# An Application of Product of Intuitionistic Fuzzy Incidence Graphs in Textile Industry

Authors: Irfan Nazeer; Tabasam Rashid; Abazar Keikha
Journal: Complexity
 (2021)

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## Abstract

In this research article, we presented the idea of intuitionistic fuzzy incidence graphs (IFIGs) along with their certain properties. The number of operations including Cartesian product (CP), composition, tensor product, and normal product in an IFIGs are also investigated. The method to compute the degree of IFIGs obtained by CP, composition, tensor product, and the normal product is discussed. Some important theorems to calculate the degree of the vertices of IFIGs acquired by CP, composition, tensor product, and normal product are elaborated. An application of CP and composition of two IFIGs in the textile industry to find the best combinations of departments expressing the highest percentage of progress and the lowest percentage of nonprogress is provided. A comparative analysis of our study with the existing study is discussed. Our study will be beneficial to comprehend and understand the further characteristics of IFIGs in detail. Another advantage of our study is that it will be helpful to find the maximum percentage of progress and minimum percentage of nonprogress in different departments of universities, garment factories, and hospitals.


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## Body

## 1. Introduction

Zadeh [1] initiated the idea of a fuzzy set (FS) to resolve difficulties in dealing with uncertainties. Since then, the theory of FSs and fuzzy logic have been examined by many researchers to solve many real-life problems involving an ambiguous and uncertain environment. FS theory becomes a sturdy area in multiple disciplines including mathematics, computer science, and signal processing. FSs were not without flaws, they only talked about the membership function (MSF) and missed the nonmembership function (NMSF). This drawback in FSs leads Attnassov [2] to present the idea of the intuitionistic fuzzy set (IFS) as a generalization of FS that handles unsure situations in a good way as its structure is not restricted to membership (MS) grades only. The concept of IFS is a preferable tool to use due to its sundry structure explaining MS as well as nonmembership (NMS) grades of an element. IFSs have been outstandingly used in different areas of life. De et al. [3] presented some applications of IFSs in medical diagnosis. Xu [4] defined some aggregation operators.A graph is an easy way of expressing information including an association between entities. The entities are shown by vertices and relations by edges. In different problems, we get incomplete information about the problem. So, there is blurriness in the explanation of the entities or their relationship or both. To tackle this form of problem, we need to design a fuzzy graph (FG) model. Zadeh’s FS provided a strong ground for the theory of FGs which have been proposed by Rosenfeld [5]. The study of FGs leads many researchers to contribute in this field; for example, Mordeson and Chang-Shyh [6] explained the various kinds of operations like composition, union, and join on FGs. Pal et al. [7] established the degree and total degree of an edge in interval-valued FGs. Some properties of highly irregular interval-valued FGs were initiated by Rashmanlou and Pal [8]. Jan et al. [9] discussed some root-level modifications in interval-valued FGs. Mathew et al. [10] introduced vertex rough graphs. Certain types of FGs such as fuzzy threshold graphs, fuzzy tolerance graphs, and Bipolar fuzzy hypergraphs were given by Samanta and Pal [11–13]. For more comprehensive and detailed work on graphs and FGs, we may refer to the reader [14–16].Parvathi and Karunambigai [17] have developed the idea of intuitionistic fuzzy graphs (IFGs). Like IFSs, some outstanding work on the theory of IFGs is also being done. Parvathi et al. [18] discussed different operations on IFGs including CP and composition. Gani and Begum [19] talked about the degree, order, and size of IFGs. Direct product, semistrong product, and strong product in IFGs were discussed by Sahoo and Pal [20]. Sahoo and Pal [21, 22] studied intuitionistic fuzzy competition graphs and intuitionistic fuzzy tolerance graphs with applications. Various kinds of products like Cartesian, tensor, normal, and their degrees in IFGs were provided by Sahoo and Pal [23]. Sahoo et al. [24] initiated new ideas in IFGs with application in the water supply system. Bozhenyuk et al. [25] presented an idea of minimal intuitionistic dominating vertex subset of an IFG.FGs and IFGs do not talk about the influence of vertices on the edges. This lack in these graphs was caused to introduce an idea of fuzzy incidence graphs (FIGs). For example, if vertices show different residence societies and edges show roads joining these residence societies, we can have a FG expressing the extent of traffic from one society to another. The society that has the maximum number of residents will have maximum ramps in society. So, if c and d are two societies and cd is a road joining them then (c, cd) could express the ramp system from the road cd to the society c. In the case of an unweighted graph, both c and d will have an influence of 1 on cd. In a directed graph, the influence of c on cd represented by (c, cd) is 1 whereas (d, cd) is 0. This concept is generalized through FIGs. Dinesh [26] presented the notion of FIGs. Malik et al. [27, 28] utilized FIGs in human trafficking. The idea of cut-pairs, fuzzy incidence trees, and strong pairs in FIGs was proposed by Mathew and Mordeson [29]. Fang et al. [30] presented an idea to find the connectivity and Wiener index of FIG.IFGs and different types of operations for IFGs including CP, composition, tensor product (TP), and normal product (NP) exist in literature but these operations are unknown for IFIGs. This motivates us to introduce IFIGs and these operations for IFIGs. IFIGs are more beneficial than FIGs due to the availability of the NMSF in IFIGs. These innovative ideas will open a new door for many researchers to study IFIGs in detail. The remainder of this article is formulated as follows: Section2 provides some preliminary results which are required to understand the remaining part of the article. The method to find the degree of a vertex in CP and composition is discussed in Section 3. In Section 4, the idea to find the degree of a vertex in TP and NP is explained. The idea of accurate domination in FIGs is presented in Section 5. A real-life application of CP and composition in the textile industry is provided in Section 5. A comparative analysis of our study with the existing study is provided in Section 6. Conclusions and prospects are elaborated in Section 7.


