# Certain Concepts of Interval-Valued Intuitionistic Fuzzy Graphs with an Application

Authors: Peng Xu; Hao Guan; A. A. Talebi; M. Ghassemi; Hossein Rashmanlou
Journal: Advances in Mathematical Physics
 (2022)

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## Abstract

Interval-valued intuitionistic fuzzy graph (IVIFG), belonging to the FGs family, has good capabilities when facing with 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 an IVIFG. The previous definitions of limitations in edge irregular FG have led us to offer new definitions in IVIFGs. Hence, in this paper, some types of edge irregular interval-valued intuitionistic fuzzy graphs (EI-IVIFGs) such as neighborly edge totally irregular (NETI), strongly edge irregular (SEI), and strongly edge totally irregular (SETI) are introduced. A comparative study between NEI-IVIFGs and NETI-IVIFGs is done. With the help of IVIFGs, the most efficient person in an organization can be identified according to the important factors that can be useful for an institution. Finally, an application of IVIFG has been introduced.


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## Body

## 1. Introduction

The FG concept serves as one of the most dominant and extensively employed tools for multiple real-word problem representations, modeling, and analysis. To specify the objects and the relations between them, the graph vertices or nodes and edges or arcs are applied, respectively. Graphs have long been used to describe objects and the relationships between them. Many of the issues and phenomena around us are associated with complexities and ambiguities that make it difficult to express certainty. These difficulties were alleviated by the introduction of fuzzy sets by Zadeh [1]. The fuzzy set focuses on the membership degree of an object in a particular set. Kaufman [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 and Nair [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 nonmembership 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). In 1999, Atanassov [10] defined intuitionistic fuzzy graph (IFG), but Akram and Davvaz investigated it in more details in [11]. Hongmei and Lianhua [12] defined interval-valued fuzzy graph and studied its properties. Karunambigai et al. [13] discussed edge regular IFG. Mishra and Pal [14] introduced product of IVIFG. Nagoorgani and Radha [15, 16] studied the concept of regular fuzzy graphs and defined degree of a vertex in FGs. Nagoorgani and Latha [17] investigated the concept of IFGs, NI-FGs, and HI-FGs in 2008. Shao et al. [18] discussed new concepts in IFG. Nandhini and Nandhini [19] described the concept of SI-FGs and studied its properties. Santhi Maheswari and Sekar defined the concepts of edge irregular FGs and edge totally irregular FGs [20]. Also, they analyzed some properties of NEI-FGs, NETI-FGs, SEI-FGs, and SETI-FGs [21, 22]. Rao et al. [23–25] studied dominating set, equitable dominating set, valid degree, isolated vertex, and some properties of VGs with novel application. Shi and Kosari [26] introduced total dominating set and global dominating set in product vague graphs. Talebi et al. [27–30] defined new concepts of irregularity in single-valued neutrosophic graphs and intuitionistic fuzzy graphs. Kou et al. [31] studied vague graphs with application in transportation systems. Kalaiarasi and Mahalakshmi [32] investigated regular and irregular m-polar fuzzy graphs. Selvanayaki [33] introduced strong and balanced irregular interval-valued fuzzy graphs. Rashmanlou et al. [34] investigate new results in cubic graphs. Poulik and Ghorai [35–37] initiated degree of nodes, detour g-interior nodes, and indices of bipolar fuzzy graphs with applications in real-life systems. Pramanik et al. [38] defined fuzzy competition graph and its uses in manufacturing industries. Muhiuddin et al. [39] introduced reinforcement number of a graph with respect to half-domination. Amanathulla et al. [40] studied on distance two surjective labeling of paths and interval graphs. Ramprasad et al. [41] investigated some properties of highly irregular, edge regular, and totally edge regular m-polar fuzzy graphs. Nazeer et al. [42] introduced an application of product intuitionistic fuzzy incidence graphs in textile industry. Bhattacharya and Pal [43] studied fuzzy covering problem of fuzzy graphs and its application. Borzooei et al. [44] defined inverse fuzzy graphs.IVIFGs have a wide range of applications in the field of psychological sciences as well as the identification of individuals based on oncological behaviors. With the help of IVIFGs, the most efficient person in an organization can be identified according to the important factors that can be useful for an institution. So, in this paper, some types of EI-IVIFGs such as neighborly edge totally irregular- (NETI-) IVIFGs, strongly edge irregular- (SEI-) IVIFGs, and strongly edge totally irregular- (SETI-) IVIFGs are introduced. Also, we have given some interesting results about EI-IVIFGs, and several examples are investigated. Finally, an application of IVIFG is presented.


