# Perfect Fuzzy Soft Tripartite Graphs and Their Complements

Authors: Kalaichelvan Kalaiarasi; L. Mahalakshmi; Nasreen Kausar; Sajida Kousar; Parameshwari Kattel
Journal: Discrete Dynamics in Nature and Society
 (2022)

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## Abstract

Fuzzy soft graphs are efficient numerical tools for simulating the uncertainty of the real world. A fuzzy soft graph is a perfect fusion of the fuzzy soft set and the graph model that is widely used in a variety of fields. This paper discusses a few unique notions of perfect fuzzy soft tripartite graphs (PFSTG), as well as the concepts of complement of perfect fuzzy soft tripartite graphs (CPFSTGs). Because soft sets are most useful in real-world applications, the newly developed concepts of perfect soft tripartite fuzzy graphs will lead to many theoretical applications by adding extra fuzziness in analysing. We look at some of their properties and come up with a few results that are related to these concepts. Furthermore, we investigated some fundamental theorems and illustrated an application of size of perfect fuzzy soft tripartite graphs in employee selection for an institution using the perfect fuzzy soft tripartite graph.


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## Body

## 1. Introduction

Zadeh’s [1] fuzzy set theory, established in 1965, is the best explanation for interacting with sources of uncertainty. In 1986, Honda and Ohsato [2] have discussed some fuzzy set concepts and applications. A graph is a simple way to communicate data and the relationship between various types of substances. Vertices represent the elements, while edges express the relationships. Rosenfield [3] created the notion of fuzzy graphs in 1975, providing an overview of fuzzy sets to graph theory. Kotzig and Rosa [4] defined the properties of magic graphs in 1970. In 1987, Bhattacharya [5] examined fuzzy graphs and made some remarkable observations. So many researchers introduced numerous maintainable and remarkable concepts in fuzzy graphs. In 1994, Mordeson and Chang-Shyh [6] introduced the concepts of the complement of fuzzy graphs.In 1999, Molodtsov [7] created a fuzzy soft theory to rectify inexact technological difficulties in social science, entrepreneurship, medicine, and the surroundings. He also applied this theory to a variation of many other areas, including crispness of function and game theory. In recent years, soft set research has been very active, with researchers from all over the world participating. In 2001, Maji et al. [8, 9] proposed the concept of fuzzy soft sets, which are a perfect combination of a fuzzy set and a soft set. In 2009, Aygünglu et al. [10] have given some ideas in mathematics with applications. In 2013, Ghosh et al. [11] proposed some ideas on the operations of intuitionistic fuzzy soft sets. In 2014, Rajesh [12] proposed the notion of soft graphs and also they investigated some of their characteristics. In the year 2015, Akram and Nawaz [13] proposed fuzzy soft graphs. Mohinta et al. developed FSG independently. They present the concept of FSG as well as a few properties related to them in their paper. All the concepts of FSG (strong, complete, and regular) have been introduced by Akram and Nawaz [14]. In 2016, Akram and Nawaz [15] presented the concepts of FSG and its applications in social networks and road networks. They also explained into different types of arcs in FSG. Al-Masarwah et al. [16] introduced some new notions in CFSG. Varkey and Shyla [17] discussed some notes on bipartite and balanced fuzzy graph structures in 2017. In 2018, Sarala and Manju [18] proposed some applications of fuzzy soft bipartite graphs. In 2019, Khan et al. [19] have described picture fuzzy soft sets and their implementation in software applications. Sarala and Tharani [20, 21] gave some ideas in bipolar and intuitionistic fuzzy soft digraph in 2019. Sarala and Tharani [22] have described fuzzy soft digraph implementation in genetic eye and hair color in 2020.Bibin et al. [23] examined vertex rough graphs and made some remarkable observations. In 2021, Paik and Mondal [24] represent and gave application of fuzzy soft set in type-2 environment. In the same year Karbasioun and Ameri [25] discussed the product of two fuzzy soft graphs. In 2016, Masarwah [26] computed genus value of fuzzy soft graphs. The next year the properties are investigated by Poubassani and Doostie [27]. Al-Masarwah and Abuqamar [28] developed the types of fuzzy soft graphs in 2017. In 2019, Sashikala and Anil [29] gave some ideas in fuzzy soft cycles. In the same year, Akilandeswari [30] investigate regular fuzzy soft graphs. Finally in 2021, Nawaz and Akram [31] presented the important application of wireless Internet services with the use of fuzzy soft graphs.Fuzzy soft bipartite graphs and fuzzy soft tripartite graphs were already discussed by several researchers along with their applications. Following that, the concepts were applied to the complement of fuzzy soft tripartite graphs. Fuzzy soft-tripartite graphs produce better results than fuzzy soft bipartite graphs. The value of the fuzzy soft tripartite graph is larger than that of the fuzzy soft bipartite graph, which gives exact maximum values. This motivates us to define and analyse the fuzzy soft tripartite graphs and their complement to form a unique mathematical model.A few main ideas of FSTG and its complement are also discussed briefly. The following is how the paper is positioned out: Section2 discusses the fundamental concepts of the fuzzy graph theory. Section 3 introduces the notion of FSTGs. Section 4 describes the concept of CFSTGs. Finally, Section 5 illustrates a model for implementing these FSTGs. Section 6 contains the conclusion.


