,dFBw−2R˜T,ST,R˜I,SI,R˜F,SF(34)⟺dT˜Au+dT˜Av−2R˜T,dTBu+dTBv−2ST,dI˜Au+dI˜Av−2R˜I,dIBu+dIBv−2SI,dF˜Au+dF˜Av−2R˜F,dFBu+dFBv−2SF≠dT˜Av+dT˜Aw−2R˜T,dTBv+dTBw−2ST,dI˜Av+dI˜Aw−2R˜I,dIBv+dIBw−2SI,dF˜Av+dF˜Aw−2R˜F,dFBv+dFBw−2SF(35)⟺dT˜Au+dT˜Av−2T˜Cuv,dTBu+dTBv−2TDuv,dI˜Au+dI˜Av−2I˜Cuv,dIBu+dIBv−2IDuv,dF˜Au+dF˜Av−2F˜Cuv,dFBu+dFBv−2FDuv≠dT˜Av+dT˜Aw−2T˜Cvw,dTBv+dTBw−2TDvw,dI˜Av+dI˜Aw−2I˜Cvw,dIBv+dIBw−2IDvw,dF˜Av+dF˜Aw−2F˜Cvw,dFBv+dFBw−2FDvw(36)⟺dT˜Cuv,dTDuv,dI˜Cuv,dIDuv,dF˜Cuv,dFDuv≠dT˜Cvw,dTDvw,dI˜Cvw,dIDvw,dF˜Cvw,dFDvw⟺dGuv≠dGvw.
Therefore, every pair of AEs has VDs, iff every node neighbor to the nodes has VDs.
Hence,G is a HI-NCG, iff G is a NEI-NCG.Theorem 5.
LetG be a CNCG and N be a CF. Then, G is HI-NCG iff G is NETI-NCG.Proof.
It is clear.Definition 11.
LetG:M,N be a CNCG. Then, G is called to be a(i)
SEI-NCG if each pair of edges has VDs (or no two edges have same degree).(ii)
SETI-NCG if each pair of edges has various TDs (or no two edges have same TD).Example 6.
Consider a graph that is both SEI-NCG and SETI-NCG.
LetG:M,N be a CNCG that is a cycle of length five where
V=u,v,w,x,y and E=uv,vw,wx,xy,yu are defined as(37)M=u,0.5,0.6,0.4,0.7,0.8,0.6,0.2,0.3,0.7v,0.2,0.4,0.2,0.5,0.7,0.8,0.6,0.9,0.6w,0.3,0.5,0.6,0.4,0.6,0.3,0.1,0.2,0.5,x,0.4,0.6,0.5,0.3,0.4,0.7,0.5,0.8,0.6,y,0.6,0.7,0.8,0.2,0.4,0.2,0.3,0.5,0.4,(38)N=uv,0.1,0.2,0.3,0.4,0.6,0.7,0.3,0.5,0.6vw,0.2,0.3,0.4,0.3,0.5,0.6,0.4,0.6,0.1wx,0.3,0.4,0.5,0.3,0.4,0.5,0.3,0.4,0.5xy,0.3,0.5,0.6,0.2,0.3,0.4,0.1,0.2,0.3yu,0.4,0.6,0.7,0.1,0.2,0.3,0.2,0.3,0.4.
From Figure6,(39)dGu=0.5,0.3,1.0,0.5,0.8,1.0,0.5,0.8,1.0,dGv=0.3,0.5,0.7,0.7,1.1,1.3,0.7,1.1,0.7,dGw=0.5,0.7,0.9,0.6,0.9,1.1,0.7,1.0,0.6,dGx=0.6,0.9,1.1,0.5,0.7,0.9,0.4,0.6,0.8,dGy=0.7,1.1,1.3,0.3,0.5,0.7,0.3,0.5,0.7,(40)dGuv=0.6,0.9,1.1,0.4,0.7,0.9,0.6,0.9,0.5,dGvw=0.4,0.6,0.8,0.7,1.0,1.2,0.6,0.9,1.1,dGwx=0.5,0.8,1.0,0.5,0.8,1.0,0.5,0.8,0.4,dGxy=0.7,1.0,1.2,0.4,0.6,0.8,0.5,0.7,0.9,dGyu=0.4,0.7,0.9,0.6,0.9,1.1,0.4,0.7,0.9.