## 2. Preliminaries

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 1 (see [5]).
A FGZ=V,ν,ξ is a nonempty 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.Definition 2 (see [11]).
An IFG is of the formG:η,ς which η=η1,η2 and ς=ς1,ς2 so that
(i)
The functionsη1:V⟶0,1 and η2:V⟶0,1 denotes the MD and NM-D of the element w∈V, respectively, and 0≤η1w+η2w≤1 for each w∈V(ii)
The functionsς1:V×V⟶0,1 and ς2:V×V⟶0,1 are the MD and NM-D of the edge xw∈E, respectively, so that ς1xw≤minη1x,η1w and ς2xw≥maxη2x,η2w and 0≤ς1xw+ς2xw≤1, for each xw in EDefinition 3 (see [11]).
An IVFG is of the formG:θ,ζ which θ=θ−,θ+ is an IVFS in V and ζ=ζ−,ζ+ is an IVFS in E⊆V×V so that ζ−xw≤minθ−x,θ−w and ζ+xw≤minθ+x,θ+w for each xw in E.All the basic notations are shown in Table1.Table 1
Some basic notations.
NotationMeaningIFGIntuitionistic fuzzy graphIVFGInterval-valued fuzzy graphIVIFGInterval-valued intuitionistic fuzzy graphIVMInterval valued membershipIV-NMInterval-valued nonmembershipI-FGIrregular fuzzy graphSIStrongly irregularHIHighly irregularNINeighborly irregularVDVarious degreeTDTotal degreeTITotal irregularMDMembership degreeNENeighbor edgeNEI-IVIFGNeighborly edge irregular interval-valued intuitionistic fuzzy graphCIVIFGConnected interval-valued intuitionistic fuzzy graphIVFSInterval-valued fuzzy setNEINeighborly edge irregularNETINeighborly edge totally irregularTERTotally edge regularSETIStrongly edge totally irregularSEIStrongly edge irregularHEIHighly edge irregularHETIHighly edge totally irregularCFConstant function