## 2. Preliminaries

A fuzzy subset (FSS)μ on a set M is a map μ:M⟶0,1. A map ν:M×M⟶0,1 is known as a fuzzy relation on μ if νa,b≤minμa,μb for each a,b∈M. A FG is a pair G=μ,ν, where μ is a FSS on a set V and ν is a fuzzy relation on μ.Definition 1 (see [23]).
An IFSA on the set X is characterized by a mapping e:X⟶0,1, which is named as a MF and f:X⟶0,1, which is said to be NMF.Definition 2 (see [23]).
An IFG is of the formG=V,E,μ,ν where μ=μ1,μ2, ν=ν1,ν2, and V=a0,a1,…,an such that μ1:V⟶0,1 and μ2:V⟶0,1 represent the degree of MS and NMS of the vertex vi∈V, respectively, and 0≤μ1+μ2≤1 for each ai∈Vi=1,2,…,n. Also, ν1:V×V⟶0,1 and ν2:V×V⟶0,1; ν1ai,aj and ν2ai,aj show the degree of MS and NMS of the edge ai,aj, respectively, such that ν1ai,aj≤∧μ1ai,μ1aj and ν2ai,aj≤∨μ2ai,μ2aj, 0≤ν1ai,aj+ν2ai,aj≤1 for every ai,aj.Example 1.
Here, we include a daily life example of six different countries. As an illustrative case, consider a network (IFG) of six vertices indicating the six different countries. The MS value of the vertices represents the percentage of people who are educated and the NMS value of the vertices indicates the percentage of those people who are uneducated. The MS value of the edges expresses the cooperation of one country with another country to enhance the percentage of educated people and the NMS value denotes the noncooperation with each other.
Figure1 shows an IFG G=V,E,μ,ν with V=p,q,r,s,u,v and(1)μp=0.5,0.3,μq=0.2,0.3,μr=0.3,0.1,μs=0.4,0.3,μu=0.6,0.4,μv=0.3,0.5,νp,q=0.2,0.3,νp,s=0.4,0.3,νp,v=0.3,0.5,νq,r=0.2,0.3,νq,v=0.2,0.5,νr,s=0.3,0.3,νr,u=0.3,0.4,νs,u=0.4,0.4.
In this paper, the minimum operator is expressed by∧ and the maximum operator is represented by ∨.Figure 1
An IFG indicating the network of six different countries.Definition 3 (see [23]).
LetG=V,E,μ,ν be an IFG and c∈V; then, its degree is represented by dGc=d1Gc,d2Gc and defined by d1Gc=∑c≠dν1c,d=∑c,d∈Eν1c,d and d2Gc=∑c≠dν2c,d=∑c,d∈Eν2c,d.Definition 4 (see [26]).
AssumeG=V,E is a graph. Then, G=V,E,I is named as an incidence graph (IG), where I⊆V×E.Example 2.
An IG is shown in Figure2. a,ab,b,ab,b,bc,c,bc,a,ac,c,ac are called incidence pairs (IPs) or pairs of an IG G=V,E,I.Figure 2
Incidence graph.Definition 5 (see [26]).
AssumeG=V,E is a graph, μ is a FSS of V, and ν is a FSS of V×V. Let ψ be a FSS of V×E. If ψa,ab≤μa∧νab for every a∈V,ab∈E, then ψ is a fuzzy incidence (FI) of G.Definition 6 (see [26]).
AssumeG is a graph and μ,ν is a fuzzy subgraph of G. If ψ is a FI of G, then G˜=μ,ν,ψ is named as FIG of G.Example 3.
A FIG withμa=0.8,μb=0.3,μc=0.1,νab=0.1,νbc=0.1 and ψa,ab=0.1,ψb,ab=0.1,ψb,bc=0.05,ψc,bc=0.07 is shown in Figure 3.Figure 3
A figure.