So,G is a SEI-NCG.(41)tdGuv=0.7,1.1,1.4,0.8,1.3,1.6,0.9,1.4,1.1,tdGvw=0.6,0.9,1.2,1.0,1.5,1.8,1.0,1.5,1.2,tdGwx=0.8,1.2,1.5,0.8,1.2,1.5,0.8,1.2,0.9,tdGxy=1.0,1.5,1.8,0.6,0.9,1.2,0.6,0.9,1.2,tdGyu=0.8,1.3,1.6,0.7,1.1,1.4,0.6,1.0,1.3.
Thus,G is a SETI-NCG.
Therefore,G is both SEI-NCG and SETI-NCG.Figure 6
G is both SEI-NCG and SETI-NCG.Example 7.
SEI-NCG need not be SETI-NCG.
LetG:M,N be a NCG on G∗:such thatV,E is a cycle of length three, V=u,v,w, and E=uv,vw,wu. We define M and N as
follows:(42)M=u,0.2,0.4,0.3,0.6,0.8,0.5,0.3,0.6,0.7,v,0.3,0.5,0.2,0.5,0.8,0.4,0.4,0.7,0.5,w,0.4,0.6,0.4,0.5,0.7,0.8,0.2,0.8,0.5.,(43)N=uv,0.2,0.3,0.1,0.5,0.7,0.3,0.4,0.6,0.5,vw,0.3,0.4,0.3,0.3,0.5,0.1,0.2,0.4,0.2,wu,0.1,0.2,0.2,0.4,0.6,0.6,0.3,0.5,0.3..
From Figure7,(44)dGu=0.3,0.5,0.3,0.9,1.3,0.9,0.7,1.1,0.8,dGv=0.5,0.7,0.4,0.8,1.2,0.4,0.6,1.0,0.7,dGw=0.4,0.6,0.5,0.7,1.1,0.7,0.5,0.9,0.5,(45)dGuv=0.4,0.6,0.5,0.7,1.1,0.7,0.5,0.9,0.5,dGvw=0.3,0.5,0.3,0.9,1.3,0.9,0.7,1.1,0.8,dGwu=0.5,0.7,0.4,0.8,1.2,0.4,0.6,1.0,0.7,(46)tdGuv=tdGwx=tdGwu=0.6,0.9,0.6,1.2,1.8,1.0,0.9,1.5,1.0.
Note thatG is SEI-NCG, since each pair of edges has VDs. Also,
G is not SETI-NCG, since all the edges have same TD. Hence,
SEI-NCG need not be SETI-NCG.Figure 7
G is SEI-NCG, but it is not SETI-NCG.Example 8.
SETI-NCG need not be SEI-NCG.
ConsiderG:M,N be a NCG so that G∗:V,E, a cycle of length four where V=u,v,w,x and E=uv,vw,wx,xu defined as(47)M=u,0.6,0.9,0.8,0.4,0.7,0.6,0.6,0.8,0.5,v,0.4,0.6,0.4,0.4,0.5,0.3,0.3,0.7,0.9,w,0.5,0.7,0.6,0.4,0.7,0.4,0.6,0.8,0.5,x,0.7,0.8,0.7,0.2,0.5,0.2,0.3,0.7,0.9,(48)N=uv,0.1,0.3,0.2,0.3,0.4,0.5,0.4,0.6,0.3,vw,0.3,0.5,0.4,0.4,0.5,0.2,0.5,0.7,0.5,wx,0.5,0.7,0.3,0.2,0.3,0.2,0.4,0.5,0.1,xu,0.6,0.8,0.7,0.1,0.3,0.3,0.2,0.4,0.1.
From Figure8,(49)dGu=0.7,1.1,0.9,0.4,0.7,0.8,0.6,1.0,0.4,dGv=0.4,0.8,0.6,0.7,0.9,0.7,0.9,1.3,0.8,dGw=0.8,1.2,0.7,0.6,0.8,0.4,0.9,1.2,0.6,dGx=1.1,1.5,1.0,0.3,0.6,0.5,0.6,0.9,0.2,(50)dGuv=dGwx=0.9,1.3,1.1,0.5,0.8,0.5,0.7,1.1,0.6,dGvw=dGxu=0.6,1.0,0.5,0.5,0.7,0.7,0.8,1.1,0.4,(51)tdGuv=1.0,1.6,1.3,0.8,1.0,0.9,1.1,1.7,0.9,tdGvw=0.9,1.5,0.9,0.9,1.2,0.9,1.3,1.8,0.9,tdGwx=1.4,2.0,1.4,0.7,1.1,0.7,1.1,1.6,0.7,dGxu=1.2,1.8,1.2,0.6,1.0,1.0,1.0,1.5,0.5.