## 3. New Concepts of Irregular IVIFGs

Definition 4.
An IVIFG is of the formG:σ,μ which σ=σ1,σ2=σ1−,σ1+,σ2−,σ2+ and μ=μ1,μ2=μ1−,μ1+,μ2−,μ2+ so that
(i)
The functionsσ1:V⟶D0,1 and σ2:V⟶D0,1 denote the degree of IVM and IV-NM of the element w∈V, respectively, so that 0≤σ1+w+σ2+w≤1, for each w∈V(ii)
The functionsμ1:V×V⟶D0,1 and μ2:V×V⟶D0,1 denote the degree of IVM and IV-NM of the edge wz∈E, respectively, are defined by the following:(i)
μ1−wz≤minσ1−w,σ1−z and μ1+wz≤minσ1+w,σ1+z(ii)
μ2−wz≥maxσ2−w,σ2−z and μ2+wz≥maxσ2+w,σ2+z
so that0≤μ1+wz+μ2+wz≤1, for each wz in E.Definition 5.
LetG be an IVIFG. Then, the degree of a node w is defined as dGw=dσ1−w,dσ1+w,dσ2−w,dσ2+w, where dσ1−w=Σz≠wμ1−w,z, dσ1+w=Σz≠wμ1+w,z, dσ2−w=Σz≠wμ2−w,z, and dσ2+w=Σz≠wμ2+w,z.Definition 6.
LetG be an IVIFG. Then, the TD of a node w is defined as tdGw=tdσ1−w,tdσ1+w,tdσ2−w,tdσ2+w which tdσ1−w=Σz≠wμ1−w,z+σ1−w, tdσ1+w=Σz≠wμ1+w,z+σ1+w, tdσ2−w=Σz≠wμ2−w,z+σ2−w, and tdσ2+w=Σz≠wμ2+w,z+σ2+w.Definition 7.
LetG be an IVIFG on. Then,
(i)
G is irregular, if there is a node which is a neighbor to nodes with VDs(ii)
G is TI, if there is a node which is a neighbor to nodes with various TDsDefinition 8.
LetG be a CIVIFG. Then, G is called an
(i)
NI-IVIFG if each pair of neighbor nodes has VDs(ii)
NTI-IVIFG if each pair of neighbor nodes has various TDs(iii)
SI-IVIFG if each pair of nodes has VDs(iv)
STI-IVIFG if each pair of nodes has various TDs(v)
HI-IVIFG if each node inG is neighbor to the nodes having VDs(vi)
HTI-IVIFG if each node inG is neighbor to the nodes having various TDsDefinition 9.
LetG be an IVIFG on. The degree of an edge wz is described as dGwz=dμ1−wz,dμ1+wz,dμ2−wz,dμ2+wz which dμi−wz=dσi−w+dσi−z−2μi−wz and dμi+wz=dσi+w+dσi+z−2μi+wz, for i=1,2.Definition 10.
LetG be an IVIFG. The TD of an edge wz is presented as tdGwz=tdμ1−wz,tdμ1+wz,tdμ2−wz,tdμ2+wz where tdμi−wz=dσi−w+dσi−z−μi−wz=dμi−wz+μi−wz and tdμi+wz=dσi+w+dσi+z−μi+wz=dμi+wz+μi+wz, for i=1,2.Definition 11.
LetG be a CIVIFG. Then, G is called an
(i)
NEI-IVIFG if each pair of NEs has VDs(ii)
NETI-IVIFG if each pair of NEs has various TDsExample 12.
Graph which is both NEI-IVIFG and NETI-IVIFG.
ConsiderG∗ which V=u,v,w,x and E=uv,vw,wx,xu.From Figure1, dGu=dGv=dGw=dGx=0.3,0.5,0.5,1.0, dGuv=dGwx=0.4,0.6,0.4,0.8, and dGvw=dGxu=0.2,0.4,0.6,1.2.Figure 1