## 3. Degree of a Vertex of CP and Composition of Intuitionistic Fuzzy Incidence Graphs

In this section, we define the IFIG, CP, and composition of two IFIGs. We also define the degree of a vertex of these products on IFIGs. Theorems are also provided to calculate the degree of a vertex in CP and composition.Definition 7.
An IFIG is of the formG=V,E,I,μ,ν,ψ where μ=μ1,μ2, ν=ν1,ν2, ψ=ψ1,ψ2, and V=a0,a1,…,an such that μ1:V⟶0,1 and μ2:V⟶0,1 represent the degree of MS and NMS of the vertex ai∈V, respectively, and 0≤μ1+μ2≤1 for each ai∈Vi=1,2,…,nν1:V×V⟶0,1 and ν2:V×V⟶0,1; ν1ai,aj and ν2ai,aj show the degree of MS and NMS of the edge ai,aj, respectively, such that ν1ai,aj≤∧μ1ai,μ1aj and ν2ai,aj≤∨μ2ai,μ2aj, 0≤ν1ai,aj+ν2ai,aj≤1 for every ai,ajψ1:V×E⟶0,1 and ψ2:V×E⟶0,1; ψ1ai,aiaj and ψ2ai,aiaj show the degree of MS and NMS of the incidence pair, respectively, such that ψ1ai,aiaj≤∧μ1ai,ν1aiaj and ψ2ai,aiaj≤∧μ2ai,ν2aiaj, 0≤ψ1ai,aiaj+ψ2ai,aiaj≤1 for every ai,aiaj.Example 4.
Here, we include a daily life example of three different shopping malls. As an illustrative case, consider a network (IFIG) of three vertices indicating the three different branches of a shopping mall. The MS value of the vertices represents the number of people who visit the shopping mall for shopping and the NMS value of the vertices indicates those people who do not come to these shopping malls for shopping. The MS value of the edges expresses their cooperation with each other and the NMS value denotes the noncooperation with each other. The MS value of the IPs shows the percentage of profit and the NMS value indicates the percentage of nonprofit of these shopping malls.
LetG=V,E,I,μ,ν,ψ be an IFIG provided in Figure 4, expressing a network of three branches of a shopping mall where μa=μ1a,μ2a, νa,b=ν1a,b,ν2a,b, and ψa,ab=ψ1a,ab,ψ2a,ab. Let V=a,b,c and μa=0.2,0.6,μb=0.1,0.1,μc=0.3,0.1, νa,b=0.1,0.1,νb,c=0.1,0.1,νa,c=0.2,0.1, and ψa,ab=0.1,0.1,ψb,ab=0.1,0.1,ψb,bc=0.1,0.1,ψc,bc=0.1,0.05,ψa,ac=0.1,0.07,ψc,ac=0.1,0.04.Figure 4
A network of different branches of the shopping mall.Definition 8.
ConsiderG=V,E,I,μ,ν,ψ is an IFIG and a∈V; then, its degree is expressed by dGa=d1Ga,d2Ga and defined by d1Ga=Σa≠b,a,ab∈I and d2Ga=Σa≠b,a,ab∈I.
Now, we are going to define the CP of two IFIGs and how to find the degree of these graphs.Definition 9.
The CP of two IFIGsGi=Vi,Ei,Ii,μi,νi,ψi and Gii=Vii,Eii,Iii,μii,νii,ψii is defined as an IFIG G=Gi×Gii=V,E,I,μi×μii,νi×νii,ψi×ψii where V=Vi×Vii, E=x1,y1,x2,y2|x1=x2,y1,y2∈Eii or y1=y2,x1,x2∈Ei, and I=x1,y1,x1,y1x1,y2|x1=x2,y1,y1y2∈Iii,y2,y1y2∈Iii or y1=y2,x1,x1x2∈Ii,x2,x1x2∈Ii with μ1i×μ1iix1,y1=μ1ix1∧μ1iiy1,∀x1,y1∈Vi×Vii, μ2i×μ2iix1,y1=μ2ix1∨μ2iiy1,∀x1,y1∈Vi×Vii.