## 2. Preliminaries

Definition 1.
[2] A fuzzy graph is an ordered triple GFVF,σF,μF where VF is a set of vertices uF1,uF2,…,uFn and σF is a fuzzy subset of VF that is σF:VF⟶0,1 and is denoted by σF=uF1,σFuF1,uF2,σuF2……,uFn,σuFn, and μF is a fuzzy relation on σF.That is,Definition 2.
[14] Let V=x1,x2,…,xn nonempty set, E (parameters set), and A⊆E. Also, let(1)
ρ:A⟶FV (collection of all fuzzy subsets in V) e⟶ρe=ρe (say) and ρe:V⟶0,1,X:⟶ρeXiA,ρ: Fuzzy soft vertex.(2)
μ:A⟶FV×V (collection of all fuzzy subsets in V×Ve⟶μe=μe (say) and μe:v×v⟶0,1.(1)xi,xj⟶μexi,xj,A,μ: Fuzzy soft edge. Then, A,ρ,A,μ is called a fuzzy soft graph iff μexi,xj≤ρexi∧ρexj for all e∈A and for all i,j=1,2,…,n and this fuzzy soft graph is denoted by GA,V.Definition 3.
[19] A fuzzy soft graph GA,V=A,ρ,A,μ is said be a fuzzy soft bipartite graph. If the vertex set V is partition into two disjoint vertex pair and μexi,xj=ρexi∧ρexj for all xi∈vi and yj∈vj.Definition 4.
[19] If a fuzzy soft graph GA,V=A,ρ,A,μ is said be a fuzzy soft bipartite graph, then, the size of the fuzzy soft bipartite graph is(2)SGA,V∪vji=∑e∈A∑xiyj∈Vi∪vjμexi,xj.