G is both NEI-IVIFG and NETI-IVIFG.Clearly, neighbor edges have VDs. Hence,G is a NEI-IVIFG.For TDs, we have the following:(1)tdGuv=tdGwx=0.5,0.8,0.7,1.4,tdGvw=tdGxu=0.4,0.7,0.8,1.6.Obviously, neighbor edges have various TDs. So,G is a NETI-IVIFG. Hence, G is both NEI-IVIFG and NETI-IVIFG.Example 13.
NEI-IVIFG needs not to be NETI-IVIFG.
ConsiderG be an IVIFG and G∗ a star includes four nodes.From Figure2, dGu=0.2,0.3,0.3,0.4, dGv=0.1,0.2,0.4,0.5, dGw=0.0,0.1,0.5,0.6, dGx=0.3,0.6,1.2,1.5, dGux=0.1,0.3,0.9,1.1, dGvx=0.2,0.4,0.8,1.0, dGwx=0.3,0.5,0.7,0.9, and tdGux=tdGvx=tdGwx=0.3,0.6,1.2,1.5.Figure 2
G is NEI-IVIFG but it is not NETI-IVIFG.Here,dGux≠dGvx≠dGwx. Hence, G is a NEI-IVIFG. But G is not a NETI-IVIFG, since all edges have same TDs.Example 14.
NETI-IVIFGs need not to be NEI-IVIFGs. The following shows this subject:
ConsiderG be an IVIFG so that G∗ a path consists of 4 nodes.From Figure3, dGu=dGx=0.05,0.20,0.15,0.25, dGv=dGw=0.15,0.60,0.45,0.75, dGuv=dGvw=dGwx=0.1,0.4,0.3,0.5, tdGuv=0.15,0.60,0.45,0.75, tdGvw=0.2,0.8,0.6,1.0, and tdGwx=0.15,0.60,0.45,0.75.Figure 3
G is NETI-IVIFG but it is not NEI-IVIFG.Here,dGuv=dGvw=dGwx. Hence, G is not a NEI-IVIFG. But G is a NETI-IVIFG, since tdGuv≠tdGvw and tdGvw≠tdGwx.Theorem 15.
SupposeG is a CIVIFG and μ is a CF. Then, G is a NEI-IVIFG if G is a NETI-IVIFG.Proof.
Assume thatμ is a CF and μwz=f,∀wzin E, which f=f1−,f1+,f2−,f2+ is constant.
Letwz and zy be pairs of neighbor edges in E; then, we have the following:
(2)dGwz≠dGzy⇔dGwz+d≠dGzy+d⇔dμ1−wz,dμ1+wz,dμ2−wz,dμ2+wz+f1−,f1+,f2−,f2+≠dμ1−zy,dμ1+zy,dμ2−zy,dμ2+zy+f1−,f1+,f2−,f2+⇔dμ1−wz+f1−,dμ1+wz+f1+,dμ2−wz+f2−,dμ2+wz+f2+≠dμ1−zy+f1−,dμ1+zy+f1+,dμ2−zy+f2−,dμ2+zy+f2+⇔dμ1−wz+μ1−wz,dμ1+wz+μ1+wz,dμ2−wz+μ2−wz,dμ2+wz+μ2+wz≠dμ1−zy+μ1−zy,dμ1+zy+μ1+zy,dμ2−zy+μ2−zy,dμ2+zy+μ2+zy⇔tdμ1−wz,tdμ1+wz,tdμ2−wz,tdμ2+wz≠tdμ1−zy,tdμ1+zy,tdμ2−zy,tdμ2+zy⇔tdGwz≠tdGzy.
Therefore, neighbor edges have VDs if they have various TDs. Hence,G is a NEI-IVIFG if G is a NETI-IVIFG.Remark 16.
LetG be a CIVIFG. If G is both NEI-IVIFG and NETI-IVIFG, then μ needs not to be a CF.Example 17.
SupposeG is an IVIFG and G∗ a path consists of four nodes.From Figure4, dGu=dGx=0.2,0.3,0.4,0.5, dGv=dGw=0.3,0.5,0.7,0.9, dGuv=0.1,0.2,0.3,0.4, dGvw=0.4,0.6,0.8,1.0, dGwx=0.1,0.2,0.3,0.4, tdGuv=0.3,0.5,0.7,0.9, tdGvw=0.5,0.8,1.1,1.4, and tdGwx=0.3,0.5,0.7,0.9.Figure 4
μ is not a CF.Here,dGuv≠dGvw and dGvw≠dGwx. Hence, G is a NEI-IVIFG. Also, tdGuv≠tdGvw and tdGvw≠tdGwx. Hence, G is a NETI-IVIFG but μ is not CF.Theorem 18.
LetG be a CIVIFG and μ a CF. If G is a SI-IVIFG, then, G is a NEI-IVIFG.Proof.