(2)ν1i×ν1iix1,y1x2,y2=μ1ix1∧ν1iiy1,y2,ifx1=x2,y1,y2∈Eii,ν1ix1,x2∧μ1iiy1,ify1=y2,x1,x2∈Ei,ν2i×ν2iix1,y1x2,y2=μ2ix1∨ν2iiy1,y2,ifx1=x2,y1,y2∈Eii,ν2ix1,x2∨μ2iiy1,ify1=y2,x1,x2∈Ei,ψ1i×ψ1iix1,y1,x1,y1x1,y2=μ1ix1∧ψ1iiy1,y1y2,ifx1=x2,y1,y1y2∈Iii,ψ1i×ψ1iix1,y2,x1,y1x1,y2=μ1ix1∧ψ1iiy2,y1y2,ifx1=x2,y2,y1y2∈Iii,ψ1i×ψ1iix1,y1,x1,y1x2,y1=ψ1ix1,x1x2∧μ1iiy1,ify1=y2,x1,x1x2∈Ii,ψ1i×ψ1iix2,y1,x1,y1x2,y1=ψ1ix2,x1x2∧μ1iiy1,ify1=y2,x2,x1x2∈Ii,ψ1i×ψ1iix1,y2,x1,y2x2,y2=ψ1ix1,x1x2∧μ1iiy2,ify1=y2,x1,x1x2∈Ii,ψ1i×ψ1iix2,y2,x1,y2x2,y2=ψ1ix2,x1x2∧μ1iiy2,ify1=y2,x2,x1x2∈Ii,ψ1i×ψ1iix2,y1,x2,y1x2,y2=ψ1iiy1,y1y2∧μ1ix2,ifx1=x2,y1,y1y2∈Iii,ψ1i×ψ1iix2,y2,x2,y1x2,y2=ψ1iiy2,y1y2∧μ1ix2,ifx1=x2,y2,y1y2∈Iii,ψ2i×ψ2iix1,y1,x1,y1x1,y2=μ2ix1∨ψ2iiy1,y1y2,ifx1=x2,y1,y1y2∈Iii,ψ2i×ψ2iix1,y2,x1,y1x1,y2=μ2ix1∨ψ2iiy2,y1y2,ifx1=x2,y2,y1y2∈Iii,ψ2i×ψ2iix1,y1,x1,y1x2,y1=ψ2ix1,x1x2∨μ2iiy1,ify1=y2,x1,x1x2∈Ii,ψ2i×ψ2iix2,y1,x1,y1x2,y1=ψ2ix2,x1x2∨μ2iiy1,ify1=y2,x2,x1x2∈Ii,ψ2i×ψ2iix1,y2,x1,y2x2,y2=ψ2ix1,x1x2∨μ2iiy2,ify1=y2,x1,x1x2∈Ii,ψ2i×ψ2iix2,y2,x1,y2x2,y2=ψ2ix2,x1x2∨μ2iiy2,ify1=y2,x2,x1x2∈Ii,ψ2i×ψ2iix2,y1,x2,y1x2,y2=ψ2iiy1,y1y2∨μ2ix2,ifx1=x2,y1,y1y2∈Iii,ψ2i×ψ2iix2,y2,x2,y1x2,y2=ψ2iiy2,y1y2∨μ2ix2,ifx1=x2,y2,y1y2∈Iii.Definition 10.
LetG=Gi×Gii=V,E,I,μi×μii,νi×νii,ψi×ψii be the CP of two IFIGs Gi=Vi,Ei,Ii,μi,νi,ψi and Gii=Vii,Eii,Iii,μii,νii,ψii. Then, the degree of V=x1,y1 is expressed by dGi×Giix1,y1=d1Gi×Giix1,y1,d2Gi×Giix1,y1 and defined by(3)d1Gi×Giix1,y1=∑x1=x2,y1,y1y2∈Iiiμ1ix1∧ψ1iiy1,y1y2+∑y1=y2,x1,x1x2∈Iiψ1ix1,x1x2∧μ1iiy1,d2Gi×Giix1,y1=∑x1=x2,y1,y1y2∈Iiiμ2ix1∨ψ2iiy1,y1y2+∑y1=y2,x1,x1x2∈Iiψ2ix1,x1x2∨μ2iiy1.Theorem 1.
LetGi=Vi,Ei,Ii,μi,νi,ψi and Gii=Vii,Eii,Iii,μii,νii,ψii be two IFIGs. If μ1i≥ψ1ii,μ2i≤ψ2ii and μ1ii≥ψ1i,μ2ii≤ψ2i, then dGi×Giix1,y1=dGix1+dGiiy1.Proof.
In CP, by the definition of the degree of a vertex, we have(4)d1Gi×Giix1,y1=∑x1=x2,y1,y1y2∈Iiiμ1ix1∧ψ1iiy1,y1y2+∑y1=y2,x1,x1x2∈Iiψ1ix1,x1x2∧μ1iiy1,=∑y1,y1y2∈Iiiψ1iiy1,y1y2+∑x1,x1x2∈Iiψ1ix1,x1x2.
Sinceμ1i≥ψ1ii and μ1ii≥ψ1i,(5)=∑x1,x1x2∈Iiψ1ix1,x1x2+∑y1,y1y2∈Iiiψ1iiy1,y1y2=d1Gix1+d1Giiy1,d2Gi×Giix1,y1=∑x1=x2,y1,y1y2∈Iiiμ2ix1∨ψ2iiy1,y1y2+∑y1=y2,x1,x1x2∈Iiψ2ix1,x1x2∨μ2iiy1=∑y1,y1y2∈Iiiψ2iiy1,y1y2+∑x1,x1x2∈Iiψ2ix1,x1x2.
Sinceμ2i≤ψ2ii and μ2ii≤ψ2i,(6)=∑x1,x1x2∈Iiψ2ix1,x1x2+∑y1,y1y2∈Iiiψ2iiy1,y1y2=d2Gix1+d2Giiy1.
Hence,dGi×Giix1,y1=dGix1+dGiiy1.Example 5.
In Figures5 and 6,μ1i≥ψ1ii,μ2i≤ψ2ii, and μ1ii≥ψ1i,μ2ii≤ψ2i. Then, by Theorem 1, we have(7)d1Gi×Giix1,y1=d1Gix1+d1Giiy1=0.2+0.1=0.3,d2Gi×Giix1,y1=d2Gix1+d2Giiy1=0.4+0.4=0.8.
So,dGi×Giix1,y1=0.3,0.8. In a similar manner, we can examine the degree of all the vertices of CP of Figures 5 and 6 provided in Figure 7.
Next, the method to explore the degree of a vertex in the composition of two IFIGs is provided.Figure 5
Gi.Figure 6
Gii.Figure 7
CP of Figures5 and 6.Definition 11.
The composition of two IFIGsGi=Vi,Ei,Ii,μi,νi,ψi and Gii=Vii,Eii,Iii,μii,νii,ψii is defined as an IFIG G=GiGii=V,E,I,μi⊙μii,νi⊙νii,ψi⊙ψii where V=Vi×Vii, E=x1,y1,x2,y2|x1=x2,y1,y2∈Eii or y1=y2,x1,x2∈Ei∪x1,y1,x2,y2|y1≠y2,x1,x2∈Ei, and I=x1,y1,x1,y1x1,y2|x1=x2,y1,y1y2∈Iii,y2,y1y2∈Iii or y1=y2,x1,x1x2∈Ii,x2,x1x2∈Ii∪x1,y1,x2,y2|y1≠y2,x1,x1x2∈Ii,x2,x1x2∈Ii