## 3. Perfect Fuzzy Soft Tripartite Graphs

Definition 5.
A FSGGA,V=A,℘,A,ℑ is said to be a FSTG, if the vertices can be partitioned into 3 disjoint vertex pairs and(3)ℑexii,yjj≤min℘exii,℘eyjj,ℑeyjj,zkk≤min℘eyjj,℘eZkk,ℑezkk,xii≤min℘ezkk,℘exii.
It is denoted byGA,Vi∪Vj∪VK.Definition 6.
A FSTG is said to be a perfect fuzzy soft tripartite graph if any one of the vertex value has one, i.e.,℘Vi=1.Definition 7.
If a FSGGA,V=A,℘,A,ℑ is said to be a FSTG, then, the size of the PFSTG is(4)SGA,Vi∪Vj∪VK=∑e∈A∑aibjcK∈ViUVjUVkℑeai,bj,ck,where SFe1=∑aibjcK∈ViUVjUVkℑe1ai,bj,ck,(5)SFe2=∑aibjcK∈ViUVjUVkℑe2ai,bj,ck,SFe3=∑aibjcK∈ViUVjUVkℑe3ai,bj,ck.Definition 8.
If a FSGGA,V=A,℘,A,ℑ is said to be a PFSTG, then the degree of the PFSTG is dai=∑e∈A∑bjck∈vj∪vkℑebj,ckdbj=∑e∈A∑ajck∈vj∪vkℑeai,ckdck=∑e∈A∑aibj∈vi∪vjℑeai,bjDefinition 9.
If a FSGGA,V=A,℘,A,ℑ is said to be a PFSTG, then the order of the PFSTG is OGA,Vi∪Vj∪Vk=∑e∈A∑aibjck∈ViUVjUVk℘eai,bj,ck,where(6)OFe1=∑aibjcK∈ViUVjUVk℘e1ai,bj,ck,OFe2=∑aibjcK∈ViUVjUVk℘e2ai,bj,ck,OFe3=∑aibjcK∈ViUVjUVk℘e3ai,bj,ck.Example 1.
ConsiderFe1, Fe2, and Fe3 defined in Figure 1–3 (Tables 1 and 2).Figure 1
Fe1.Figure 2
Fe2.Figure 3
Fe3.Table 1
Relation between vertices and parameters of perfect fuzzy soft tripartite graphs.
ρa1a2a3b1b2b3c1c2c3e10.10.300.4100.80.90e210.90.20.80.60.400.70.1e300.80.7010.90.20.50.4Table 2
Relation between vertices and edges of perfect fuzzy soft tripartite graphs.
μe1e2e3a1b10.0100a1b20.020.50a1c200.60a2b100.70a2b20.200.7a2b300.30.4a2c1000.1a2c20.100a2c300.010a3b200.30.5a3b300.10.6a3c3000.4b1c20.300b1c300.020b2c10.700.01b2c200.020.3b2c3000.2b3c200.030The size of the PFSTG is(7)SFe1=1.33,SFe2=2.47,SFe3=3.21,∴SGA,Vi∪Vj∪Vk=7.01.The order of the PFSTG is(8)OFe1=3.5,OFe2=4.7,OFe3=4.5,∴OGA,Vi∪Vj∪Vk=12.7.The degree of the PFSTG is(9)da1=1.13,da2=2.51,da3=1.53,db1=0.32,db2=1.62,db3=0.03,dc1=0.51,dc2=1.35,dc3=0.63.