AssumeG is a CIVIFG, μ is a CF, and μwz=f,∀wzin E, which f=f1−,f1+,f2−,f2+ is constant.
Letwz and zy be any two NEs in G. Assume that G is a SI-IVIFG. Then, each pair of nodes in G has VDs, and hence,
(3)dGw≠dGz≠dGy⇒dσ1−w,dσ1+w,dσ2−w,dσ2+w≠dσ1−z,dσ1+z,dσ2−z,dσ2+z≠dσ1−y,dσ1+y,dσ2−y,dσ2+y⇒dσ1−w,dσ1+w,dσ2−w,dσ2+w+dσ1−z,dσ1+z,dσ2−z,dσ2+z−2f1−,f1+,f2−,f2+≠dσ1−z,dσ1+z,dσ2−z,dσ2+z+dσ1−y,dσ1+y,dσ2−y,dσ2+y−2f1−,f1+,f2−,f2+⇒dσ1−w+dσ1−z−2f1−,dσ1+w+dσ1+z−2f1+,dσ2−w+dσ2−z−2f2−,dσ2+w+dσ2+z−2f2+≠dσ1−z+dσ1−y−2f1−,dσ1+z+dσ1+y−2f1+,dσ2−z+dσ2−y−2f2−,dσ2+z+dσ2+y−2f2+⇒dσ1−w+dσ1−z−2μ1−wz,dσ1+w+dσ1+z−μ1+wz,dσ2−w+dσ2−z−2μ2−wz,dσ2+w+dσ2+z−2μ2+wz≠dσ1−z+dσ1−y−2μ1−zy,dσ1+z+dσ1+y−2μ1+zy,dσ2−z+dσ2−y−2μ2−zy,dσ2+z+dσ2+y−2μ2+zy⇒dμ1−wz,dμ1+wz,dμ2−wz,dμ2+wz≠dμ1−zy,dμ1+zy,dμ2−zy,dμ2+zy⇒dGwz≠dGzy.
Hence, neighbor edges have VDs. Thus,G is a NEIIVIFG.Theorem 19.
LetG be a CIVIFG on G∗ and μ a CF. If G is a SI-IVIFG, then G is a NETI-IVIFG.Proof.
It is similar to Theorem18.Remark 20.
Converse of Theorem19 is not generally true.Example 21.
LetG be an IVIFG so that G∗ consists of four nodes.From Figure5, dGu=dGx=0.1,0.3,0.2,0.4 and dGv=dGw=0.2,0.6,0.4,0.8.Figure 5
G is both NEI-IVIFG and NETI-IVIFG, not SI-IVIFG.Here,G is not a SI-IVIFG. dGuv=0.1,0.3,0.2,0.4, dGvw=0.2,0.6,0.4,0.8, dGwx=0.1,0.3,0.2,0.4, tdGuv=0.2,0.6,0.4,0.8, tdGvw=0.3,0.9,0.6,1.2, and tdGwx=0.2,0.6,0.4,0.8.Hence,dGuv≠dGvw and dGvw≠dGwx. Furthermore, tdGuv≠tdGvw and tdGvw≠tdGwx. Hence, G is both NEI-IVIFG and NETI-IVIFG. But G is not a SI-IVIFG.Theorem 22.
LetG be a CIVIFG and μ a CF. Then, G is a HI-IVIFG if G is a NEI-IVIFG.Proof.
AssumeG is a CIVIFG and μ is a CF. Consider μwz=f,∀wz in E, which f=f1−,f1+,f2−,f2+ is CF.
Letwz and zy be any two neighbor edges in G. Then,
(4)dGw≠dGy⇒dσ1−w,dσ1+w,dσ2−w,dσ2+w≠dσ1−y,dσ1+y,dσ2−y,dσ2+y⇒dσ1−w,dσ1+w,dσ2−w,dσ2+w+dσ1−z,dσ1+z,dσ2−z,dσ2+z−2f1−,f1+,f2−,f2+≠dσ1−z,dσ1+z,dσ2−z,dσ2+z+dσ1−y,dσ1+y,dσ2−y,dσ2+y−2f1−,f1+,f2−,f2+⇒dσ1−w+dσ1−z−2f1−,dσ1+w+dσ1+z−2f1+,dσ2−w+dσ2−z−2f2−,dσ2+w+dσ2+z−2f2+≠dσ1−z+dσ1−y−2f1−,dσ1+z+dσ1+y−2f1+,dσ2−z+dσ2−y−2f2−,dσ2+z+dσ2+y−2f2+⇒dσ1−w+dσ1−z−2μ1−wz,dσ1+w+dσ1+z−μ1+wz,dσ2−w+dσ2−z−2μ2−wz,dσ2+w+dσ2+z−2μ2+wz≠dσ1−z+dσ1−y−2μ1−zy,dσ1+z+dσ1+y−2μ1+zy,dσ2−z+dσ2−y−2μ2−zy,dσ2+z+dσ2+y−2μ2+zy⇒dμ1−wz,dμ1+wz,dμ2−wz,dμ2+wz≠dμ1−zy,dμ1+zy,dμ2−zy,dμ2+zy⇒dGwz≠dGzy.
Therefore, neighbor edges have VDs, if each node neighbor to the nodes has VDs. Hence,G is a HIIVIFG, if G is a NEIIVIFG.Theorem 23.