Obviously,dGuv=dGwx. Hence, G is not SEI-NCG.
ButG is SETI-NCG, since tdGuv≠tdGvw≠tdGwx≠tdGxu.
Hence, SETI-NCG need not be SEI-NCG.Figure 8
G is SETI-NCG but it is not SEI-NCG.Theorem 6.
LetG be a CNCG on G∗ and N be a CF. Then, G is a SEI-NCG, iff G is a SETI-NCG.Proof.
AssumeN=C,D=T˜C,TD,I˜C,ID,F˜C,FD is a CF. Let Nuv=R˜,S, for all uv in E, in which R˜,S=R˜T,ST,R˜I,SI,R˜F,SF is constant.
Letuv and xy be any pair of edges in E. Then,(52)dGuv≠dGxy⟺dGuv+R˜,S≠dGxy+R˜,S⟺dT˜Cuv,dTDuv,dI˜Cuv,dIDuv,dF˜Cuv,dFDuv+R˜T,ST,R˜I,SI,R˜F,SF≠dT˜Cxy,dTDxy,dI˜Cxy,dIDxy,dF˜Cxy,dFDxy+R˜T,ST,R˜I,SI,R˜F,SF(53)⟺dT˜Cuv+R˜T,dTDuv+ST,dI˜Cuv+R˜I,dIDuv+SI,dF˜Cuv+R˜F,dFDuv+SF≠dT˜Cxy+R˜T,dTDxy+ST,dI˜Cxy+R˜I,dIDxy+SI,dF˜Cxy+R˜F,dFDxy+SF(54)⟺dT˜Cuv+T˜C,dTDuv+TD,dI˜Cuv+I˜C,dIDuv+ID,dF˜Cuv+F˜C,dFDuv+FD≠dT˜Cxy+T˜C,dTDxy+TD,dI˜Cxy+I˜C,dIDxy+ID,dF˜Cxy+F˜C,dFDxy+FD(55)⟺tdT˜Cuv,tdTDuv,tdI˜Cuv,tdIDuv,tdF˜Cuv,tdFDuv≠tdT˜Cxy,tdTDxy,tdI˜Cxy,tdIDxy,tdF˜Cxy,tdFDxy⟺tdGuv≠tdGxy.
So, each edge has different degree if and only if it has different total degrees. Hence,G is SEI-NCG iff G is a SETI-NCG.Remark 3.
LetG be a CNCG. If G is both SEI-NCG and SETI-NCG, then
N need not be a CF.Example 9.
LetG:M,N be a NCG so that G∗:V,E is graph for Example 6 (Figure 9). As seen in that example,
each pair of edges in G has VDs. Hence, G is a SEI-NCG.
Also, note that each pair of edges inG has various TDs. Hence, G is a SETI-NCG. Therefore, G is both SEI-NCG and SETI-NCG. But N is not a CF.Figure 9
NC-digraph (influence graph).Theorem 7.
LetG be a NCG on G∗. If G is a SEI-NCG, then G is a NEI-NCG.Proof.
LetG:M,N be a NCG. Assume that G is a SEI-NCG. Then, each pair of edges in
G has VDs. So, each pair of AEs has VDs. Hence,
G is a NEI-NCG.Theorem 8.
LetG be a NCG. If G is a SETI-NCG, then, G is a NETI-NCG.Proof.
LetG:M,N be a NCG. Suppose that G is a SETI-NCG; then, each pair of edges in
G has various TDs. So, each pair of AEs has various TDs.
Hence, G is a NETI-NCG.Remark 4.
The inverse of Theorems7 and 8 is not generally true.Example 10.
ConsiderG:M,N be a NCG so that G∗:V,E is graph for Example 4 (Figure 4). As seen in that example,
dGuv≠dGvw and dGvw≠dGwx. Hence, G is a NEI-NCG. But G is not a SEI-NCG, since dGuv≠dGwx. Also, note that tdGuv≠tdGvw and tdGvw≠tdGwx. Hence, G is a NETI-NCG. But G is not a SETI-NCG, since tdGuv≠tdGwx.Theorem 9.
LetG be a CNCG and N be a CF. If G is a SEI-NCG, then G is an irregular NCG.Proof.
LetG:A,B be a CNCG. Assume that N=C,D=T˜C,TD,I˜C,ID,F˜C,FD is a CF, and let Nuv=R˜,S, for all uv in E, in which R˜,S=R˜T,ST,R˜I,SI,R˜F,SF is constant.