## 4. Complement of Perfect Fuzzy Soft Tripartite Graph

Definition 10.
A FSTGGA,Vi∪Vj∪Vk is known as a strong fuzzy soft tripartite graph (SFSTG) if(10)ℑexii,yjj=min℘exii,℘eyjj,ℑeyjj,zkk=min℘eyjj,℘eZkk,ℑezkk,xii=min℘ezkk,℘exii∀e∈A.and is complete FSTG if(11)ℑexii,yjj=min℘exii,℘eyjj,ℑeyjj,zkk=min℘eyjj,℘eZkk,ℑezkk,xii=min℘ezkk,℘exii∀e∈A.Definition 11.
LetGA,Vi∪Vj∪Vk be a FSTG. Then, GA,Vi∪Vj∪Vk is called isolated fuzzy soft tripartite graph (IFSTG) if ℑexii,yjj=0,ℑeyjj,zkk=0andℑezkk,xii=0∀xii,yjj,zkk∈V×Vande∈A.Definition 12.
LetGA,Vi∪Vj∪VK be a FSTG. The CFSTG is defined as G¯A,Vi∪Vj∪VK where(12)ℑe¯xii,yjj=min℘exii,℘eyjj−ℑexii,yjj,ℑe¯yjj,zkk=min℘eyjj,℘ezkk−ℑeyjj,zkk,ℑe¯zkk,xii=min℘ezkk,℘exii−ℑezkk,xii.Definition 13.
A CFSTG is called a complement of perfect fuzzy soft tripartite graph if any one of the vertex value has one, i.e.,℘¯Vi=1.Definition 14.
A CFSTGG¯A,Vi∪Vj∪VK is called SFSTG if(13)ℑe¯xii,yjj=min℘exii,℘eyjj,ℑe¯yjj,zkk=min℘eyjj,℘ezkk,ℑe¯zkk,xii=min℘ezkk,℘exii∀e∈A,and is complete FSTG if(14)ℑe¯xii,yjj=min℘exii,℘eyjj,ℑe¯yjj,zkk=min℘eyjj,℘ezkk,ℑe¯zkk,xii=min℘ezkk,℘exii∀e∈A.Proposition 1.
(15)SG¯A,Vi∪Vj+SGA,Vi∪Vj≤2∑e∈A∑xii≠yjj℘exii∧℘eyjj,SG¯A,Vj∪Vk+SGA,Vj∪Vk≤2∑e∈A∑yjj≠zkk℘eyjj∧℘ezkk,SG¯A,Vk∪Vi+SGA,Vk∪Vi≤2∑e∈A∑xii≠zkk℘ezkk∧℘exii.Proof.
Since(16)ℑe¯xii,yjj=℘exii∧℘eyjj−ℑexii,yjj,ℑexii,yjj≤℘exii∧℘eyjj.
Also,(17)ℑe¯xii,yjj≤℘exii∧℘eyjj.
From (16) and (17), we have(18)ℑexii,yjj+ℑe¯xii,yjj≤2℘exii∧℘eyjj.
Now,∑e∈A∑xii≠yjjℑexii,yjj+ℑe¯xii,yjj≤∑e∈A∑xii≠yjj2℘exii∧℘eyjj(19)∑e∈A∑xii≠yjjℑexii,yjj+∑e∈A∑xii≠yjjℑe¯xii,yjj≤2∑e∈A∑xii≠yjj℘exii∧℘eyjj.
Hence,(20)SGA,Vi∪Vj+SG¯A,Vi∪Vj≤2∑e∈A∑xii≠yjj℘exii∧℘eyjj.
Similarly,(21)SGA,Vj∪VK+SG¯A,Vj∪VK≤2∑e∈A∑yjj≠zkk℘eyjj∧℘ezkk,SGA,VK∪Vi+SG¯A,VK∪Vi≤2∑e∈A∑xii≠zkk℘exii∧℘ezkk.Example 2.
ConsiderFe1, Fe2, and Fe3 defined in Figures 4–6.Figure 4
Fe1.Figure 5
Fe2.Figure 6
Fe3.(22)SG¯A,Vi∪Vj=1.66,SG¯A,Vj∪VK=1.64,SG¯A,VK∪Vi=0.49,SGA,Vi∪Vj=4.04,SGA,Vj∪VK=1.46,SGA,VK∪Vi=1.21,SG¯A,Vi∪Vj=5.7,SG¯A,Vj∪VK=3.1,SG¯A,VK∪Vi=1.7,∑℘exii∧℘eyjj=5.7,∑℘eyjj∧℘ezkk=3.1,∑℘ezkk∧℘exii=1.7,∴SG¯A,Vi∪Vj+SGA,Vi∪Vj≤2∑e∈A∑xii≠yjj℘exii∧℘eyjj.Properties of complement of perfect fuzzy soft tripartite graph.(1)