with(8)μ1i⊙μ1iix1,y1=μ1ix1∧μ1iiy1,∀x1,y1∈Vi⊙Vii,μ2i⊙μ2iix1,y1=μ2ix1∨μ2iiy1,∀x1,y1∈Vi⊙Vii,ν1i⊙ν1iix1,y1x2,y2=μ1ix1∧ν1iiy1,y2,ifx1=x2,y1,y2∈Eii,ν1ix1,x2∧μ1iiy1,ify1=y2,x1,x2∈Ei,ν1ix1,x2∧μ1iiy1∧μ1iiy2,ify1≠y2,x1,x2∈Ei,ν2i⊙ν2iix1,y1x2,y2=μ2ix1∨ν2iiy1,y2,ifx1=x2,y1,y2∈Eii,ν2ix1,x2∨μ2iiy1,ify1=y2,x1,x2∈Ei,ν1ix1,x2∨μ1iiy1∨μ1iiy2,ify1≠y2,x1,x2∈Ei,ψ1i⊙ψ1iix1,y1,x1,y1x1,y2=μ1ix1∧ψ1iiy1,y1y2,ifx1=x2,y1,y1y2∈Iii,ψ1i⊙ψ1iix1,y2,x1,y1x1,y2=μ1ix1∧ψ1iiy2,y1y2,ifx1=x2,y2,y1y2∈Iii,ψ1i⊙ψ1iix1,y1,x1,y1x2,y1=ψ1ix1,x1x2∧μ1iiy1,ify1=y2,x1,x1x2∈Ii,ψ1i⊙ψ1iix2,y1,x1,y1x2,y1=ψ1ix2,x1x2∧μ1iiy1,ify1=y2,x2,x1x2∈Ii,ψ1i⊙ψ1iix1,y1,x1,y1x2,y2=ψ1ix1,x1x2∧μ1iiy1∧μ1iiy2,ifx1≠x2,y1≠y2,x1,x1x2∈Ii,ψ1i⊙ψ1iix2,y2,x1,y1x2,y2=ψ1ix2,x1x2∧μ1iiy1∧μ1iiy2,ifx1≠x2,y1≠y2,x2,x1x2∈Ii,ψ1i⊙ψ1iix1,y2,x1,y2x2,y1=ψ1ix1,x1x2∧μ1iiy1∧μ1iiy2,ifx1≠x2,y1≠y2,x1,x1x2∈Ii,ψ1i⊙ψ1iix2,y1,x1,y2x2,y1=ψ1ix2,x1x2∧μ1iiy1∧μ1iiy2,ifx1≠x2,y1≠y2,x2,x1x2∈Ii,ψ2i⊙ψ2iix1,y1,x1,y1x1,y2=μ2ix1∨ψ2iiy1,y1y2,ifx1=x2,y1,y1y2∈Iii,ψ2i⊙ψ2iix1,y2,x1,y1x1,y2=μ2im1∨ψ2iiy2,y1y2,ifx1=x2,y2,y1y2∈Iii,ψ2i⊙ψ2iix1,y1,x1,y1x2,y1=ψ2ix1,x1x2∨μ2iiy1,ify1=y2,x1,x1x2∈Ii,ψ2i⊙ψ2iix2,y1,x1,y1x2,y1=ψ2ix2,x1x2∨μ2iiy1,ify1=y2,x2,x1x2∈Ii,ψ2i⊙ψ2iix1,y1,x1,y1x2,y2=ψ2ix1,x1x2∨μ1iiy1∨μ1iiy2,ifx1≠x2,y1≠y2,x1,x1x2∈Ii,ψ2i⊙ψ2iix2,y2,x1,y1x2,y2=ψ2ix2,x1x2∨μ1iiy1∨μ1iiy2,ifx1≠x2,y1≠y2,x2,x1x2∈Ii,ψ2i⊙ψ2iix1,y2,x1,y2x2,y1=ψ2ix1,x1x2∨μ1iiy1∧μ1iiy2,ifx1≠x2,y1≠y2,x1,x1x2∈Ii,ψ2i⊙ψ2iix2,y1,x1,y2x2,y1=ψ2ix2,x1x2∨μ1iiy1∧μ1iiy2,ifx1≠x2,y1≠y2,x2,x1x2∈Ii.Definition 12.
LetG=GiGii=V,E,I,μi×μii,νi×νii,ψi×ψii be the composition of two IFIGs Gi=Vi,Ei,Ii,μi,νi,ψi and Gii=Vii,Eii,Iii,μii,νii,ψii. Then, the degree of V=x1,y1 is represented by dGiGiix1,y1=d1GiGiix1,y1,d2GiGiix1,y1 and defined by(9)d1GiGiix1,y1=∑x1=x2,y1,y1y2∈Iiiμ1ix1∨ψ1iiy1,y1y2+∑y1=y2,x1,x1x2∈Iiψ1ix1,x1x2∧μ1iiy1+∑y1≠y2,x1,x1x2∈Iiψ1ix1,x1x2∨μ1iiy1∨μ1iiy2,d2GiGiix1,y1=∑x1=x2,y1,y1y2∈Iiiμ2ix1∨ψ2iiy1,y1y2+∑y1=y2,x1,x1x2∈Iiψ2ix1,x1x2∨μ2iil1+∑y1≠y2,x1,x1x2∈Iiψ1ix1,x1x2∨μ1iiy1∨μ1iiy2.Theorem 2.
LetGi=Vi,Ei,Ii,μi,νi,ψi and Gii=Vii,Eii,Iii,μii,νii,ψii be two IFIGs. If μ1i≥ψ1ii,μ2i≤ψ2ii and μ1ii≥ψ1i,μ2ii≤ψ2i, then dGiGiix1,y1=ViidGix1+dGiiy1.Proof.
In the composition of two IFIGs by definition of degree of a vertex, we have(10)d1GiGiix1,y1=∑x1=x2,y1,y1y2∈Iiiμ1ix1∧ψ1iiy1,y1y2+∑y1=y2,x1,x1x2∈Iiψ1ix1,x1x2∧μ1iiy1+∑y1≠y2,x1,x1x2∈Iiψ1ix1,x1x2∧μ1iiy1∧μ1iiy2,=∑y1,y1y2∈Iiiψ1iiy1,y1y2+∑y1=y2,x1,x1x2∈Iiψ1ix1,x1x2+∑y1≠y2,x1,x1x2∈Iiψ1ix1,x1x2=∑y1=y2,x1,x1x2∈Iiψ1ix1,x1x2+∑y1≠y2,x1,x1x2∈Iiψ1ix1,x1x2+∑y1,y1y2∈Iiiψ1iiy1,y1y2=Viid1Gix1+d1Giiy1.d2GiGiix1,y1=∑x1=x2,y1,y1y2∈Iiiμ2ix1∨ψ2iiy1,y1y2+∑y1=y2,x1,x1x2∈Iiψ2ix1,x1x2∨μ2iiy1+∑y1≠y2,x1,x1x2∈Iiψ2ix1,x1x2∨μ1iiy1∨μ2iiy2,=∑y1,y1y2∈Iiiψ2iiy1,y1y2+∑y1=y2,x1,x1x2∈Iiψ2ix1,x1x2+∑y1≠y2,x1,x1x2∈Iiψ2ix1,x1x2=∑y1=y2,x1,x1x2∈Iiψ2ix1,x1x2+∑y1≠y2,x1,x1x2∈Iiψ2ix1,x1x2+∑y1,y1y2∈Iiiψ2iiy1,y1y2=Viid2Gix1+d2Giiy1.
This impliesdGiGiix1,y1=ViidGix1+dGiiy1.Example 6.
The composition of graphs given in Figures5 and 6 is shown in Figure 8. Here, μ1i≥ψ1ii,μ2i≤ψ2ii and μ1ii≥ψ1i,μ2ii≤ψ2i; then, by Theorem 2, d1GiGiix1,y1=Viid1Gix1+d1Giiy1=2×0.2+0.1=0.5, d2GiGiix1,y1=Viid2Gix1+d2Giiy1=2×0.4+0.4=1.2. So, dGiGiix1,y1=0.5,1.2. In the same way, we can examine the degree of the remaining vertices given in Figure 8.Figure 8
Composition of IFIGs given in Figures5 and 6.