SupposeG is a SEI-NCG. Then, each edge has VD. Let
uv and vw be AEs in G having VDs, and hence(56)dGuv≠dGvw⟹dT˜Cuv,dTDuv,dI˜Cuv,dIDuv,dF˜Cuv,dFDuv≠dT˜Cvw,dTDvw,dI˜Cvw,dIDvw,dF˜Cvw,dFDvw(57)⟹dT˜Au+dT˜Av−2T˜Cuv,dTBu+dTBv−2TDuv,dI˜Au+dI˜Av−2I˜Cuv,dIBu+dIBv−2IDuv,dF˜Au+dF˜Av−2F˜Cuv,dFBu+dFBv−2FDuv≠dT˜Av+dT˜Aw−2T˜Cvw,dTBv+dTBw−2TDvw,dI˜Av+dI˜Aw−2I˜Cvw,dIBv+dIBw−2IDvw,dF˜Av+dF˜Aw−2F˜Cvw,dFBv+dFBw−2FDvw(58)⟹dT˜Au+dT˜Av−2R˜T,dTBu+dTBv−2ST,dI˜Au+dI˜Av−2R˜I,dIBu+dIBv−2SI,dF˜Au+dF˜Av−2R˜F,dFBu+dFBv−2SF≠dT˜Av+dT˜Aw−2R˜T,dTBv+dTBw−2ST,dI˜Av+dI˜Aw−2R˜I,dIBv+dIBw−2SI,dF˜Av+dF˜Aw−2R˜F,dFBv+dFBw−2SF(59)⟹dT˜Au+dT˜Av,dTBu+dTBv,dI˜Au+dI˜Av,dIBu+dIBv,dF˜Au+dF˜Av,dFBu+dFBv−2R˜T,ST,R˜I,SI,R˜F,SF≠dT˜Av+dT˜Aw,dTBv+dTBw,dI˜Av+dI˜Aw,dIBv+dIBw,dF˜Av+dF˜Aw,dFBv+dFBw−2R˜T,ST,R˜I,SI,R˜F,SF(60)⟹dT˜Au+dT˜Av.,dTBu+dTBv,dI˜Au+dI˜Av,dIBu+dIBv,dF˜Au+dF˜Av,dFBu+dFBv≠dT˜Av+dT˜Aw,dTBv+dTBw,dI˜Av+dI˜Aw,dIBv+dIBw,dF˜Av+dF˜Aw,dFBv+dFBw⟹dT˜Au,dTBu,dI˜Au,dIBu,dF˜Au,dFBu+dT˜Av,dTBv,dI˜Av,dIBv,dF˜Av,dFBv≠dT˜Av,dTBv,dI˜Av,dIBv,dF˜Av,dFBv+dT˜Aw,dTBw,dI˜Aw,dIBw,dF˜Aw,dFBw⟹dGu+dGv≠dGv+dGw⟹dGu≠dGw.
So, there exists a nodev which is neighbor to nodes u and w having VDs. Hence, G is an irregular NCG.Theorem 10.
LetG be a CNCG and N be a CF. If G is a SETI-NCG, then G is an irregular NCG.Proof.
Proof is similar to Theorem9.Remark 5.
The inverse of Theorems9 and 10 is not generally true.Example 11.
LetG:M,N be a NCG so that G∗:V,E is graph for Example 5 (Figure 5). As seen in that example,
G is an irregular NCG. Also, it is noted that
dGuv=dGwx. Hence, G is not a SEI-NCG. Also, tdGuv=tdGwx. Hence, G is not a SETI-NCG.Theorem 11.
LetG be a CNCG and N be a CF. If G is a SEI-NCG, then, G is a HI-NCG.Proof.
LetG:M,N be a CNCG. Assume that N=C,D=T˜C,TD,I˜C,ID,F˜C,FD is a CF. Let Nuv=R˜,S, ∀uv in E, in which R˜,S=R˜T,ST,R˜I,SI,R˜F,SF is constant.