The order ofG¯A,Vi∪Vj∪VK is equal to the order of GA,Vi∪Vj∪VK.(23)OG¯A,Vi∪Vj∪VK=OGA,Vi∪Vj∪VK=12.2.(2)
The proportion of edge set elements ofG¯A,Vi∪Vj∪VK is not exactly or equivalent to the proportion of edge set elements of GA,Vi∪Vj∪VK.(3)
Node set ofG¯A,Vi∪Vj∪VK is same as the node set of GA,Vi∪Vj∪VK. The number of vertices in the complement of fuzzy soft tripartite graph is equal to the number of vertices in the perfect fuzzy soft tripartite graph.(4)
SG¯A,Vi∪Vj+SGA,Vi∪Vj=∑e∈A∑xii,yjj∈ℑ℘exii∧℘eyjj,(24)SG¯A,Vj∪VK+SGA,Vj∪VK=∑e∈A∑yjj,zkk∈ℑ℘eyjj∧℘ezkk,SG¯A,Vk∪Vi+SGA,Vk∪Vi=∑e∈A∑xii,zkk∈ℑ℘ezkk∧℘exii,SG¯A,Vi∪Vj=1.66,SG¯A,Vj∪VK=1.64,SG¯A,VK∪Vi=0.49,SGA,Vi∪Vj=4.04,SGA,Vj∪VK=1.46,SGA,VK∪Vi=1.21,SG¯A,Vi∪Vj=5.7,SG¯A,Vj∪VK=3.1,SG¯A,VK∪Vi=1.7,∑℘exii∧℘eyjj=5.7,∑℘eyjj∧℘ezkk=3.1,∑℘ezkk∧℘exii=1.7.(25)ℑe¯xii,yjj=min℘exii,℘eyjj,ℑe¯yjj,zkk=min℘eyjj,℘ezkk,ℑe¯zkk,xii=min℘ezkk,℘exii.Theorem 1.
The complement of a PSFSTG is also SFSTG.Proof.
LetGA,Vi∪Vj∪VK be a SFSTG. ℑe¯xii,yjj=min℘exii,℘eyjj(26)ℑe¯yjj,zkk=min℘eyjj,℘ezkk,ℑe¯zkk,xii=min℘ezkk,℘exii∀e∈A.By the definition of complement of perfect soft tripartite is defined as(27)ℑe¯xii,yjj=min℘exii,℘eyjj−ℑexii,yjj,ℑe¯yjj,zkk=min℘eyjj,℘ezkk−ℑeyjj,zkk,ℑe¯zkk,xii=min℘ezkk,℘exii−ℑezkk,xiiand any one of the vertex values has one.Now consider,(28)ℑe¯xii,yjj=min℘exii,℘eyjj−ℑexii,yjj,=min℘exii,℘eyjj−min℘exii,℘eyjj∀xii,yjj∈V,e∈A,=0,ℑexii,yjj>0,min℘exii,℘eyjjℑexii,yjj=0.∴ℑe¯xii,yjj=0,ℑe¯xii,yjj=min℘exii,℘eyjj∀xii,yjj∈V.Similarly,(29)ℑe¯yjj,zkk=0,min℘eyjj,℘ezkk,ℑe¯zkk,xii=0,min℘ezkk,℘exii.∴The complement of a PSFSTG is also a SFSTG.Theorem 2.
The complement of a perfect complete FSTG is also complete FSTG.Proof.
LetGA,Vi∪Vj∪VK be a compete FSTG.(30)ℑe¯xii,yjj=min℘exii,℘eyjj,ℑe¯yjj,zkk=min℘eyjj,℘ezkk,ℑe¯zkk,xii=min℘ezkk,℘exii∀e∈A.By the definition of complement of perfect soft tripartite graph is defined as(31)ℑe¯xii,yjj=min℘exii,℘eyjj−ℑexii,yjj,ℑe¯yjj,zkk=min℘eyjj,℘ezkk−ℑeyjj,zkk,ℑe¯zkk,xii=min℘ezkk,℘exii−ℑezkk,xii,and any one of the vertex value has must be one.Now consider,(32)ℑe¯xii,yjj=min℘exii,℘eyjj−ℑexii,yjj,=min℘exii,℘eyjj−min℘exii,℘eyjj∀xii,yjj∈V,e∈A,=0,ℑexii,yjj>0,min℘exii,℘eyjjℑexii,yjj=0,∴ℑe¯xii,yjj=0,ℑe¯xii,yjj=min℘exii,℘eyjj∀xii,yjj∈V.Similarly,(33)ℑe¯yjj,zkk=0min℘eyjj,℘ezkk,ℑe¯zkk,xii=0min℘ezkk,℘exii.