## 4. Degree of a Vertex of TP and NP of Intuitionistic Fuzzy Incidence Graphs

This section explained the idea of the degree of a vertex in TP and NP of two IFIGs.Definition 13.
The TP of two IFIGsGi=Vi,Ei,Ii,μi,νi,ψi and Gii=Vii,Eii,Iii,μii,νii,ψii is defined as an IFIG, G=Gi⋄Gii=V,E,I,μi⋄μii,νi⋄νii,ψi⋄ψii, where V=Vi×Vii, E=x1,y1,x2,y2|x1,x2∈Ei,y1,y2∈Eii, and I=x1,y1,x1,y1x1,y2|x1,x1x2∈Ii,x2,x1x2∈Ii,y1,y1y2∈Iii,y2,y1y2∈Iii with(11)μ1i⋄μ1iix1,y1=μ1ix1∧μ1iiy1,∀x1,y1∈Vi⋄Vii,μ2i⋄μ2iix1,y1=μ2ix1∨μ2iiy1,∀x1,y1∈Vi⋄Vii,ν1i⋄ν1iix1,y1,x2,y2=ν1ix1,x2∧ν1iiy1,y2∀x1,x2∈Ei,y1,y2∈Eii,ν2i⋄ν2iix1,y1,x2,y2=ν2ix1,x2∨ν2iiy1,y2∀x1,x2∈Ei,y1,y2∈Eii,ψ1i⋄ψ1iix1,y1,x1,y1x2,y2=ψ1ix1,x1x2∧ψ1iiy1,y1y2∀x1,x1x2∈Ii,y1,y1y2∈Iii,ψ1i⋄ψ1iix2,y2,x1,y1x2,y2=ψ1ix2,x1x2∧ψ1iiy2,y1y2∀x2,x1x2∈Ii,y2,y1y2∈Iii,ψ1i⋄ψ1iix1,y2,x1,y2x2,y1=ψ1ix1,x1x2∧ψ1iiy2,y1y2∀x1,x1x2∈Ii,y2,y1y2∈Iii,ψ1i⋄ψ1iix2,y1,x1,y2x2,y1=ψ1ix2,x1x2∧ψ1iiy1,y1y2∀x2,x1x2∈Ii,y1,y1y2∈Iii,ψ2i⋄ψ2iix1,y1,x1,y1x2,y2=ψ2ix1,x1x2∨ψ2iiy1,y1y2∀x1,x1x2∈Ii,y1,y1y2∈Iii,ψ2i⋄ψ2iix2,y2,x1,y1x2,y2=ψ2ix2,x1x2∨ψ2iiy2,y1y2∀x2,x1x2∈Ii,y2,y1y2∈Iii,ψ2i⋄ψ2iix1,y2,x1,y2x2,y1=ψ2ix1,x1x2∨ψ2iiy2,y1y2∀x1,x1x2∈Ii,y2,y1y2∈Iii,ψ2i⋄ψ2iix2,y1,x1,y2x2,y1=ψ2ix2,x1x2∨ψ2iiy1,y1y2∀x2,x1x2∈Ii,y1,y1y2∈Iii.Definition 14.
LetG=Gi⋄Gii=V,E,I,μi×μii,νi×νii,ψi×ψii be the TP of two IFIGs Gi=Vi,Ei,Ii,μi,νi,ψi and Gii=Vii,Eii,Iii,μii,νii,ψii. Then, the degree of V=x1,y1 is shown by dGi⋄Giix1,y1=d1Gi⋄Giix1,y1,d2Gi⋄Giix1,y1 and defined by(12)d1Gi⋄Giix1,y1=∑x1,x1x2∈Ii,y1,y1y2∈Iiiψ1ix1,x1x2∧ψ1iiy1,y1y2,d2Gi⋄Giix1,y1=∑x1,x1x2∈Ii,y1,y1y2∈Iiiψ2ix1,x1x2∨ψ2iiy1,y1y2.Theorem 3.
LetGi=Vi,Ei,Ii,μi,νi,ψi and Gii=Vii,Eii,Iii,μii,νii,ψii be two IFIGs. If ψ1ii≥ψ1i,ψ2ii≤ψ2i, then dGi⋄Giix1,y1=dGix1 and if ψ1i≥ψ1ii,ψ2i≤ψ2ii, then dGi⋄Giix1,y1=dGiiy1.Proof.
Supposeψ1ii≥ψ1i,ψ2ii≤ψ2i; then,(13)d1Gi⋄Giix1,y1=∑x1,x1y1,x2,x2y2∈Iψ1i⋄ψ1iix1,y1x2,y2=∑x1,x1x2∈Ii,y1,y1y2∈Iiiψ1ix1,x1x2∧ψ1iiy1,y1y2=∑ψ1ix1,x1x2=d1Gix1,d2Gi⋄Giix1,y1=∑x1,x1x2∈Ii,y1,y1y2∈Iiiψ1ix1,x1x2∨ψ2iiy1,y1y2=∑ψ2ix1,x1x2=d2Gix1.
This impliesdGi⋄Giix1,y1=dGix1. In a similar way, if ψ1i≥ψ1ii,ψ2i≤ψ2ii, then dGi⋄Giix1,y1=dGiiy1.Example 7.
LetGi and Gii be two IFIGs as shown in Figures 9 and 10, and their TP is provided in Figure 11 with ψ1ii≥ψ1i,ψ2ii≤ψ2i; then, by Theorem 3, d1Gi⋄Giix1,y1=0.1=d1Gix1 and d2Gi⋄Giix1,y1=0.4=d2Gix1. Hence, dGi⋄Giix1,y1=0.1,0.4.
Finally, we discussed the way to explore the degree of a vertex in NP of two IFIGs.Figure 9
Gi.Figure 10
Gii.Figure 11
TP of IFIGs given in Figures9 and 10.Definition 15.
The NP of two IFIGsGi=Vi,Ei,Ii,μi,νi,ψi and Gii=Vii,Eii,Iii,μii,νii,ψii is defined as an IFIG G=Gi⊕Gii=V,E,I,μi⊕μii,νi⊕νii,ψi⊕ψii where V=Vi×Vii, E=x1,y1,x2,y2|x1=x2,y1,y2∈Eii or y1=y2,x1,x2∈Ei∪x1,y1,x2,y2|x1,x2∈Ei,y1,y2∈Eii, and I=x1,y1,x1,y1x1,y2|x1=x2,y1,y1y2∈Iii,y2,y1y2∈Iii or y1=y2,x1,x1x2∈Ii,x2,x1x2∈Ii∪x1,y1,x2,y2|x1,x1x2∈Ii,x2,x1x2∈Ii,y1,y1y2∈Iii,y2,y1y2∈Iii