Letv be any node neighbor with u, w, and x. Then, uv, vw, and vx are AEs in G. Let G be a SEI-NCG. Then, each pair of edges in
G has VDs. So, each pair of AEs in G has VDs. Hence,(61)dGuv≠dGvw≠dGvx⟹dT˜Cuv,dTDuv,dI˜Cuv,dIDuv,dF˜Cuv,dFDuv≠dT˜Cvw,dTDvw,dI˜Cvw,dIDvw,dF˜Cvw,dFDvw≠dT˜Cvx,dTDvx,dI˜Cvx,dIDvx,dF˜Cvx,dFDvx(62)⟹dT˜Au+dT˜Av−2T˜Cuv,dTBu+dTBv−2TDuv,dI˜Au+dI˜Av−2I˜Cuv,dIBu+dIBv−2IDuv,dF˜Au+dF˜Av−2F˜Cuv,dFBu+dFBv−2FDuv≠dT˜Av+dT˜Aw−2T˜Cvw,dTBv+dTBw−2TDvw,dI˜Av+dI˜Aw−2I˜Cvw,dIBv+dIBw−2IDvw,dF˜Av+dF˜Aw−2F˜Cvw,dFBv+dFBw−2FDvw≠dT˜Av+dT˜Ax−2T˜Cvx,dTBv+dTBx−2TDvx,dI˜Av+dI˜Ax−2I˜Cvx,dIBv+dIBx−2IDvx,(63)dF˜Av+dF˜Ax−2F˜Cvx,dFBv+dFBx−2FDvx⟹dT˜Au+dT˜Av−2R˜T,dTBu+dTBv−2ST,dI˜Au+dI˜Av−2R˜I,dIBu+dIBv−2SI,dF˜Au+dF˜Av−2R˜F,dFBu+dFBv−2SF≠dT˜Av+dT˜Aw−2R˜T,dTBv+dTBw−2ST,dI˜Av+dI˜Aw−2R˜I,dIBv+dIBw−2SI,dF˜Av+dF˜Aw−2R˜F,dFBv+dFBw−2SF≠dT˜Av+dT˜Ax−2R˜T,dTBv+dTBx−2ST,dI˜Av+dI˜Ax−2R˜I,dIBv+dIBx−2SI,dF˜Av+dF˜Ax−2R˜F,dFBv+dFBx−2SF(64)⟹dT˜Au+dT˜Av,dTBu+dTBv,dI˜Au+dI˜Av,dIBu+dIBv,dF˜Au+dF˜Av,dFBu+dFBv≠dT˜Av+dT˜Aw,dTBv+dTBw,dI˜Av+dI˜Aw,dIBv+dIBw,dF˜Av+dF˜Aw,dFBv+dFBw≠dT˜Av+dT˜Ax,dTBv+dTBx,dI˜Av+dI˜Ax,dIBv+dIBx,dF˜Av+dF˜Ax,dFBv+dFBx(65)⟹dT˜Au,dTBu,dI˜Au,dIBu,dF˜Au,dFBu+dT˜Av,dTBv,dI˜Av,dIBv,dF˜Av,dFBv≠dT˜Av,dTBv,dI˜Av,dIBv,dF˜Av,dFBv+dT˜Aw,dTBw,dI˜Aw,dIBw,dF˜Aw,dFBw≠dT˜Av,dTBv,dI˜Av,dIBv,dF˜Av,dFBv+dT˜Ax,dTBx,dI˜Ax,dIBx,dF˜Ax,dFBx(66)⟹dT˜Au,dTBu,dI˜Au,dIBu,dF˜Au,dFBu≠dT˜Aw,dTBw,dI˜Aw,dIBw,dF˜Aw,dFBw≠dT˜Ax,dTBx,dI˜Ax,dIBx,dF˜Ax,dFBx⟹dGu≠dGw≠dGx.
Therefore, the nodev is neighbor to the nodes with VDs. Hence,
G is a HI-NCG.Theorem 12.
LetG be a CNCG and N be a CF. If G is a SETI-NCG, then G is a HI-NCG.Proof.
Proof is similar to Theorem11.Remark 6.
The inverse of Theorems11 and 12 is not generally true.Example 12.
LetG:M,N be a NCG so that G∗:V,E is graph for Example 5 (Figure 5). As seen in that example,
G is a HI-NCG. Note that dGuv=dGwx. So, G is not a SEI-NCG. Also, tdGuv=tdGwx. Thus, G is not a SETI-NCG.


## 4. Application of Neutrosophic Cubic-Influence Digraph to Find the Most Effective
Person in a School

School is one of the most important places for education and training of students. The
approved goals of the study courses are established and managed in accordance with the rules
and instructions of the Ministry of Education. At school, a student interacts with his/her
classmates and tries to learn the necessary scientific points. The physical, psychological,
and educational environment of the school is one of the issues that can have an important
and significant reflection on the structure of mental and intellectual growth and
development, as well as creativity and mental health of students. The transfer of the basic
values of the society is the main focus of the educational system, in such a way that the
school commits the students to internalize the values of the society. In schools, values are
taught in a variety of subjects, and the effectiveness of teaching values in each subject
depends on the teacher’s understanding of the objectives of the subject. By
recognizing the possibilities, the teacher equips the educational environment and by
recognizing the interests and abilities of the students guides them in the right direction
of learning because success in schools requires teachers to accept the opinions of others.