∴ The complement of a complete PFSTG is also a complete FSTG.Example 3.
In the graph (see Figure7) the number of vertices in the complement of perfect fuzzy soft tripartite graph is equal to number of vertices in the complete fuzzy soft tripartite graph.Figure 7
Fe1.Moreover, the above fuzzy graph is a complement fuzzy soft tripartite graph. It is also satisfy the condition for the complete fuzzy soft tripartite graph.Theorem 3.
IfGA,Vi∪Vj∪VK=A,℘,A,ℑ is a PFSTG. Then, GA,Vi∪Vj∪VK is an IFSTG if CPFSTG is a complete FSTG.Proof.
GivenGA,Vi∪Vj∪VK is an IFSTG. Now, consider GA,Vi∪Vj is an IFSTG. Then,(34)ℑexii,yjj=0.
We know that(35)ℑe¯xii,yjj=min℘exii,℘eyjj−ℑexii,yjj.Substitute (33) in (34),(36)ℑe¯xii,yjj=min℘exii,℘eyjj.Similarly,(37)ℑe¯yjj,zkk=min℘eyjj,℘ezkk&ℑe¯zkk,xii=min℘ezkk,℘exii.Hence, the CPFSTG is a complete FSTG.Conversely, given the CPFSTG is a complete FSTG, i.e.,(38)ℑe¯xii,yjj=min℘exii,℘eyjj.To prove, GA,Vi∪Vj∪Vk is an IFSTG.We know that(39)ℑe¯xii,yjj=min℘exii,℘eyjj−ℑexii,yjj⇒ℑe¯xii,yjj=min℘exii,℘eyjj−ℑexii,yjj,=min℘exii,℘eyjj−min℘exii,℘eyjj.By3,ℑexii,yjj=0.Similarly,(40)ℑe¯yjj,zkk=0&ℑe¯zkk,xii=0.Hence, GA,Vi∪Vj∪VK is an IFSTG.Theorem 4.
IfGA,Vi∪Vj∪VK=A,℘,A,ℑ is a FSTG. Then, GA,Vi∪Vj∪VK is an IFSTG if CPFSTG is a SFSTG.Proof.
GivenGA,Vi∪Vj∪VK is an IFSTG. Now, consider GA,Vi∪Vj∪VK is an IFSTG.
Then,(41)ℑexii,yjj=0.We know that(42)ℑe¯xii,yjj=min℘exii,℘eyjj−ℑexii,yjj.
Substitute (40) in (41),(43)ℑe¯xii,yjj=min℘exii,℘eyjj.Similarly,(44)ℑe¯yjj,zkk=min℘eyjj,℘ezkk&ℑe¯zkk,xii=min℘ezkk,℘exii.
Hence, the CPFSTG is a SFSTG.Conversely,
Given the CPFSTG is a SFSTG, i.e.,(45)ℑe¯xii,yjj=min℘exii,℘eyjj.To prove,
GA,Vi∪Vj∪VK is an IFSTG.We know that(46)ℑe¯xii,yjj=min℘exii,℘eyjj−ℑexii,yjj,⇒ℑe¯xii,yjj=min℘exii,℘eyjj−ℑexii,yjj,=min℘exii,℘eyjj−min℘exii,℘eyjjBy3,ℑexii,yjj=0.Similarly,(47)ℑe¯yjj,zkk=0&ℑe¯zkk,xii=0.
Hence,GA,Vi∪Vj∪Vk is an IFSTG.