with(14)μ1i⊕μ1iix1,y1=μ1ix1∧μ1iiy1,∀x1,y1∈Vi⊕Vii,μ2i⊕μ2iix1,y1=μ2ix1∨μ2iiy1,∀x1,y1∈Vi⊕Vii,ν1i⊕ν1iix1,y1x2,y2=μ1ix1∧ν1iiy1,y2,ifx1=x2,y1,y2∈Eii,ν1ix1,x2∧μ1iiy1,ify1=y2,x1,x2∈Ei,ν1ix1,x2∧ν1iiy1,y2,ifx1≠x2,y1≠y2,x1,x2∈Eiandy1,y2∈Eii,ν2i⊕ν2iix1,y1x2,y2=μ2ix1∨ν2iiy1,y2,ifx1=x2,y1,y2∈Eii,ν2ix1,x2∨μ2iiy1,ify1=y2,x1,x2∈Ei,ν2ix1,x2∨ν2iiy1,y2,ifx1≠x2,y1≠y2,x1,x2∈Eiandy1,y2∈Eii,ψ1i⊕ψ1iix1,y1,x1,y1x1,y2=μ1ix1∧ψ1iiy1,y1y2,ifx1=x2,y1,y1y2∈Iii,ψ1i⊕ψ1iix1,y2,x1,y1x1,y2=μ1ix1∧ψ1iiy2,y1y2,ifx1=x2,y2,y1y2∈Iii,ψ1i⊕ψ1iix1,y1,x1,y1x2,y1=ψ1ix1,x1x2∧μ1iiy1,ify1=y2,x1,x1x2∈Ii,ψ1i⊕ψ1iix2,y1,x1,y1x2,y1=ψ1ix2,x1x2∧μ1iiy1,ify1=y2,x2,x1x2∈Ii,ψ1i⊕ψ1iix1,y1,x1,y1x2,y2=ψ1ix1,x1x2∧ψ1iiy1,y1y2,ifx1≠x2,y1≠y2,x1,x1x2∈Ii,y1,y1y2∈Iii,ψ1i⊕ψ1iix2,y2,x1,y1x2,y2=ψ1ix2,x1x2∧ψ1iiy2,y1y2,ifx1≠x2,y1≠y2,x2,x1x2∈Ii,y2,y1y2∈Iii,ψ1i⊕ψ1iix1,y2,x1,y2x2,y1=ψ1ix1,x1x2∧ψ1iiy2,y1y2,ifx1≠x2,y1≠y2,x1,x1x2∈Ii,y2,y1y2∈Iii,ψ1i⊕ψ1iix2,y1,x1,y2x2,y1=ψ1ix2,x1x2∧ψ1iiy1,y1y2,ifx1≠x2,y1≠y2,x2,x1x2∈Ii,y1,y1y2∈Iii,ψ2i⊕ψ2iix1,y1,x1,y1x1,y2=μ1ix1∨ψ2iiy1,y1y2,ifx1=x2,y1,y1y2∈Iii,ψ2i⊕ψ2iix1,y2,x1,y1x1,y2=μ1ix1∨ψ2iiy2,y1y2,ifx1=x2,y2,y1y2∈Iii,ψ2i⊕ψ2iix1,y1,x1,y1x2,y1=ψ2ix1,x1x2∨μ1iiy1,ify1=y2,x1,x1x2∈Ii,ψ2i⊕ψ2iix2,y1,x1,y1x2,y1=ψ2ix2,x1x2∨μ1iiy1,ify1=y2,x2,x1x2∈Ii,ψ2i⊕ψ2iix1,y1,x1,y1x2,y2=ψ2ix1,x1x2∨ψ2iiy1,y1y2,ifx1≠x2,y1≠y2,x1,x1x2∈Ii,y1,y1y2∈Iii,ψ2i⊕ψ2iix2,y2,x1,y1x2,y2=ψ2ix2,x1x2∨ψ2iiy2,y1y2,ifx1≠x2,y1≠y2,x2,x1x2∈Ii,y2,y1y2∈Iii,ψ2i⊕ψ2iix1,y2,x1,y2x2,y1=ψ2ix1,x1x2∨ψ2iiy2,y1y2,ifx1≠x2,y1≠y2,x1,x1x2∈Ii,y2,y1y2∈Iii,ψ2i⊕ψ2iix2,y1,x1,y2x2,y1=ψ2ix2,x1x2∨ψ2iiy1,y1y2,ifx1≠x2,y1≠y2,x2,x1x2∈Ii,y1,y1y2∈Iii.Definition 16.
LetG=Gi⊕Gii=V,E,I,μi⊕μii,νi⊕νii,ψi⊕ψii be the NP of two IFIGs Gi=Vi,Ei,Ii,μi,νi,ψi and Gii=Vii,Eii,Iii,μii,νii,ψii. Then, the degree of V=x1,y1 is shown by dGi⊕Giix1,y1=d1Gi⊕Giix1,y1,d2Gi⊕Giix1,y1 and defined by(15)d1Gi⊕Giix1,y1=∑x1=x2,y1,y1y2∈Iiiμ1ix1∧ψ1iiy1,y1y2+∑y1=y2,x1,x1x2∈Iiψ1ix1,x1x2∧μ1iiy1+∑x1,x1x2∈Ii,y1,y1y2∈Iiiψ1ix1,x1x2∧ψ1iiy1,y1y2,d2Gi⊕Giix1,y1=∑x1=x2,y1,y1y2∈Iiiμ2ix1∨ψ2iiy1,y1y2+∑y1=y2,x1,x1x2∈Iiψ2ix1,x1x2∨μ1iiy1+∑x1,x1x2∈Ii,y1,y1y2∈Iiiψ2ix1,x1x2∨ψ2iiy1,y1y2.Theorem 4.
LetGi=Vi,Ei,Ii,μi,νi,ψi and Gii=Vii,Eii,Iii,μii,νii,ψii be two IFIGs. If μ1i≥ψ1ii,μ2i≤ψ2ii,μ1ii≥ψ1i,μ2ii≤ψ2i,ψ1i≤ψ1ii and ψ2i≥ψ2ii, then dGi⊕Giix1,y1=ViidGix1+dGiiy1.Proof.
Letμ1i≥ψ1ii,μ2i≤ψ2ii,μ1ii≥ψ1i,μ2ii≤ψ2i,ψ1i≤ψ1ii, and ψ2i≥ψ2ii; then, we have(16)d1Gi⊕Giix1,y1=∑x1=x2,y1,y1y2∈Iiiμ1ix1∧ψ1iiy1,y1y2+∑y1=y2,x1,x1x2∈Iiψ1ix1,x1x2∧μ1iiy1+∑x1,x1x2∈Ii,y1,y1y2∈Iiiψ1ix1,x1x2∧ψ1iiy1,y1y2,=∑y1,y1y2∈Iiiψ1iiy1,y1y2+∑y1=y2,x1,x1x2∈Iiψ1ix1,x1x2+∑x1,x1x2∈Iiψ1ix1,x1x2=∑y1=y2,x1,x1x2∈Iiψ1ix1,x1x2+∑x1,x1x2∈Iiψ1ix1,x1x2+∑y1,y1y2∈Iiiψ1iiy1,y1y2=Viid1Gik1+d1Giil1,d2Gi⊕Giix1,y1=∑x1=x2,y1,y1y2∈Iiiμ2ix1∨ψ2iiy1,y1y2+∑y1=y2,x1,x1x2∈Iiψ2ix1,x1x2∨μ2iiy1+∑x1,x1x2∈Ii,y1,y1y2∈Iiiψ2ix1,x1x2∨ψ2iiy1,y1y2,=∑y1,y1y2∈Iiiψ1iiy1,y1y2+∑y1=y2,x1,x1x2∈Iiψ2ix1,x1x2+∑x1,x1x2∈Iiψ2ix1,x1x2=∑y1=y2,x1,x1x2∈Iiψ2ix1,x1x2+∑x1,x1x2∈Iiψ2ix1,x1x2+∑y1,y1y2∈Iiiψ1iiy1,y1y2=Viid2Gix1+d1Giiy1.
This impliesdGi⊕Giix1,y1=ViidGix1+dGiiy1.Example 8.
LetGi and Gii be two IFIGs as shown in Figures 12 and 13 and their NP is given in Figure 14 with μ1i≥ψ1ii,μ2i≤ψ2ii,μ1ii≥ψ1i,μ2ii≤ψ2i,ψ1i≤ψ1ii, and ψ2i≥ψ2ii; then, by Theorem 4, d1Gi⊕Giix1,y1=0.6=d1Gix1 and d2Gi⊕Giix1,y1=0.9=d2Giy1. Hence, dGi⊕Giix1,y1=0.6,0.9.Figure 12
Gi.Figure 13
Gii.Figure 14
NP of IFIGs provided in Figures12 and 13.


