Fuzzy sets and fuzzy relations were formulated by L.A. Zadeh in 1965
Cen Zuo et al. in 2019
The interesting development in the field of applied mathematics is the emergence of Plithogenic logic and Plithogenic sets by Smarandache in 2017
Being motivated by the discussions on Plithogenic fuzzy graphs and their applications in
As the number of attributes for a PPFG is greater than or equal to four, it is more reliable and realistic representation of vagueness in certain reallife situations. It is also more efficient in obtaining higher precision in our perception of uncertain scenarios in the decisionmaking process. PPFG model can be utilized to represent and analyze social network of persons, medical diagnosis of patients, students’ performance, resource/computer networking, psychological and sociological issues, weather forecast, etc. where the units involved are to be characterized with many attributes, and the relations between them need to be analyzed depending on the corresponding attribute values allotted to them. Since social networks play a vital role in human life, in section 7 of this paper, we have applied the PPFG model to analyze a smallsized social network.
In this section we mainly recall definitions of FGs, degrees of a vertex, order and size of a fuzzy graph, Product fuzzy graphs, Plithogenic fuzzy sets (PFSs) and PFGs which are relevant to the study on PPFGs.
PPFGs has been defined in this section. Basic concepts and properties of PPFGs are newly introduced
V_{i }= X_{i }(s_{1}, s_{2} ,…, s_{k})
V_{j}_{ }= X_{j}_{ }(t_{1}, t_{2} ,…, t_{k})
E_{c}_{ }= Y_{c}_{ }(c_{1}, c_{2} ,…, c_{k}) where for any c_{d}
Since
Since
(i.e.)
We know that
Hence the proposition is proved.
Conversely, let
If
We have the supremum and the infimum of each Pvertex, the supremum and the infimum of each Pedge, and the central Pvertex of
Pvertices 2 1 1.4 2.7 1.4 1.3 1.7 2.3 1.7 Pedges 0.25 0.22 1.53 0.36 0.47 0.89 0.45 1.31 0.93 0.51 0.78 1.06
In this section, Porder, Psize, Pvertex range, Pedge range, degree of a Pvertex, total degree of a Pvertex, the minimum degree and the maximum degree of a PPFG are defined and discussed with examples.
The difference between the maximum Pweight
For some
If some
For some
For some
If
In the above PPFG, by computation we have the Pweight of each Pvertex as follows:
Here
A network is a representation of different entities and their relations or connections. A social network is a representation of a social phenomenon of a group of people, organizations, structures, events, countries, places, animals, etc. In any social network all the units not necessarily have the same importance. The density of connections at every unit is not always the same. The strength of connection between every pair of units varies due to various reasons. Therefore, all these need to be measured to decide upon the stability of any social network
Since PPFG model has four or more attribute values associated with vertices, a better prediction about the quality of Pvertices possible in PPFG model. The computation of the attribute values for the Pedges in PPFG model reduces the error to the minimum in our prediction regarding strength of connectivity. Similarly the connectivity parameters discussed in this paper are better tools to investigate connectedness in the network. In this section a new mathematical model based on PPFGs has been utilized to analyze a social network of persons. P vertices and Pedges in the network are characterized by four attributes which have the corresponding attribute values from [0,1]. Some of the other terms in PPFGs and their corresponding meanings in our example of social network are as follows:



Total number of persons 

Total number of connections 
Pweight of a Pvertex 
Strength of sociability of a person 
Pweight of a Pedge 
Strength of connectivity between two persons 
Strong Pvertex 
Important or influential person 
Weak Pvertex 
Unimportant or uninfluential person 
Highly strong Pedge 
Highly cohesive connection 
Strong Pedge 
Cohesive connection 
Weak Pedge 
Weak connection 

Central and significant persons of the network 

Total strength of sociability of persons in the network 

Total strength of connectivity in the network 

The maximum difference in the strengths of sociability of persons 

The maximum difference in the strengths of connectivity of persons 

Strength of direct connectivity of a person 

Strength of direct connectivity and sociability of a person 

Average strength of sociability in the network 
Strength of a PPFP 
Strength of indirect connectivity 

Maximum strength of indirect connectivity 
To extract the overall impression regarding the cohesiveness of their connections.
To find the types of relations among them in the network.
To identify the most social, sociable, influential and central persons in the network.
Any other relational information about the persons in the network
Let us consider a PPFG,
The attributes to the persons and their connections are (i) Number of visits made
The corresponding PPFG,











8 
2 
1 
0 
0.25 
0.18 
0.17 
0 
0.6 

7 
3 
2 
1 
0.22 
0.27 
0.33 
0.11 
0.93 

8 
4 
2 
4 
0.25 
0.36 
0.33 
0.44 
1.38 

9 
2 
1 
4 
0.28 
0.18 
0.17 
0.44 
1.07 

32 
11 
6 
9 
1 
0.99 
1 
0.99 
3.98 
The attribute values of Pedges are calculated using the usual product operator by the method shown in the definition 3.1. The attribute values of Pedges concerning








0.06 
0.06 
0.07 
0.06 
0.06 
0.07 

0.05 
0.06 
0.03 
0.1 
0.05 
0.06 

0.06 
0.06 
0.03 
0.11 
0.06 
0.06 

0 
0 
0 
0.05 
0.05 
0.19 

0.17 
0.18 
0.13 
0.32 
0.22 
0.38 
Degree and total degree of Pvertices are computed and shown in Table 6.






0.48 
0.71 
0.88 
0.73 

1.08 
1.64 
2.26 
1.8 
The strengths of sociability of
The degree and the total degree of
The overall impression regarding the strength of sociability of persons in the network seems to be very good as three out of four persons have better strengths of sociability. Since the grading of strength of connectivity among them depends on their strengths of sociability, based on the data in Table 4 we conclude that
PPFGs has been newly defined and its properties are investigated with results and examples in this paper. It is proved that every PPFG is a PFG, but the converse need not be true. It is also verified that every subgraph of a PPFG is PPFG. Finally, the utility of PPFG and its application in a smallsized social network is demonstrated to understand the advantage of it in decision making process. As future work, A largescale practical network can be modeled using PPFG to analyze a reallife problem under uncertain environment. Operations on PPFGs are also areas of scope for future research to understand better the nature of PPFGs.