## 5. Application

To accomplish objectives in their day-to-day existence, they need the best school and the understudies assume a significant part in professional openings in any organization. Recruiting the best school and the best understudy, they pick the best programming organization is a challenging process. A study is carried out here using PFSTG. The study’s goal is to find the best match between colleges, students, and software companies.Consider colleges, students, and software companies to be three sets of disjoint vertex sets, with matching qualities as parameters and preferences for colleges, students, and software companies as edges. LetV=vi∪vj∪vk=vi:a1,a2,a3,vj:b1,b2,b3,vk:c1,c2,c3 are set of all three disjoint vertices and A=e1,e2,e3,e4 are parameter set.Identified qualities of colleges are given below.a1= Good communication and a comfortable environment.a2= Professional advancement and on-site opportunity.a3= On-site opportunity and a comfortable environment.Identified qualities of students are given below.b1= Self-discipline and responsible.b2= Professional skills and responsible.b3= Confident and professional skills.Identified qualities of software companies are given below.c1= Good salary and friendly work environment.c2= Transparency and good salary.c3= Professional advancement and friendly work environment.Also, the parameters aree1={college-a comfortable environment, student-confident, company-good salary}e2={college-professional advancement, student-responsible, company-Friendly work environment}e3={college-on-site opportunity, student-professional skills, company-Transparency}e4={college-good communication, student-self-discipline, company-professional advancement}Example 4.
ConsiderFe1, Fe2, Fe3, and Fe4 defined in Figures 7–10 (Table 3).Figure 8
Fe2.Figure 9
Fe3.Figure 10
Fe4.Table 3
Relation between vertices and parameters of perfect fuzzy soft tripartite graphs.
ρa1a2a3b1b2b3c1c2c3e10.10.300.40.500.80.91e210.90.20.80.60.400.70.1e300.80.7010.90.20.50.4e40.50.10.40.610.80.70.30.9μe1e2e3e4a1b10.01000.4a1b20.020.500.3a1b300.0200a2b10.10.700.01a2b20.20.50.70a2b300.30.40.02a3b100.100a3b200.010.50.1a3b300.10.60.3b1c10.3000.5b1c20.30.500.2b1c30.030.0200b2c10.0400.010.6b2c20.10.20.30b2c30.40.010.020.8b3c1000.10b3c200.30.20.2b3c300.30.30.7(48)SFe1=1.5SFe2=3.29SFe3=3.13SFe4=4.13.Most of the best colleges taught good communication skills, software companies hired self-disciplined students, and students preferred software companies with good professional advancement. According to the above discussion, “the most favourable matching occurs between good communication in college, self-discipline student, and professional advancement in company.”


## 6. Comparison Analysis

Only the principles and applications of fuzzy soft bipartite graphs were discussed previously. Later, we discovered a novel size concept that was only used for perfect fuzzy soft tripartite graphs. Following that, the concepts were applied to the complement of perfect fuzzy soft tripartite graphs. Perfect fuzzy soft tripartite graphs produce better results than fuzzy soft bipartite graphs. The value of the perfect fuzzy soft tripartite graph is larger than that of the fuzzy soft bipartite graph, which gives exact maximum values.


## 7. Conclusion

PFSTG and its complements are a modern phenomenon based on the combination of fuzzy graphs. The introduction of these PFSTG is a novel idea that can be expanded into a variety of graph hypothetical concepts. We introduced this PFSTG and its complement, explored their properties, and established related theorems to contribute to the theoretical aspect of the fuzzy graph theory. The PFSTG and its complement, as well as several of its fundamental properties, have been defined. The order, size, and degree of PFSTG have been defined with appropriate examples. We have used this PFSTG in a practical application. Since 1965, several generalizations of fuzzy sets are introduced including picture fuzzy, interval-valued fuzzy, Pythagorean fuzzy, and spherical fuzzy sets. The theory established in this article can be extended to these generalizations. Graph theory and fuzzy sets are two important areas with wide range applications in real life combining the two will broaden the application by integrating the uncertain and imprecise information.



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