SupposeG is a CIVIFG and μ is a CF. Then, G is HI-IVIFG if and only if G is NETI-IVIFG.Proof.
It is clear.Theorem 24.
LetG be an IVIFG on G∗, a star K1,n. Then, G is a TER-IVIFG. If the degrees of IVM and IV-NM of no two edges are similar, then G is a NEI-IVIFG.Proof.
Supposev1,v2,v3,⋯,vn are the nodes neighbor to the node x. Assume e1,e2,e3,⋯,en are the edges of a star G∗ in that order having degrees of IVM m1,m2,m3,⋯,mn and degrees of IV-NM n1,n2,n3,⋯,nn which mi=mi−,mi+ and ni=ni−,ni+, for i=1,2 so that 0≤mi++ni+≤1, for each 1≤i≤n.
Then,(5)tdGei=tdμ1−ei,tdμ1+ei,tdμ2−ei,tdμ2+ei=dμ1−ei+μ1−ei,dμ1+ei+μ1+ei,dμ2−ei+μ2−ei,dμ2+ei+μ2+ei=∑k=1nmk−−mi−+mi−,∑k=1nmk+−mi++mi+,∑k=1nnk−−ni−+ni−,∑k=1nnk+−ni++ni+=∑k=1nmk−,∑k=1nmk+,∑k=1nnk−,∑k=1nnk+.
All edgesei, 1≤i≤n, have same TDs. Hence, G is a TER-IVIFG. Now, if mi−≠mj−, mi+≠mj+, ni−≠nj−, and ni+≠nj+, for each 1≤i,j≤n, then we have the following:
(6)dGei=dμ1−ei,dμ1+ei,dμ2−ei,dμ2+ei=dσ1−x+dσ1−vi−2μ1−xvi,dσ1+x+dσ1+vi−2μ1+xvi,dσ2−x+dσ2−vi−2μ2−xvi,dσ2+x+dσ2+vi−2μ2+xvi=∑k=1nmk−+mi−−2mi−,∑k=1nmk++mi+−2mi+,∑k=1nnk−+ni−−2ni−,∑k=1nnk++ni+−2ni+=∑k=1nmk−−mi−,∑k=1nmk+−mi+,∑k=1nnk−−ni−,∑k=1nnk+−ni+,foreach1≤i≤n.
Therefore, all edgesei,1≤i≤n, have VDs. Hence, G is a NEI-IVIFG.Definition 25.
LetG be a CIVIFG on G∗. Then, G is called to be a:
(i)
SEI-IVIFG if each pair of edges has VDs(ii)
SETI-IVIFG if each pair of edges has various TDsExample 26.
Graph that is both SEI-IVIFG and SETI-IVIFG.
LetG be a CIVIFG on G∗ which is a cycle of length five.From Figure6, dGu=0.5,0.8,0.5,0.8, dGv=0.3,0.5,0.7,1.1, dGw=0.5,0.7,0.6,0.9, dGx=0.6,0.9,0.5,0.7, dGy=0.7,1.1,0.3,0.5, dGuv=0.6,0.9,0.4,0.7, dGvw=0.4,0.6,0.7,1.0, dGwx=0.5,0.8,0.5,0.8, dGxy=0.7,1.0,0.4,0.6, and dGyu=0.4,0.7,0.6,0.9.Figure 6
G is both SEI-IVIFG and SETI-IVIFG.Thus,G is a SEI-IVIFG.tdGuv=0.7,1.1,0.8,1.3, tdGvw=0.6,0.9,1.0,1.5, tdGwx=0.8,1.2,0.8,1.2, tdGxy=1.0,1.5,0.6,0.9, and tdGyu=0.8,1.3,0.7,1.1.The above calculations show that each edge has various TD. Therefore,G is a SETI-IVIFG.So,G is both SEI-IVIFG and SETI-IVIFG.Example 27.
SEI-IVIFG needs not be SETI-IVIFG.
LetG be an IVIFG so that G∗, a cycle of length three.From Figure7, dGu=0.3,0.5,0.9,1.3, dGv=0.5,0.7,0.8,1.2, dGw=0.4,0.6,0.7,1.1, dGuv=0.4,0.6,0.7,1.1, dGvw=0.3,0.5,0.9,1.3, dGwu=0.5,0.7,0.8,1.2, and tdGuv=tdGvw=tdGwu=0.6,0.9,1.2,1.8.Figure 7
G is SEI-IVIFG but it is not SETI-IVIFG.Note thatG is SEI-IVIFG, since each pair of edges has VDs. Also, G is not SETI-IVIFG, since all the edges have same TDs. Hence, SEI-IVIFG needs not to be SETI-IVIFG.Example 28.