## 5. Real-Life Application of CP and Composition of IFIGs

Here, we include an application of CP and the composition of two IFIGs. As an illustrative case, consider two networks (IFIGs)Gi and Gii with respective two and two vertices showing different departments of different branches of a textile industry. The MS value of the vertices shows the percentage of the working staff in a department and the NMS value of the vertices represents the nonworking staff. The MS value of the edges shows the mutual collaboration of the departments with each other and the NMS value of the edges shows the noncollaboration of the departments with one another. The MS value of IPs shows the percentage of progress and NMS value shows the percentage of nonprogress of the departments. As in Figure 7 and Figure 8, CP and composition of Gi and Gii show the percentage of progress of department x1 with departments y1 and y2 has the maximum MS value and the percentage of nonprogress of department x1 with departments y1 and y2 has the lesser NMS value. Therefore, these are the most appropriate combinations of departments showing the maximum percentage of progress and a minimum percentage of nonprogress of the textile industry among all departments.


## 6. Comparative Analysis

We have applied the model on CP of two IFIGs provided in Figure7 and we get the following results. In Figure 7, there are two possible paths from vertex x1,y1 to vertex x1,y2 such as P1:x1,y1,x1,y1,x1,y1x1,y2,x1,y1x1,y2,x1,y2,x1,y1x1,y2,x1,y2 and P2:x1,y1,x1,y1,x1,y1x2,y1,x1,y1x2,y1,x2,y1,x1,y1x2,y1,x1,y2. The incidence strength (IS) of P1 is the minimum value of all MS values of all the IPs of P1 and the maximum value of all NMS values of all the IPs of P1. Therefore, the I SP1=∧0.1,0.2=0.1,∨0.4,0.4=0.4=0.1,0.4 and ISP2=0.1,0.4. The greatest IS∞ is the maximum value of MS values of ISP1 and ISP2 and the minimum values of NMS values of ISP1 and ISP2. This implies IS∞x1,y1,x1,y2=0.1,0.4. Similarly, the IS∞x1,y1,x2,y1=0.1,0.4, IS∞x1,y2,x2,y2=0.1,0.4, and IS∞x2,y1,x2,y2=0.1,0.4. Here, all the IS∞ are the same but, in our model, the degrees of all the vertices of CP of two IFIGs dGi×Giix1,y1=0.3,0.8=dGi×Giix1,y2 and dGi×Giix2,y1=0.2,0.8=dGi×Giix2,y2 are not all same.By applying the model on the composition of two IFIGs given in Figure8, we get IS∞x1,y1,x1,y2=0.1,0.4, IS∞x1,y1,x2,y1=0.1,0.4, IS∞x1,y2,x2,y2=0.1,0.4, and IS∞x2,y1,x2,y2=0.1,0.4. In our model, the degrees of the vertices of the composition of two IFIGs dGi⊙Giix1,y1=0.5,1.2=dGi⊙Giix1,y2 and dGi⊙Giix2,y1=0.3,1.2=dGi⊙Giix2,y2 are not all the same. It can be observed that the degree of the vertices dGi⊙Giix1,y1=0.5,1.2=dGi⊙Giix1,y2 shows the percentage of progress of department x1 with departments y1 and y2 has the maximum MS value and the percentage of nonprogress of departments y1 and y2 has the lesser NMS value. Due to this reason, the existing model is not helpful to decide which departments have the percentage of progress that has the maximum MS value and the percentage of nonprogress that has the minimum value. The method to find IS and IS∞ of a single network of all possible paths between vertex xi and yi is explained in the existing model but there is no clue to find IS and IS∞ of the product of networks. The existing model is beneficial for single networks but not suitable to explain the overall impact of products of different networks. But in our model, we can discuss the overall impact of the combination of different networks with the help of CP, composition, TP, and NP. Our model deals with not only a single network but also more than one network. In this way, we can talk about the overall impact of the products of different networks. Therefore, our proposed method is better than the existing method.


## 7. Conclusion

This study makes explicit the several structures of IFIGs and their properties. Further research can be carried out to explore the structure of IFIGs which would have applications in different fields. In this research article, we examined the degree of the vertices of the IFIGsGi×Gii,GiGii,Gi⋄Gii, and Gi⊕Gii under certain terms and conditions and demonstrated them with various examples. We also established some new results on the degree of a vertex in terms of theorems. The degree of any graph plays a key role in studying the foundational characteristic of a graph in detail. This study is beneficial to explore different kinds of properties of IFIGs such as Gi×Gii,GiGii,Gi⋄Gii, and Gi⊕Gii. An application of CP and composition of IFIGs in the textile industry is also provided. Our study will be instrumental to apprehend the further attributes of IFIGs at length. The main benefit of our study is that it will be supportive to find the percentage of progress that has maximum MS value and percentage of nonprogress that has minimum NMS value in different universities, garment factories, and hospitals. Another advantage of our study is that we do not have any need to calculate the IS and IS∞ of all possible paths from vertex ui to vertex vi to find the neutrosophic incidence cut-pair. We have to just calculate the degree of the vertices to find the percentage of progress having maximum MS value and percentage of nonprogress having minimum NMS value of different departments. The vertex having maximum MS value and minimum NMS value will be the required vertex. Therefore, our proposed study is better than the existing neutrosophic IGs. The authors propose to extend further these concepts to other types of IFIGs such as soft IFIGs, competition IFIGs, interval-valued IFIGs, threshold IFIGs, Hamiltonian IFIGs, the coloring of IFIGs, and bipolar IFIGs.



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