Therefore, considering the importance of schools in shaping the personality and behavior of
students, we have tried to determine the most effective person in a school according to its
performance. To do this, we consider the nodes of this influence graph as the staff of a
school and the edges as the influence of one employee on another employee. The names of the
staff and their specialization in the school are shown in Table2. For this school, the staff is as follows:(67)A=Eskandari,Momeni,Rouhi,Salimi,Jafari.(i)
Momeni has been working with Salimi for 16 years and values his views on issues.(ii)
Eskandari has been the head of the school, and not only Salimi but also Jafari is
very satisfied with Eskandari’s performance.(iii)
Taking care of the educational and moral affairs of students is one of the most
important issues. Rouhi is an expert for this.(iv)
Rouhi and Jafari have a long history of conflict.(v)
Jafari is a very effective person in communicating with students’ parents and
teachers in school.Table 2
The names of the staff and their specialization in the school.
NameServicesEskandari (Es)HeadmasterMomeni (Mo)School deputyRouhi (Ro)Educational instructorSalimi (Sa)Representative of the teacher’s councilJafari (Ja)Representative of the parents and teachers associationConsidering the above points, the influence graph can be very important. But such a graph
cannot show the power of employees within a school and the degree of influence of employees
on each other. Since power and influence do not have defined boundaries, they can be
represented as a neutrosophic cubic set. On the other hand, there can be no fair
interpretation of the power and influence of individuals because evaluations are always
accompanied by skepticism. So, here we use the neutrosophic cubic degrees, which is very
useful for influence and conflicts between employees. The neutrosophic cubic set of
employees is shown in Table3.Table 3
Employee power.
EskandariMomeniRouhiSalimiJafariT˜A[0.3, 0.4][0.2, 0.4][0.2, 0.3][0.2, 0.3][0.2, 0.3]TB0.50.30.50.30.3I˜A[0.2, 0.3][0.3, 0.5][0.3, 0.5][0.3, 0.4][0.4, 0.5]IB0.40.40.40.50.3F˜A[0.3, 0.5][0.3, 0.4][0.4, 0.5][0.1, 0.3][0.2, 0.4]FB0.30.50.30.60.5We have shown the influence of persons in the NC-digraph with an edge. This graph is shown
in Figure9.School staff are the vertices of the NC-digraph of Figure9. The weight of the vertices, respectively, indicates the power of speech, the
degree of interaction with students, and the extent of their management in school affairs.
For example, Mr. Rouhi has 20% to 30% of eloquence, but does not have
30% of the power to interact with students. Also, his power in processing school
affairs is between 20% and 40%. Edges represent the extent of friendship,
cultural, and political relationships, respectively. For example, Mr. Momeni has between
20% and 30% friendship with Mr. Jafari, but cultural differences between them
are equal to 30%. Similarly, the rate of political relations between them is equal to
20% to 30%.In Figure9, it is clear that Mr. Eskandari controls
the school deputy, Mr. Momeni, the representative of the parents and teachers association,
Mr. Jafari, and educational instructor, Mr. Rouhi. Clearly, Mr. Eskandari has the most
influence in the organization because he has an impact on three school staff and also has
the highest level of management among other employees.


## 5. Conclusions

Neutrosophic cubic graphs have various uses in modern science and technology, especially in
the fields of neural networks, computer science, operation research, and decision making.
Also, they have a wide range of applications in the field of psychological sciences as well
as the identification of individuals based on oncological behaviors. Therefore, in this
research, some types of EI-NCGs such as NETI-NCGs, SEI-NCGs, and SETI-NCGs are introduced. A
comparative study between NEI-NCGs and NETI-NCGs is presented. Finally, an application of
neutrosophic cubic digraph to find the most effective person in a school has been
introduced. In our future work, we will introduce connectivity index, Wiener index, and
Randic index in neutrosophic cubic graphs and investigate some of their properties. Also, we
will study some types of edge irregular neutrosophic cubic graphs such as neighborly edge
totally irregular, strongly edge irregular, and strongly edge totally irregular neutrosophic
cubic graphs.



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*Source: 6738962-2022-06-14.xml*