SETI-IVIFG needs not to be SEI-IVIFG.
SupposeG is an IVIFG so that G∗, a cycle of length four.From Figure8, dGu=0.7,1.1,0.4,0.6, dGv=0.4,0.8,0.7,0.9, dGw=0.8,1.2,0.6,0.8, dGx=1.1,1.5,0.3,0.5, dGuv=dGwx=0.9,1.3,0.5,0.7, dGvw=dGxu=0.6,1.0,0.5,0.7, tdGuv=1.0,1.6,0.8,1.1, tdGvw=0.9,1.5,0.9,1.2, tdGwx=1.4,2.0,0.7,1.0, and dGxu=1.2,1.8,0.6,0.9.Figure 8
G is SETI-IVIFG but it is not SEI-IVIFG.It is noted thatdGuv=dGwx. So, G is not SEI-IVIFG.ButG is SETI-IVIFG, since tdGuv≠tdGvw≠tdGwx≠tdGxu. Hence, SETI-IVIFG needs not to be SEI-IVIFG.Remark 29.
LetG be a CIVIFG on G∗. If G is both SEI-IVIFG and SETI-IVIFG, then μ needs not to be a CF.Example 30.
ConsiderG be an IVIFG so that G∗ is a cycle of length five.From Figure9, dGu=0.5,0.8,0.5,0.8, dGv=0.3,0.5,0.7,1.1, dGw=0.5,0.7,0.6,0.9, dGx=0.6,0.9,0.5,0.7, and dGy=0.7,1.1,0.3,0.5. Also, dGuv=0.6,0.9,0.4,0.7, dGvw=0.4,0.6,0.7,1.0, dGwx=0.5,0.8,0.5,0.8, dGxy=0.7,1.0,0.4,0.6, and dGyu=0.4,0.7,0.6,0.9.Figure 9
μ is not a CF.Clearly, each edge inG has VD. Therefore, G is a SEI-IVIFG.Also,tdGuv=0.7,1.1,0.8,1.3, tdGvw=0.6,0.9,1.0,1.5, tdGwx=0.8,1.2,0.8,1.2, tdGxy=1.0,1.5,0.6,0.9, and tdGyu=0.8,1.3,0.7,1.1.Thus, each edge inG has various TD. So, G is a SETI-IVIFG. Hence, G is both SEI-IVIFG and SETI-IVIFG. But μ is not a CF.Theorem 31.
LetG be an IVIFG on G∗. If G is a SEI-IVIFG, then G is a NEI-IVIFG.Proof.
LetG be an IVIFG on G∗ that is SEI-IVIFG. Then, each edge in G has VD. Thus, each neighbor edge has VD. So, G is a NEI-IVIFG.Theorem 32.
LetG be an IVIFG on G∗. If G is a SETI-IVIFG, then G is a NETI-IVIFG.Proof.
LetG be an IVIFG on G∗ that is SETI-IVIFG. Then, each pair of edges in G has various TDs. Hence, each pair of neighbor edges has various TDs. Therefore, G is a NETI-IVIFG.Remark 33.
The inverse of Theorems31 and 32 is not generally true.Example 34.
LetG be an IVIFG so that G∗ is a path with four nodes.From Figure10, dGu=dGx=0.2,0.3,0.4,0.5, dGv=dGw=0.3,0.5,0.7,0.9, dGuv=0.1,0.2,0.3,0.4, dGvw=0.4,0.6,0.8,1.0, dGwx=0.1,0.2,0.3,0.4, tdGuv=0.3,0.5,0.7,0.9, tdGvw=0.5,0.8,1.1,1.4, and tdGwx=0.3,0.5,0.7,0.9.Figure 10
G is NEI and NETI-IVIFG but it is not SEI and SETI-IVIFG.Here,dGuv≠dGvw and dGvw≠dGwx. Hence, G is a NEI-IVIFG. But G is not a SEI-IVIFG, since dGuv≠dGwx. Also, note that tdGuv≠tdGvw and tdGvw≠tdGwx. So, G is a NETI-IVIFG. But G is not a SETI-IVIFG, since tdGuv≠tdGwx.


## 4. Application of IVIF Influence Digraph to Find the Most Effective Person in a Hospital

Serving the people has always been an important duty of any government, and this has also played a significant role in the growth and prosperity of any country, because if the people are satisfied with the government of their country, then they will perform their duties in the best possible way. As a result, a healthier society will be formed with more progress. One of these very important services is taking care of people’s health. Hospitals and medical centers must also serve patients in the best possible way and not neglect to admit and treat patients. But a very important issue that can be important in the service and treatment of patients in the fastest possible time is the rapid and correct management of hospital wards and health centers. If a manager can properly guide the staff under her supervision and give them the necessary training to treat patients, then the service will be provided in the best possible way. Therefore, in this section, we try to introduce the most effective staff of a hospital with the help of an IVIFG. To do this, we consider the nodes of this influence graph as the staff of a hospital and the edges as the influence of one employee on another employee. The names of the staff and their specialization in the hospital are shown in Table2. For this hospital, the staff is as follows:
(7)E=Jafari,Mohseni,Alavi,Samie,Rezai,Ghoreishi,Khorami.(i)
Mohseni has been working with Samie for 12 years and values his views on issues(ii)
Rezai has been the head of the hospital, and not only Mohseni but also Samie is very satisfied with Rezai’s performance(iii)
The safety of hospital staff and also the care of hospital equipment is a very important issue. Mohseni is an expert for this(iv)
Alavi and Ghoreishi have a long history of conflict(v)
Samie is very effective in laboratory and radiology affairs of the hospitalTable 2
The names of the staff and their specialization in the hospital.
NameServicesJafari (Ja)Medical equipment expertMohseni (Mo)Head of security guardAlavi (Al)Head of admissionsSamie (Sa)Expert radiology and laboratoryRezai (Re)Hospital headGhoreishi (Gh)IT expertKhorami (Kh)Head of financeConsidering the above points, the influence graph can be very important. But such a graph cannot show the power of employees within a hospital and the degree of influence of employees on each other. Since power and influence do not have defined boundaries, they can be represented as an interval-valued fuzzy 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 interval-valued intuitionistic fuzzy degrees, which is very useful for influence and conflicts between employees. The interval-valued intuitionistic fuzzy set of employees is shown in Table3.Table 3
Employee power based on degree of membership and nonmembership.
JafariMohseniAlaviSamieRezaiGhoreishiKhoramiσ1−0.850.850.750.650.550.550.45σ1+0.950.950.850.750.650.650.55σ2−000.050.050.250.250.25σ2+0.050.050.150.150.350.350.35We have shown the influence of persons in the IVIF digraph with an edge. This graph is shown in Figure11, and its adjacency matrix is shown in Table 4.Figure 11
IVIF digraph (influence graph).Table 4
Adjacency matrix corresponding to Figure11.
JaMoAlSaReGhKhJa00.75,0.85,0,0.0500000Mo000.65,0.75,0,0.050000.15,0.25,0.25,0.35Al0000000.35,0.45,0.25,0.35Sa000000.45,0.55,0.25,0.350.25,0.35,0.25,0.35Re0.55,0.65,0.25,0.350.55,0.65,0.25,0.3500000.25,0.35,0.25,0.35Gh0000000.15,0.25,0.25,0.35Kh0000000Hospital staff are the vertices of the IVIF digraph of Figure11, and their strength in terms of conditions, degrees of IVM, and IV-NM is that it can also be expressed as a percentage. For example, Mr. Alavi has 75% to 85% of power and between 5% and 15% does not have this power. Also, the edges of this graph indicate the influence of one person on another. Degree of IVM and IV-NM can be described in terms of positive and negative influence. For example, between 55% and 65% of the time, Mr. Jafari is influenced by Mr. Rezaei’s thoughts and ideas, but 25% to 35% of the time he is not influenced by his opinions.In Figure11, it is clear that Mr. Rezaei controls both the medical equipment experts: Mr. Jafari and the head of security guard, Mr. Mohseni. His influence on both of them is the same. Because the IV membership rate in both of them is (0.55, 0.65), i.e., 55% to 65%. But in the case of Mr. Rezaei and Mr. Mohseni, the degree of doubt is between 0% and 20% because
(8)1−0.65−0.35,1−0.55−0.25=0,0.20,and in the case of Mr. Jafari and Mohseni, it is between 10% and 25% because
(9)1−0.85−0.05,1−0.75−0=0.10,0.25.The implication is that Mr. Jafari is more skeptical than Mr. Rezaei. Clearly, Mr. Rezaei has the most influence in the organization, because he dominates both the equipment expert and the security guard; these two people have the most power in the hospital, i.e., between 85% and 95%.


## 5. Conclusions

Interval-valued intuitionistic fuzzy 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. With the help of IVIFGs, the most effective person in an organization can be identified according to the important factors that can be useful for an institution. Therefore, in this research, some types of EI-IVIFGs such as NETI-IVIFGs, SEI-IVIFGs, and SETI-IVIFGs are introduced. A comparative study between NEI-IVIFGs and NETI-IVIFGs is presented. Finally, an application of IVIF influence digraph has presented. In our future work, we will introduce connectivity index, Winer index, and Randic index in interval-valued intuitionistic fuzzy graphs and investigate some of their properties. Also, we will investigate some types of energy, including Laplacian and skew Laplacian in both interval-valued intuitionistic fuzzy graphs and interval-valued intuitionistic fuzzy digraphs.



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